Mathematical Consideration

[[p.1]]
Mathematical Consideration
15.1 INTRODUCTION
Mathematics is the science of numbers dealing with their operation and relationship between .them,
space, configuration, structure, measurementetc. It has different branches and has varied application in
medical sciences. Insofar the blood banking and transfusion medicine is concerned its application is
mainly limited to arithmeticand statistics,we therefore,restrict our discussion to these two spheres only
as applied to blood banking and transfusion practices.
Arithmetic is the branch of mathematics that deals with computation and calculation of numbers.
This is the first branch of mathematicsto be introduced into medicine by Bayle, who counted the pulse
and respiratry movement in his patients.
Statistics is the branch of mathematics that deals with statistical method (classification,analysis and
drawing of conclusion)andinterpretationof massesof numericaldata.This wasthougha lateintroduction
(in medical science), has permeatedalmost all branches of medical science as on today.
In blood banking and transfusion medicine, the profiles and parameters are mostly expressed for
preciosity in quantitative terms and assessed by statistical probability or numerical facts in relation to
observation, analysis, investigation for drawing, conclusion.
Arithmetic is applied for computation and calculation expressed in terms of quantity through units
of measurement of mass, volume, temperature etc., mass (quantity or aggregate of matter present per
unit) expressed in grams (g) as per international recommendation, i.e., SI units kilo (103), milli (I03),
micro (1~),
nano (10-9) , pico (10-12).
.
Length is expressed in units of metre (m), deci (10-1), centi. (10-2), milli (10-3) , micrometer (10-6),
nano (10-9).
Volume(spaceoccupiedby liquid,gas or anyothersubstanceas measuredby cubicunits)is expressed
in litre, decilitre (10-1) , milli (10-3) , micro (10-6), nano (10-9) , pico (10-12) etc., derived unit of
measurement.
Temperature is unit for heat and is expressed in centigrade (c) - derived unit of measurement.
Time is expressed in seconds. The above description are for some basic measurements required in
all laboratory particularly·inblood bank laboratory.The other measurementssuch as electricity,radiation
are sparsely used in blood banking. The most commonly used substance in blood banking are liquid
425'

[[p.2]]
426
BlOod Bankl.f!lg and TransfusiM
(blood and components) solutions, reagents etc. Their definition, expression etc. has already been
discussed in. previous' chapter and again briefly appended below.
1-5.2 SOLUTIONS AND ITSCONCENTRATION CAN BE EXPRESSED IN VARIETY OF WAYS
MOLARITY, NORMALITY AND WN
(a) Molarity is number of moles of dissolvedsubstance (solute) per unit volume (litre) of solution.
(b) Normality is equal to the number of equivalents of dissolved substances per unit volume of
solution.
(c) WN is equal to the amount of solid substance dissolved to make up to 100 ml in a particular/
required solvent and is expressed in percentage.
Exempte
(a) Molarity M is equal to molesllitre ofsolution: Toprepare 1MNaCIsolution, dissolve, 58.5 gms
(molecularwt, of NaCI)of Sodium Chloride and add redistilled water to make upto 1litre or if
we want to prepare 9 gllitre, then calculate as follows :
9/58.5=0.154 molJIitre
(
milligramIiitre )
=154 m molllitre
GMW
where GMW is gram molecular weightofsubstance,one gram molecularweightof a substance
is also called 1 mole of the.subsrance, so it can be expressed in gms/litre or molllitre'or mmoV
litre.
(b) Normality (N) : One equivalent weight of a substance equals to one gram equivalent weight
(GEW) :
Nb
f eoui
g-
g
urn er 0 equivalents = GEW - (GMW/Valence)
Molecular weight of NaOH is 40 g (23 + 1 + 16)
GMW
40
GEW = Valence =-1
Valency of NaOH is 1 as it dissociates with 1 mol of H+.
So, to prepare I. N NaOH, we will re,quire 40 gms of NaOH made upto 1000 ml by redistilled
water.
The oneequivalentweightof an acidor a baseis theweightthatcontainsone moleof replaceable
H+or one mole of replaceable 'OH' respectively.
For example, H2S04 IDol wt. 98 (2 + 32 +64)
GEW=98/2 =49
(I equivalent weight contains 1 mole of replaceable H+ in H2S04, there are 2).
(c)
WIV expressed in % : To prepare 10%solution of A, we require 10gm of 'A' to bedissolved in
required solvent or distilled water to make upto 100 mI.

[[p.3]]
All that are described aboveareconcentration based on volume. There·are solutions,concentration
of whichare based on weight (w/w). It is equal to the weight of dissolved substance per 100 gin of
solution, i.e, w/w per cent and expressed as molalityuse is infrequentin bloodbank practice,
.
The weights and measure described· are those whichfind its utility in blood banking services.. The
approved SI units adopted in 1960 by the general conference of weights and measures and later on
accepted by International Federation of Clinical Chemistry (IFCC, 1967, 1974 - modificatioa) ·and
International Union of Pure and AppliedChemistry (1973).. This international system which was ·also
acceptedby International Committee for Standardisation in Haematology (197.3) and by WH·O in 1977
is summarised here in Table 15.1..
15.3 THE SEVEN BASE UNITS ARE AS FOLLOWS
·:·TabIe: 15.1:...Derives. ,,~,:ilitiairrOrrn
'Frend.:_~:
..·1nlemafioftat UNite
, Quantity
.SIIJDim
Symbol .
Length
Mass
"me
Electric aJrrent
Temperature
Amountof substance
Lummous intensity
Metre
KiJo.gmm
SecoAd
Ampere
Kelvin
Mol
Candela
.M
Kg
S
A
K
Mol
Cd
Thereare alsocertainunitsderivedfrom.SI unit andare usedto facilitate the'works in clinicaland blood
bank laboratory as ·well. The .SI derivedunitscallbe coherent, i.e, deriveddirectlyfrom 81unitwithout .
the'useof conversion factor, or non-coherent, l.e.,derivedfrombase unitbutcontainsa numerical factor
for better adaptability. The SI derivedunitsare appended in Table15..2.
To standardise the .multiple .and .submultiples underthis system, prefixesareproposed.
For example, if 3 kilo voltis to bewritten, itcan be expressed as 3 kVorf millio.nths of an ampere
can be expressed ~JlA
·and Iikewise,
The descriptions .ofunits of measurement (Table 15.2)are to facilitate the arithmetical application
in blood bank, laboratory works and statistical work,
.
In laboratory arithmetic is applied to obtain quantitative expres.sion of .experimental observation
through computation and calculation of numbers and formations (geometric formation and aJgebric
formation) by help of four basics; addition, subtraction, division,multiplication (Table 15..3)..
1..5.4 NU'MBERS ARE MAINLY
Wholenurnberrconsists of integer(negativeinteger zero and positiveinteger makes the set) .: Positive
integerlike 1, 2., 3.alsoknownas countingnumbersor natural numbers. The natural numberon addition
producesanothernaturalnumber'butthisdoesnotholdgood in case of.subtraction (inverseofaddition)..
It has been observedthat byintrodueing zero and negative integers(-1, --:2 and so on) the defectcan be
takencare of..Againincaseof multipJication,productof multiplication oftwointegers(positive,negative
and zero) will be.aninteger,
.

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428
Blood Banking and Transfusion
Table 15.2. 81 derived units
Quantity
SI unit
Symbol
Relation to
base units
Area
Sq. metre (coherent)
m2
Volume
Cubic metre (coherent)
m3
Liter (non-coherent)
L=dm3
dm3
Velocity
Metre/second
mls
m!S-l
Mass
Kilogramllitre
KgIL
Substance
Mole/litre
Mol/L
Molality
Molelkilogram
Mollkg
Density
Kilogramllitre
KgIL
Kglm3
Mass fraction
KilogramlKilogram
Kg/kg
?vIole fraction
Mole/mole
mole/mole
Number concentration
Numberllitre
L-l
Temperature
Degree 'centigrade
°C="K - 273.15
Pressure
Pa (coherent)
Pa
Kg/ms2
Frequency (cycles per second)
Hertz (coherent)
Hz
S-1
Clearance
Litre/second
US
Electrical potential
Volt (coherent)
V
W/A (Kgm2S3A)
Energyof work
Joule (coherent)
J
Nm (Kgm2/S2)
Radioactivity
Becqueree
BQ
S-1
Power
Watt
'W
JIS
Force
Newton
N
(MIkg 5-2)
Table 15.3•.Prefix, abbreviation, factors
Prefix
Abbreviation
Factor
Exa
E
10
18
IS
Peta
P
1012
Tera
T
1°9
Giga
G
10
6
Mega
M
10
3
Kilo
K
10
Hectc
H
10
2
1
Deca
Da
10
-1
Deci
D
10-2
Centi
C
10
-3
Mill
l\~
10
-6
Micro
JL
10
-9
Nano
11
10-12
Pico
P
10

[[p.5]]
Mathematical Consideration
429
Integers=(-2, -1, 0, 1, 2).
The operation of integer in respect of addition, subtraction and multiplication is thus clear.But
division (inverseof multiplication) required introduction of a new number for its operation, e.g., if 'p' is
an integer and 'q' is a non-zero integer (division by zero is yet to be defined) and the p/q when multiplied
by
Iq' gives the result 'p'This p/q x q=P, where q=0 and 'p' is an integer, the new number p/q is known
as rational number. The integers (zero, negative, positive) can be written in the form of rational number
-0=011, -2 = -211, 7=7/1.Every positive rational number is represented by a negative rational number.
Rational numbers has line property and a field of rational number, this is beyond the scope of present
discussion. The rational numbers can be represented as terminating decimal or non-terminating decimal
viz. 3/2 =1.5 or 20/6 = 3.333. But the rational number couldnot be applied universally to all measurement
orformation, Whenit was applied to basic geometry, the pythagorus theorem (the square ofthe hypotenuse
of right angled triangle is equal to the" sum of the squares of other two sides) could not be solved. So
again, a new number has to be introduced to solve the problem. This new number is ~2
is not a rational
number, also known as irrational number. So ~2,
~3
(the imperfect squares and cubes) are known as
irrational numbers and unlike rational numbers, they are non-terminating and non-repeating viz. 0.101001
(observe the number of zero goes on increasing in succession between two one, i.e. 1). The relation
between rational number and irrational number can be expressed as :
0.1 < X <: 0.2(where 'X' is irrational number and 0.1 and 0.2 are rational numbers) and putting the
above on a line L, it was observed that :
(a) Irrational number can be approximated closer and closer by rational number.
(b) The irrational number corresponds to a point formed by approximation of successive segments.
This is an oversimplified account given and for details consult textbook mathematics.
In laboratory the different forms of numbers commonly applied are fraction, mixed number and
exponential forms.
Fraction of a number indicates one or more equal parts or division of one number. For example, II
4 + 1/4 + 1/2=1.
Mixed number consists of a whole"number and a decimal fraction. A decimal is based on numbers of
10, or system which is reckoned by tens. It has a decimal point and decimal fraction. The dot at the left
of decimal fraction is decimal point. The decimal fraction is the fraction which is less than 1 and is
placed to the right of decimal point, e.g. 0.16, O~ 18etc. or it may even be a mixed number which consists
of a whole number and a fraction in which denominator is a power of 10, e.g., 4.16 (a mixed number) =
4 and 16/100.
The exponential form of numbers, where the symbol/number is written above and to the right of a
mathematical expression to signify, how many times it is to be used as a factor and they are guided by
law of integral exponents, i.e. for any two real numbers 'A' and 'B', where a, b ¢ 0 and for any two
positive number m and n :
(i)
aman =a(m + n)
(ii) am + an := a(m - n)
(iii) (am)D =amn
(iv) (~r =::
(v) (ab)D = anbn

[[p.6]]
430
Blood Banking and Transfusion
In the aboveform the conditionis a, b, ¢ 0, the condition in which a < 0 and 'N' is a positiveinteger,
the Nth root of 'A' is yet to be defined: but when a > 0 and 'N' is a natural number, the existence of a
unique number 'x' principal Nth root of 'a' (positivereal number) could be defined (for details interested
reader may consult textbookof mathematics). Thus x" =a. The principal Nth root of 'a' is representedby
aJ/n =n".}a.
The laws for exponents are equally applicable to both real and rational exponents because in real
number a > 0 and in rational number p/q, where q > o.
aplq=(aP) lIq
Substituting by number say P =3 and q = 2
a312= (a3)112=~a,
a, a
But this doesnot apply to irrational number such as "'2,'3~2,
~3,
4V5 etc. If nth power cannot be
expressed as a rationalnumber,then n~a ought to be an irrational number. The nth root of this irrational
number is named as surd or radical n~ (radical sign) is called the order of surd and 'a' is called the
radicand and are guided by law of radicals and not by law of exponent.
Law of radicals
(i) nvan..Jb = nvab
(ii) (Ova)o"=a
(iii) !iff;
n ta ( h
d·
I
f
d )
~ = Vb
were two ra tea s are 0 same Oli er
(iv) rnnva = nvm~a,
(where nand m are positive integer)
(v) mnvanm, (where In and ~ are positive integer and is a rational number).
Forms
Geometric formation deals with measurement of solid, surfaces, lines and angles, their relations and
property.Actuallythe wordgeometrycan betracedtogreek languagewheregeo meansearthandmeteron
stands for measuring. Indian veda has reference for geometric formation but the origin of geometry
could be traced to Euklid (300 Be) and Pythagorns around 500 Be."
Algebric formation :AIgebra derived from an arabic word al- jabr, uses letters that represents
numbers. Lettersherearethesymbolsthatexploresrelationshipsbetweenthenumbersandtheiroperation.
Both the above forms are uncommonlyused in laboratory work except in some higher research and
as such the discussionis restricted to operation of numbers frequentlyand commonly used in laboratory
work.
Operation of numbers (fundamental)
Addition : Operation of combining numbers are also known as summation is dorieto obtain the sum of
numbers, sign is (+).
Subtraction : This is an inverse operation of addition and centres round the process of deduction or
deducing the numberssign is (-).

[[p.7]]
Mathematical Consideration
431
Multiplication: Operationof increasinga number by certain number of times, e.g. 4 x 3=12,sign
is (x) or (.).
Division : Operation of finding how many one number of quantity "is contained in the other. An
inverse operationof multiplication, e.g., 6 + 2 =3, sign is (+).
It is always advisable that while operating the number (addition, subtraction, multiplication and
division), retain thesignificantfigures:
50~253
1.255
51.478
Ans: 51.48
Observe the answer the significantfigures are contained in the quantity.
Some of the calculationinvolvesoperationwithlarge numbersof data and applicationof exponential
form (already described) and logarithm makes such calculation easier.
Concept of logarlthim and application
Logarithim is the exponentthat indicatesthe power to which a base number is raised to produce a given
number.
For a positive real number 'x' (x "* I)and a rational number 'n', let x" = y, where y is another real
number. In logarithimic expressionsubstitutingby number log 28 =3 (Since 23 =8).
There are two types of lorarithim :
(a) Natural or Naperian logarithim based on exponential number 'e', approximate value is 2.718
and is expressed as loge' log is the abbreviationof :'ord logarithim.
(b) Common logarithimof a number is the exponent which is usually applied to the base 10, also
known as logarithim to base 10,expressed as log 10.
The later is commonly used for laboratory work. Some of the examples are given :
Log lOt
= 0
Since 10°
=
1
'Log 1010
=
1
Since 101
=
10
Log 10100 = 2
Since 102
=
I~O
Log 101000 = 3
Since'I03
= 1000
Log 10°.1
= -1
Since 10-1
=
0.1
Log 10.01
= -2
Since 10-2
= 0.01
From above,it followsthat ifn is integralpower of 10,it is easy'tocalculatelogn. Log table has thus
been·prepared by mathematicians with that idea in mind and from the log table, the logarithmicvalue of
numbers between I to 10 can be ascertained. For the laboratorypurpose, it is sufficient.
Law of ,Iogarlthim
(i) Log a(mn) =Log am + log an
(ii) Log a(mln) =Log am - Log an
(iii) Log am"=n log am

[[p.8]]
432
Blood Banking and Transfusion
Operation of decimal form
Say 'X' is a number lying between 10 and 100 :
X::; lOX 10::;Y X 101
X
:. X = Y x lOP (where 'p' is an integer - (+), (0), (-) and 1 ~ Y< 10).
Substituting by number:
32.6= 3:~6
xl0 = 3.26X 10
1
(where 'X' is 32.6 and y is 3.26).
It thus follows that we can express number in decimal form as product of integral power of 10 and
number lying between 1 and 10.
In general the working principle can be summarised as :
(i) According to necessitymove the decimal point to the right or left to ensure a non-Zerodigit on
the left of decimal point,
(ii) Multiply by 1Q-Pin cases of shifting of 'p' places to the right.
(iii) Multiply by lOP, if you move 'p' places to the left.
(iv) If no movement multiply by 10°.
Characteristic and mantissa
(i) Characteristicis an integer to the left of decimal point, the integer maybe (+), 0, (-).
(ii) Mantissa is never negative, less than I always and is placed to right of decimal point.
Example
x = y x lOP, where 1 ~ Y< 10 (in previous illustration)
So
log x = log y x log lOP
=log Yx Plog 10
By application law of logarithim and log 10
= P + log y
Now 'y' lies between 1 and 10and 'p' is an integer,the logy is expected to lie between 0 and 1 i.e. 0
Sy<l.
So
log x = P + 0 Slog y < I
Characteristic Mantissa
So to read value of :
log x = P + log Y
= 6 + 0.335 (substitutingby numbers p = 6, log Y= 0.335)
log x =6.335.

[[p.9]]
Mathematical' Consideration
433
se, we are to find out the value of log (6.335) from log table if 'p' is anegative integer, then it is
represented as 6.335(where p = -6) log Table15.4.Row starts witha two digit number andcolumns are
headed by 0 to 9 and on extreme right is the mean difference - 'column headed by 1 to 9.
Example
Table
15.4~
N
0
1
2
3
4
5
6
7
8
9
123
456
789
63
7993
8000
8007 . 8014
8021
8028
8035
8041
8048
8055
112
334
566
Since we are to find log 6.335 only required part from the log table is given above.
log (6.330)=0.8014
But actually we require to find oU,t value of log (6.335)and our fourth digit is 5, we look up for 5 in
the section of mean difference which gives the reading of 3.
So
log (6.335) = 0.8017
If in the above procedure, we have found out the value of log x, now, if we want to find out the value
of 'x' only, then how do we proceed, say log x = n.
:. x = antilog n
The anti-log of n can be obtained from antilog table. The antilog table is same as log table but here
the column N starts with 00 unlike 10 in log table and rest all is same.
Substituting by number say
log x =n = 2.0~78
So, for log x = 2 is characteristic and .0878 is mantissa
We' are to find outthe antilog of mantissa.
So the antilog of .0878can be obtained from part of the antilog table (Table 15.5).
'Table 15.5.
N
0
1
2
3
4
5
6
7
8
9
123
456
789
08
1202
1205
1208
1211
1213
1216
1219
1222
1225
1227
011
112
223
Anti log of..0878 = 1222+ 2 =1224
The log x =2 =-2 = 10-2 =0.01
So
x = 1.224 X 10-2
=0.0]24
Logarithims are mainly applied in numerical calculation of various datas in clinical chemistry,
biochemistry, microbiology, statistics and population genetics.
(a) For example in clinical chemistry :
pH = log 10IH+.

[[p.10]]
434
Blood Banking and Transfusion
(b) In biochemistry: Handerson-.Haselbach equation (to calculateacid to salt ratio to a desiredpH
in bufferedsystem) :
,
(A+)'
pH=pK + log (HA)
(c) In microbiology : Bacterial susceptibility testing and.'assays particularly in agar dilution
susceptibility testing.
(d) In population statistics: Geometric progression.
(e) In population genetics: Used alongwith statistical method.
(f) Measurement ofcolours (colorimetry) :
log 10
(A) Absorbance = -1- = I-log ,T
(g) .Measurementof blood gas analysis.
(h) Measurementof radioactivityand decay of isotopes.
(i) Activity and ionic strength.
(j) Colloidal particle size determination.
(k) Donnan membrane equilibrium aridosmotic pressure.
(1) Bio-energetics:
Oxidation-Reduction reaction
Enzyme kinetics etc.
Above are the various applicationsof arithmetic in laboratoryand related fields. For our purpose,
we restrict ourself to use in numericalcalculation and geometric progression etc.
Illustration numerical calculation
Find 7.525 x 2.313
Log x = 7.525 x 2.313
Log x = log (7.525 x 2.313)
= log 7.525 + log 2.313
Now
log 7.5?,f\ =
0.~765
(from log table)
log 2.313 =, 0.3642 (log table)
So
0.8765 + 0.3642 = 1.2407
= 1741 (from antilog table)
1
= 17.41
Take anti-log
Antilog of .2407
Characteristic of log x =
So x =1.741 X 101

[[p.11]]
Mathematical Consideration
435
15.5 STATISTICS
In Latin status means "a political state" and the statistics originally was related to numerical data or
information regarding the peoples of state and different aspects of their life viz. birth, death etc. The
organisation, presentation (graphical and tabular) and summarisation of data is in short known as
descriptive statistics.Viralstatisticsthat relateswith events of life and has very important use in modem
medicineis a partof descriptivestatistics.This trendof statisticscontinuedtill the beginningof twentieth
century approximately. However, with the advent of science and technology, the scope of statistics
broadenedfurther, thebranchofmodemstatisticsknownas statistical inferencecameintobeing.Statistical
inference is drawn from the logical conclusion and analysis of collected data. It is now becoming
increasingly relevantto clinical and laboratorypractice of medicine,because medicine has'now become
more quantitative and medical researchers, technologistsbelieves more in statistical interpretation and
methodology, e.g., analysis of a sample of bloodfrom where a logical conclusion is drawn regarding a'
largerpopulation,similarlyfor urineor experimentwith newdrugsetc.This is possiblethrough statistical
inference of computed data obtained from samples.·Statistical inference of computed data revolves
around a concept of probability or theory of mathematical probability. To understand the theory of
probability,we need to understandsomebasic terminologyand expressionof statisticsand herein below
is a brief discussion and application of statistical methodology in the field of laboratory medicine and
transfusion medicine.
To start we must know the fundamental characteristicof statistics :
(a) Statisticsin fact is always a quantitativeexpression,
(b) It is sum total of observationof datas collected purposively.
(c) In any experiment,we can compare the data.
The basic thing in statistics is collection and presentation of data. The collection of data can be
primary or secondaryin nature. The former is more reliable because it is collected by investigatorfrom
the source whereasin secondarythe data is obtained from the record or documentscollectedby another
person. Statisticsis an applied mathematicswhere the collection of data is done through-measurements
(abstracts or reality) in the form of numbers (1, 2, 3, 4 etc.) or continuousmeasurement(eM, weighing.
by scale or balance, scalar etc.),
Collection of data, thus involves operation of significant figures those are accurate andlorprecise
for correct statistical inference.
Significant figures are numerals contained within the quantity and are closely nearest to the true
value, e.g., if a length is stated to 80.0174562 em, then it-may sound incredible because accuracy in
measurement uptothat degreeis not possible,but if it is statedas 80.01,it is acceptablebecauseaccuracy
upto that degree is possible.This measurementof 80.01 em that is contained within the quantity is said
to have four significantfigures. So any calculation or measurement that" is represented by true value +
errors (positive or negative)of operationandlor measurementwill thus contain the significantfigures in
the quantity,
Lesser or smallerthe error, as described above, the expression will be more closer to the true value
and thus the exactnessof expressionwill bemore accurate.Accuracy. thus means measurementwhichis
unbiased and precise. Precise means highly accurate that refers to closer agreement between different
readings/observation obtained by repeated operation/measurement." Precision and accuracy are not the
same thing, the measurement whichis precisemayor may not be accurate.It refers to a degree of spread
of a series of observation.

[[p.12]]
436
Blood Banking and Transfusion
After the collection of data, the next step of operation is presentation of data, which is done in
manner that facilitates the study of salient features. As described earlier, the collected datas may be
either numerical discrete (corresponding to some sort of counting of numbers), e.g., bacterial colony
count,pulsecountwithina timeperiod,cellcountundermicroscope etc.or.it maybe informofcontinuous
observation/measurement, where the observation has to fall along the spectrum of continuation. Here
the observationis restrictedby degreeof accuracy, whereasin former,it is the particularvalueof integers
or numerical discrete. Thus one is a numerical discrete and the other is numerical continuous. The
numerical continuous is commonly used in expressing height, weight and blood pressure, age, serum
levels etc. Datas, thus obtained, are haphazardor ungrouped and has to be arranged in a proper manner
so as to facilitate the studyof salientfeatureseasily.So we can arrangethedatas first whichmaybe in an
ascending or descendingorder to obtain a more clear picture.The knowledgeof variateand frequency is
essential to arrangethedata properly. Variate is thequantitythat weobtainfrommeasurementof different
observation,e.g. in an experimentalstudy,themeasurementis 'X'and'Y' in a groupof 4 to 6 respectively.
This 'x' and·'y' when represented in mathematical form is known as variate and the later is known as
frequency or morepreciselythe frequency of the variate. It has been observedthat with a greaternumber
of observation, a more ciear picture can be obtained, if we summarise and present the data in form of
frequency distribution,either in tabular or graphical pattern. It can be further simplifiedby condensing
them in two classes or groups, known as class interval. Again, this class interval hasa true upper limit
and true lower limit and midvalue,This is done to overcomethe shortcomings in presentationand make
it more implied.
Example 1
Let usconsider datacollectedfrom 10(ten)experimentalobservationstates asfollowsnumericaldiscrete,
count per minute or pulse.
82,68,60,72,82,84,54,72, 100, 100(ungroupeddata).
Ascending arrangement :
54,60,68,72,72,80,82,82,100,100.
Example 2
To have a clearer picture, we can arrange this grouped data in tabularform of frequency distribution
(Table 15.6).
Example 3
Let us consider an observationon height of a person from frequencydistribution table arrangedin class
interval.
(a) Height: Stated in centimeter.
(b) Accuracy of measurement will be given by the nearest centimeter, say 80.48 em, could be
written as 80.5 em, since it is not possible to measure 80.48 em accurately.
(c) Number of classes in which the datas are grouped should be in general, between 10 to 20 or
nearer to 10 or 20.
(d) Class intervals preferablybe arranged:
(i) Finer to broader classes.

[[p.13]]
Mathematical Consideration
437
(ii) .Be of equal width.
(iii) Measurement is done accurately as described above, say if we want to express an interval
of 80 em to 80.48 cm and since accuracy to measure. 48 em is normally not possible, so it
can be written as 80 to 80.5 cm.
.(iv) Arrangement to bemutually exclusive, say 1 to 5, then 6 to 10 etc., because, if we write as
1 to 5and 5 to 10, then it is difficult to distinguish an observation of 5, viz., whether it falls
in 5 or within 5 (slightly above 5 and less than 6).
(v) Any reading stated as equal to or greater is considered open and such open interval creates
problem in further numerical manipulation or graphical representation.
(e) Limits are tabulated limit (actually used in table) and true limit (underlying continuous
measurement). If limit is tabulated as 80 to 80.5 em, then the true lower limit and true upper
limit respectively will be 79.95 and 80.55em that means any value above 79.95 and any value
below 80.55has to be considered correct to the nearesttenth, i.e., a reading of 79.98 is written
as 80 ern.
. .
True lower limit + True upper limit _ 79.95 + 80.55 _ 80 25
MId value =
-
-
·
2
2
'Ta~l~
,15.6. PUlsec~lJ~t,
:'a~~,
~:~r8qu~"cy.
Pulse count
Frequency
54
1
60
1
68
1
72
2
80
1
82
2
100
2
From above description the data can be tabulated as follows.
Measurement of height in centimeter for 80.48 em can be written as 80.5 em that is to the nearest of
a centimeter, tabulated limit of class interval is 80 to 80.5, true limit is 79.95 em to 80.55 em, mid value
is 80.25 ern.
Tables should be so made that it is clear, condense, but not complex, accuracy maintained, base for
percentage and units indicated.
To make a proper table of frequency distribution :
(i) Decide the maximum and minimum value-from the data of any observation.
(ii) Find out class interval by dividing the maximum - minimum value by the number of classes, we
desire to make. The class interval includes both the minimum and maximum value, e.g., maximum
- .minimum (also known as range).
.
.
Substitute by number 60 - 0 =60.

[[p.14]]
438
Blood Banking and Transfusion
Say, we desire to make 12 classes:
:. Class interval is 60/12 = 5.
1 to 5, 6 to 10 and likewise.
(iii) Find out frequency distribution by putting each item from data against respective class interval
and total of which gives the frequency in that particular class. Total of all the frequency equals
total number of observation.
Table 15.7•. Frequency
calcu.l~tion·.of
class intervals
Class interval of class
Item fromdata belonging to this class
Frequency
6 to 10
N{J*
5
11 to 16
N{J* III
8
16 to 20
III
3
Total
16
Class interval is 5, observe intervals. are mutually exclusive.
* Itemaredenoted by 1 and5 of such IIII makesa bunch.
Once the frequency distribution is made, it can be represented in graphical forms, viz., Histogram
and Frequency polygon. There are other types of graphs such as bar chart, component bar chart, line
diagram and pie chart.
Histograms are also known as frequency histogram are the graphical representation of frequency
distribution. Frequency polygon less commonly used are made by joining the midpoints of adjacent
block or rectangles at its top in a histogram. The midpoints are joined by line segments which is later on
made continuous.
Shaded area shows cholesterol at 75 to 100 age group. Having organised the datas in tabular form,
our next step is graphical representation of organised data which we have already done. Now the third
step is computation of the estimate of the population parameter. Several characteristic may be examined
but the most common are :
A. Measures of central tendency or location.
B. Measurement of dispersion, i.e., spread or variation.
A. Measure of central tendency
These are also known as statistical average and there are three most common measures for central
tendency or location mean, mode, median.
Mean : This is defined as :
x = _L_~
= _X.;;...l_+_x_2_+_ X_ 3_+_··_··_·x_
D
n
n
{
sum of values of observation}
(number of observation)

[[p.15]]
'Mathematical .Consideration
439
I
>0-
I
India
~
Brazil
~
France
~
(Note as per scale)
o
-- X--
Fig. 15.1. Histogram and frequency polygon.
Rectanglelblocks arranged vertically represents
histogram basedon class interval (x axis),
frequency along (y axis). The area of each block
is proportional to its frequency. Frequency
polygon is represented by broken line obtained
by joining midpoint of each block.
Fig. 15.2. Bar chart (multiple). Also known as a
compound bar chart. In this chart two bars are
}grouped together, e.g., systolic (shaded) and
diastolic (unshaded) blood pressure in a particular
population of diabeties. N.B. Scale here is not
starting at zero, so cautions approach to
interpretation is required.
-----=-------
Substitute by number
---. Cholesterol Level at age
group of 75 to 100 years
(Note as per scale)
Fig. 15.3. Pie chart. Here the areas/segments lof a
circle is compared, segments are dependent on the
angle formed at the'centre. In above chart circle di~ded
into four segments of 0 to 25, 25 to 50, 50 to 75 and
75 to 100 age groups.
n
n
So x, i.e. mean== m + e.
Where x represents mean, m represents true
value and e represents errors of measurement.
m + m· 1'"m + et + e2 + e3
3
3
4 +3 + 5
0.24+ 0.42 +0.15
= ---+-------
3
3
·Fig. 15.4. Simple bar chart. Length of each bar is
proportional to magnitude represented, each bar is
separated by a minimal space which are equal but the
space has no statistical significance axis should be
clea~ly
labelled and. units indicated.
12
0.81
=-+--
3
3
=4 + 0.27
=4.3 (round back to number of significant
figure)
Calculation of mean has certain advantages
and disadvantages which are to be noted :
(a) Single aberrant observation may affect the
mean, say, mean or x bar i.e., .x of
ungrouped data collected viz., 10, 26, 32,
34, 28 is 130 + 5 + 26, observe that 10
has seriously affected the mean and at
Vertical
(Not as per scale)
Horizontal

[[p.16]]
440
Blood Banking and Transfusion
least four observation is either ~ 26. The
lOis then an abberant observation.
(b) Amenability to mathematical treatment is
the main advantage of the mean. This is
usefully employed while performing
statistical inference based on theory of
probability.
~
E
"~
I
=~
1ft
c
"iiia:
om
-x-
(Note as per Scale)
Fig. 15.5. Component bar chart. Component bar chart,
each bar here is divided into two parts representing
two different items and they are proportional to
magnitUdeof that item. Rainfall at Cherrapunjee, years
in 'x' axis, rainfall in mm "I' axis, bars showing white,
rainfall, dark, yearly net accretion.
Median : It is the middlemost observation or
the value of middle observation. Say, in above odd
numbered observation of 5 items, i.e., 10, 26, 32,
34,28,it is easy to find the middlemost observation.
In the present case, it is 32 because two figures are
on either side and 32 is on the middle. But if it is
aneven numbered observation of, say 8 items, viz.,
10, 12, 14, 16, 18, 20, 22, 30, then we find out
median from the average of two middle value,
therefore (16 + !8) + 2 = 17 is the median in the
series. Median and mean are again not the same
and there is difference. Any abberant observation can seriously affect the mean but not the median. For
example, a data collected from six observation states 5, 7, 7,8,11,100. Mean and median repectively
are 23 (138 + 5) and 7.5 (7 + 8 + 2). The sixth value has affected the mean but not the median, We can
thus infer that median is more representative or nearer to truth (Table 15.8).
Mode: It is most frequent amongobservation or "most frequently occurring value" in a distribution
(one peak, two peak) of data, e.g., 4, 8, 9, 8, 11, 6, 7, 8, 10. In this distribution '8' is most frequently
occurring value and hence the mode. The main disadvantge of mode is that its exact location is difficult
to determine and neither it is clearly defined. As such mode is not suitable for use in medical statistics.
It is rarely used in scientific literature and in performing statistical. inference, its usefulness if almost nil.
B. Measurement of dispersion/spread or· variation
There are several measures of variation and commonly used are :
(a)
Range
Table 15.8. Comparison between' mean, rt:ledian'and' mode
Measure
Value affected
Commonly used in
Characteristic
Location
Mean
Seriously affected by
Performing statistical
More mathematically
Arithmetic mean
extreme value
inference
amenable
Median
Not affected by
(a) Descriptive statistics
Less mathematically
Middle most
extreme observation
(b) In those with prone-
amenable
observation
ness to extreme value
Frequently
Mode
Not affected by
Rarely used in
Lesser amenable
occurring
extreme observation
descriptive measure
mathematically
observation

[[p.17]]
Mathematical Consideration
441
(b) Mean deviation
(c) Standard deviation
Range is the simplest measure of variation and is the difference between highest and lowest value,
let us take a series of ten observation 10, 20, 25, 28, 30, 32, 35, 40, 50, 45, the range as per definition is
50 - 10 = 40 and it is expressed as 10 to 50. Range has a tendency to increase with increase in number
of observation and to express range one employs two extreme observation neglects dispersion between
the two values. Therefore, the range as a measure is practically less important though it may be fruitful
to use it in deriving statistical inference.
Mean deviation is defined as :
L(X-X)
n
where
~
= Sign of summation
x = Arithmetic mean
x
== Individual observation
N = Number of observation
So to find out mean deviation, one has to find out the difference between arithmetic mean and
individual observation, all the values so obtained are totalled up and divided by number of observation
will give the mean deviation. But, it has been observed that it is difficult to manipulate any absolute
value. Mean deviation is absolute value, and therefore, not much useful in performing statistical inference.
c. Standard deviation
SD = JL(X-X)2
0-1
Simplified way in which it can be defined is "Root-mean-square-deviation" or square root ofvariance.
Note that 0-1 appears as denominator in place' of n and has been observed that 0-1 performs betterin
deriving the statistical inference. The details are available in larger text books and beyond .the scope of
this discussion. Standard deviation is one of the most important expression in statistics. To calculate the
standard deviation one has to find out the arithmetic mean and then total up the square of individual
deviation of each observation from the arithmetic mean, divide the total of squared value by n-l. Square
root of value so obtained gives .standard deviation.
N,orma.1 doistrib&,ation
~ondo
confidence limit
Normal distribution also known as normal curve is important in statistics. Alarge number of data collected
from large number of peoples and then make narrow class interval later on graphically represent the
frequency distribution, we get a smooth curve which is symmetrical. Shape of the curve is dependent on
mean and standard deviation. Such curve is known as normal distribution curve.
Standard curve is that which has a total area 1, mean '0' and SD is 1. It is a unimodal bell-shaped
curve which is symmetrical. The curve is symmetrical about mean thatwhich falls under the highest

[[p.18]]
442
Blood Banking and Transfusion
point on the curve and coincides to zero. The one standard deviation on.eitherside on mean includes: x
± IS is approximately 68%and the area between 2S on either side of mean is approximately 95% and
covers the maximum of the values and area between 3s on either side of mean includes 99.7%. The
limits described above which are placed on either side of mean are known as "confidence limit". The
confidence limit determines the probability. By saying 95% of area of standard curve, we mean 95% of
values will be within the area between 2S on either side'ofthe mean, so there will beonly 5% probability
(5/100 = 20) or 1 in 20 chances of areading falling outside the confidence limit. For details consult
Textbook of Statistics.
Fig. 15.6. Diagrammatic representation of
standard curve.
(Note as per scale)
Standard error of mean
This is also known as Standard Error (S.E.) and is defined as:
S.E. x =S/~n
where S.E. is the standard error
X - arithmetic mean
S .- standard deviation
N·- number of observation
This is calculated to find out how much correct is the
mean of the sample. So to calculate this standard error of
mean is first determined .and then set up the confidence
limit. The true mean shouldbewithin the confidencelimit.
Let usconsidera 36observation/samplesin a particular
age group. Pulserateof thisgroup,the mean valueof which
is 78/min and a deviation (S.D.) of 3 substitute by Dumber.
S.E.(x) =3/~36
7= 3/6 = 0.05
Now, if we set out the confidence limit at 95%, i.e.,
twice the standard (2 S on either side of the mean) error,
the range of mean will lie within 78 ± (2 x .05), l.e.,
77/min to 79/min. Chances of falling out of the limit is
1 : 20, means probability:
p.: 0.05
Negative
Values-Va
o
X
Positive
Values+Va
Estimation of probability
This is done by applying the formulae of the relative deviate :
(x -x)
SD
For example, consider arithmetic mean of samples of ESR is 25 and standard deviation is 3 in a.
group of healthy person (female). Now,if we want to find out that a female choosen at random would be
found to have ESR, say 37 or above.
(37 - 25) =!3. =4
3
3

[[p.19]]
Mathematical Consideration
443
Say relative deviate that corresponds to an area 'a' in the normal standardcurve which covers, say
1/2 of total area. So area beyond 'a' is equal to 0.5 - a = 'b', 'b' is thus the probability out of 100,000
individual to have ESR 37 or above.
Standard error of difference between two mean
Thisiscommonlyemployed inexperimental biologicaVclinical researchworks.whereweneedtocompare
the results between two groups, say control and experimental. This is employed to find out that the
differencebetweentwomeansaresignificantstatistically. This is donebycomputingthe threeparameters
namely number,·mean and standard deviation of the two groups to be compared, i.e. control and
experimental.To find out the standarderror of differenceapply the formulaeas given below :
S.D. of 1
S.D. of 2
SE (d) =
+
so SE (d) = Y
nl
n2
where
SE (d) Standard error of difference
SD1 SD of control, SD2-SD of experimental
n Number
y Square root value
Now to find out the actual error of difference between two means of control and experimental
group:
X, - X2 = actual error of differencebetween two means
If this actual error of differencebetween two means is two times or more greater than the value of
SE (d), the test is consideredsignificant.
Standard error of proportion
This is calculated to find out the proportion of one item present ill the other,e.g., proportionof samplein
universe.
Standarderror of proportion=Ji!
where
p Proportionof sick actually present
q Proportionfound by random' sampling
n Number of universe
.
[pq
Let us consider= V~
Standard error Z on either side 'p' as criterion i.e, :
p + 2 (Z) = at
p-2(Z)=~
Thus the range of proportionin the said sampleand universeis betweena1and az. Find out the 95%
confidence limit and see whether the range so obtained falls withinthe 95% confidence limit. But if the

[[p.20]]
444
Blood Banking and Transfusion
observed valuetvalue obtained by random sampling) is outside the confidence limit, then we consider
the difference between '0' observed and 'E' (expected)'as statistically significant
Also find out the relative deviate of 5 :
p-o
Z
where
P = Proportion of sick present
o =Observed proportion found after random sampling.
The relative deviate, if exceeds 2, then it is considered significant(more than two times).
Standard error of difference between proportion
This is calculated to find out standard error of difference between two proportion and its significanceby
applying the formulae asgiven below:
Plql + P2Q2
SE (d) proportion =
nl
n2
where
'p' is proportion of sick
'q' is proportion of healthy
'n' is number/size of sample
Say SE (d) proportion=Z
The observed difference between two groups, i.e. 1 and 2 is PI - P2 =y.
The observed difference 'y' may be less than twice of SD (2XZ) or it may be more than 2xz. H the
observed difference is less, we consider them insignificant, but if it is more than 2xz, we infer that it is
significant and the observed difference is not due to chance.
'CHI-square test
This is an alternative method to find out SE (d) of proportion between two or more groups.
It involves the concept of degree of freedom 'df and null hypothesis.
.Deqreeot freedom :tdf')
In standard deviation the denominator mentioned is n-l where n represents-the number of observation
and n-l is the degree of freedom. By degree" of freedom, it actually means the number of independent
quantity in the whole series of observationor the total number of quantity minusthe number of restriction
imposed upon.
To find out the degree of freedom in statistics, the sum of the squares corresponding to individual
degree of freedom is subtracted from the total sum of squares of measurement.
:EX2..:: :E(X)2.
[TOtalsum of squares J_[sums of sq.uare corresponding ]
n
i.e.
of measurement
to individual degree offreedom

[[p.21]]
Mathematical Consideration
445
This can be expressed otherwise as one degree of freedom for the mean. The rem.aining degree of
freedom, therefore,willcorrespondto the value of square of difference between two measurement.That
indicates us to find out the error of variance.From the above we can conclude that sum of squarescan be
divided as.follows :
Sum of square = (Degree of freedom for mean) + (Degree of freedom of variance)
Sum of squares has thus been divided into two component each represent one degree of freedom
viz., one degree of freedom for mean and one degree of freedom for variance.
Wehave discussionon mean earlier but what about the term variance.In describing the measuresof
variation that deals with sources of variation, we are more concerned with quantitative clinical
measurement and its affinity to different sources of variation. Main sources of variation could be :
(i) Biological: .Genetic factor; sex, age etc.
(ii) Temporal: State of activity viz. mental (anxiety emotion), environmental, family atmosphere
and various other.
(iii) Measurement error: Already described and is produced by different measurement under same
experimental condition.
For laboratorypurpose we can neglect the temporal source of variation and consider the other two,
i.e., biological and measurement error.
We can find out by repeatedly ascertaining the values from each individual sample, viz. twice or
thrice and then calculate the mean of each such determination.Putthe mean value in a raw (preferably
last raw) in respect of observationfor each samples, put in successive columns. The main observationis
totalled up and resultant variance among the mean is computed by the formulae :
LX2 _(LX)z/n
(0-1)
Thus this intersubjectbiological variation has component of errors of measurement and to estimate
the true biological variation, we can segregate the error of measurement as follows :
Say we have taken three determination, calculate the variance by applying the formulae as given
above.
Say each determinationrespectively are a, band c :
~X2_.
(~X)2
2
b2
2
(a+b+c)2
~
a +
+c ------
___............n_=
n
=z
n-l
0-1
Similarly find out variances of all three determination irrespect of each sample.All are summed up
and divided by total number of samples to get the mean variance.
Say for sample 'M', 'N; and '0' variance are 'Z', 'Y', 'X' respectively.
Z+Y+X
· ·
3
=Mean variance
This mean variance gives the component of errors of measurement.
So biologicalvariationthatconsistsof twocomponentnamelytrue biologicaland measurementerror
has been segregated. As discussed earlier, the SD is the square root of mean variance given by 'V' :

[[p.22]]
446
Blood Banking and Transfusion
LX2 -(LX)Z IN
S.D. =~V =-----
n-l
where rx2
=Sum of squares of observation
Lx2/n = Correction factor for the mean
n - 1 = Number of degree of freedom or (df)
After gaining some knowledgeon mean and variance,it willbe appropriate,if we make a comparison
of variances andlor means. It has been observed that if a large number of samples are drawn from same
populationand thatboththeparameters(varianceandmean)differsfrom sampleto sample.This variability
apparently becomes smaller with increase in number of samples. To determine this we are to estimate
the likelihood of the variances coming from the single population arid if our variance is from the same
population what is the likelihood of getting 'F
I value as largeor largerthan that obtained. If this is small,
we assume.our hypothesis of COlTIlnOn population variance was wrong. In other words, the variances are
really different coming from different population.
L(X. -
Xl)~
/(0 -1)
F = L(X2 - x2)2/(m - 1)
Varianceratioor IF'is the estimatedlikelihoodof the twovariancescoming from the samepopulation,
the larger variance is put into the numerator,and the 'F value is computed from the ratio between degree
of freedom for variance in numerator (larger variance) and 'df for variance in denominator at a certain
probability level viz., 5%, 1% etc. (consult Textbookof.Statistics for 'F value table).
Particular
'----... value of F
Regionof null
hypothesisrejection
Fig. 15.7. Diagrammatic representation of IF value
distribution.
.
Null hypothesis
Null hypothesis is based on many statistical
calculation and by itself is a statistical test of
significance on a mean, where a hypothesis is
set up with no difference in variance .and
inference on claim about the population mean :
(a) Ifsample valueis incompatiblewithnull
hypothesis and concludes that Jlo is not
population mean, then null hypothesis
is
rejected
which is
considered
statistically significant.
(b) If sample is compatible with null hypo-
thesis and concludes that flo may be the
population mean, then null hypothesis
is not rejected which is considered not statistically significant.
Comparison of averages
This is more important from laboratory point of viev.. By previous description of mean, we measure the
success ofthe process.Butin this,wecollectrandomsamplefromsamepopulationandmakeacomparison
between control (group treated-by standard method) and experimental method. To obtain a clear cut
difference between two groups is difficult and there occurs certain overlap.

[[p.23]]
Mathematical'Consideration
447
This problem is solved by following procedure. Fromthe observed measurement of control and
experimental XI andx2 respectively that contains a component of measurement e1TQr 'e' we are to find
out the differenceof 'E' (experimentsl populationmean) and '0' (control population mean) and that can
begiven by 't' test.
The 't' is calculatedby applying the formula:
X1-X2
l:";(XI - xt)2+ L(X~_-=-
X2)2 (lin +lIn)
n+m-2
t = ---r=============-----
where
n = Numberof control
m =Number of experimental
(n + m) - 2 =Degree of freedom
xl = Observationor data in control group
x2 =Observationor data in experimentalgroup
x, = Mean0' contro~
Xl =Mean of experimental
To simplify the above, we can state it as follows :
Differencebetweenmeans 9f control and experimentalgroup
S.D. of the difference
Here also, if we consider that both control and experimentalgroup were randomly collected from
same population and.variationhas occurred in sampling,we compute their probabilityof occurring and
again, if it is small or incompatible, we may reject the hypothesis statisticallysignificant.
Referring back to our earlierstatementthat differencemay occur in respectof varianceand mean of
population under consideration. So, it is advisableto ron IF test first followed by 't' :
!:(X1-X1)7. / n -1
F= ~
z
L.(X2 -x2) /m-l
If 'F' is statisticallysignificant't' need not be computed.
Normal distribution and binomial distribution
When any statisticalinferenceis made from a series of data and induce a conclusion,the mathematical
way for describing it or the populationin a smoothcurve form is known as normaldistribution.This has
already been describedearlier and graphically represented.
Normal distributioncan be defined mathematically as :
y =J;1tS e-1I4(X ~ XJ

[[p.24]]
448
Blood Banking and Transfusion
where
e = A mathematical constant approximately equivalent t02.71828
x =Mean
S = Standard deviation range from 0 to 1 where '0' is the mean, 1 is standard deviation to the
right or left of mean (see graphical presentation of normal distributioncurve)
For further details Textbook ofStatistics may be consulted.
Binomial distribution
The word binomial is a mathematical expression consisting of two terms connected by sign (+) or (-)
and the binomial distribution is thus defined by :
. (P + Q)n
P =Probability of independent events occurring
Q
=: Probability of independent events not occurring
n
= Total number of test
The question that logically arises is "probability of an event". Probability of an event occurring can
be computed by taking into consideration the number of ways the event can happen or occur and total
number of possible outcome, e.g., if we toss an unbiased coin having head at one face and tail at other,
the probability of face appearing (event occurring) in order is 1/2 or 0.5. It require little explanation, if
we say, probability is zero or '01, it indicates to event that certainly, doesnot occur and by probability 111
(unity) indicates to event that are certain to occur. The other probability will thus fall between two
extremes, '0' and '1'. The probability 1/2 or 0.5 described above is due to orderly occurring but if we
disregard the order then what could be the outcome, if we happen to toss the unbiased coin twice.
HH
+
2HT
+
IT
~V2
+
2(~~)
+
~~
Note in the above expression the coefficient ofH (head) and T (tail) are similar to binomial expression.
(H + T)0
Generally all possible probability of In' independent events are indicated by (P + Q)0. The expansion
of binomial distribution is easy when 'n' is small but difficult when 'n' is large.
After gaining some basic knowledge on the items involved in "chi-square test", let us proceed to
find out the difference in proportion by application of "CHI-Square test".
Formulae of "CHI Square-Test"
x2= L L(O-E)2
E
where
~ =Summation
o =Observed number
E = Expected number
This test can be considered as a statistical test to find out :
(a) Goodness offit : Degree of closeness of distribution of events observed with the corresponding
distribution of events expected.

[[p.25]]
Mathematical Consideration
449
(b) Independence : The events occurring independently or the distribution of events are non-
compatible with each other.
Statistical inference is drawn from the result, we obtain after .solving the formulae, To draw the
inference, a table that contain critical value for "CHI - Square distribution" is consulted and 'P' value is
obtained (probability -P) which could be as great or greater that indicates the events occurring could be
due to chance alone, i.e., the distributionobserved is equivalent to the binomial distribution of expected.
The null hypothesis is thus supported.
The statistical inference is reversed, if the (P) value is small, the difference between observed and
expected is too great and the events couldnot be due to chance alone, the null hypothesis is not supported
rejected, i.e., statisticallysignificant.For clinical and laboratory purpose level of significance (generally
considered) is P < 0.5, the level of P < 0.1 is a more conservative value. While the consulting the table,
we are to take the 'df into consideration because the table is arranged according to 'df',
'df =(total no. of column-I) (total no. of rows-I)
e.g., in 2 x 2 distribution (2 - 1) (2 - 1) = 1 'df
again in 3 x 3 distribution (3 - 1) (3 - 1)=2 x 2 =4 'df
Let us now consider some examples of 2 x 2 distribution.
We. can adopt anyone of the following depending on the conditionality :
(a) Develop an exact test provided that each of two sample follows a binomial distribution.
(b) Approximatetest where there is a normal approximationto binomial distribution.The former is
known as Fisher's exact test and is applicable when ~ne
of the frequency is '0' and 'n' is 20or
less.
Let us examine the behaviourof unknown antisera with known cells (Table 15.9).
~
..'
'
.
.Table· ·~6~9
Known cells
Unknown Sera
D
d
+
6a
ob
6 (a + b)
-
oc
5d
5 - + (c + d)
(a+c)
(h+d)
(a« b) +(c + d)
6
5
lin
(a + b)! (e + d)f (a + e)f (b + d)
P=
of a! b! e!d
=
6!5!6!5 =900 =0.0021645
11!6!0!0!5
.330
The 'P' value is very small, the possibility that outcome is due to chance only is refuted and we
conclude that almost certainly the unknown serum contains anti-'D', critical value for this 2 x 2 is taken
from chi-square table at 1 'df since (2 - 1) (2 - 1) = 1.

[[p.26]]
450
Blood Banking and Transfusion
In case none of the frequency is '0' a rearrangement of Table 15.10and 15.11 is made by reducing
each of the two diagonal figure having the lowest value by one but retaining the same marginal value.
The probabilities are then summed up.
Table 15.10. Arrangement of table'with non-eO' frequency
Known
Cells
0
D
+
5a
3b
8(a+b)
-
4c
4d
8 (c+d)
(a + c) 9
(a + d) 7
16n
(a';' b) (c .+ d) (a + c) (b + d)
pI =
=Y
na!b!c!d
Table··15.11:
Known
Cells
D
d
+
6a
2b
8(a+b)
-
3c
5d
8(c+d)
(a + c) 9
(a + d) 7
16n
(a + b) (c + d) (a + c) (b + d)
1
2
p2 =
= z; p + p = y + Z = x
na!b!c!d
Continue till one of frequency becomes '0'
Pi + P2+ P3+P4=Y+ Z + ..... =x
Interpretation of probability valueis obtained from the standard chi-square table.
2 x 2 distribution can also be used to find out genetic association two antigens, if we happen to
suspect such association in a population group and is given by formulae.
.t~bI8·,·15~1·2~.
2 X" ·2.:table for genetic:.a~~QatlOi1'
Reactionof anti-unknown
Distribution of antigen
+
-
a
b'
a+b
c
d
c+d
(a+c)
(b+d)
(a + b) + (c + d) =n
Heren=a+b+c+d

[[p.27]]
Mathematical Consideration
451
Formulae for calculation
x
2 = (a + b) (c + d){a + c)(b + d)
Conditionality anya, b, c or d equals to zero.
Substitutingby number.
Table. 1~.13.~
.:·.Subs~iti.rting
..by ..nUmber·,
Reaction of anti-unknown
Distribution of antigen
9a
ob
(a + b) = 9
4c
Sd
(c +d)=9
13
5
n 18
2 _ (9.5-4.0)2 xI8
x -
9!9! 1315
(45_4)2 xI8
=
5265
=30258,= 5.7'
5265
So at 1 'df the x2 valueis between0.2 to 0.1,. therefore,it is statistically significantandthereactions
not independentsuggesting associationbetweenantigen(linked), If on other hand axd is smallthen bxc,
it suggests a negativeassociation (allelicgene).However, these are only indicatorand not confinnative.
2. Approximation test
It is commonly employed :
(a) Normal approximation to the binomial distribution.
(b) Chi-square with 1 'df and continuity correction.
(c) Comparison of two proportion in independent. or paired sample.
(d) Comparison of more than two proportion in independent sample.
(e) Chi-square with -I 'df or more than two.proportion in matched sample.
Discussion of all these is beyond scope of this book. In laboratory, it is commonly employed to find
out difference between control and experimental group - subject's response or reaction, if independent,
null hypothesis is rejected and there is no difference. In blood banking and transfusion, it canbefruitfully
utilised to compare the two sets of data.
1. Arrange datas in tabulated form (as given in Table 15.14).
2. The expected number for each cell is to be found out.
(O-E)2
·3. For each cell compute
E
where
'0' = Observed number

[[p.28]]
452
Blood Banking and Transfusion
Table 15.14. Comparison of data of Rh
Rh type
A (0)
M (0)
TQtal
A (E)
M(E)
A
M
Others
Total
a
b
c
d
e
x
f
g
h
j
Y
a+f
b+g
c+h
d + i
e +j
z
a+f
x --
z
b+g
x __
z
c+h
x --
z
d + i
x --
z
e +j
x
z
a+f
y
---r-
Z
b+g
y --
Z
c+h
y --
z
d + i
y --
z
e+ j
Y
z
a+f
a-x--
z
x (a+ 0 / Z
b+g
b-x __
Z
x (b + g) / z
c+h
c-x--
z
x(c+h)/z
d + i
d-x--
z
x(d+i)/z
e+y
e-x __
z
x (e+j) / z
a+f
f-y - z
y (a + f) / Z
b+g
g-y-
Z
y (b +g) / z
c+h
h-y-
z
y (c +~)
/ z
d+i
i-y --
z
Y(d + i) / z
.
e +j
J-Y - z
y (e+j) / z
(Row total) (Column total)
E = Expected number
d
1
gran
tota
4. Summation of above value :
Since
Let us compare a series of data for Rh in a population with different origin, say, Indian population
with Aryan origin (a) and Mongoloid origin (M).
(O-E)2
For each cell compute the value of x2, i.e.
E
separately for A-Aryan and M-Mongoloid. The
A (0) stands for Aryan (observed) and A (E) stands for Aryan (expected), similarly M (0) and M (E),
e.g., RIf; the X2 value is given by :
a+f
a-x--
Z
a+f
Z

[[p.29]]
Mathematical Consideration
453
=(o-Ef =X2
E
During actual calculation, the letters are substituted by number and sum up the values for each cell.
Next we are to find out the 'df followed by reading of 'P' value at the concerned 'df from "chi-square"
table. From the value of 'P', we can infer whether it is statistically significant or not.
Other application of statistics
,Screening ofblood donors by application of normal distribution (Prince and Gershon quoted by Theodore
Colton in "Statistics in Medicine" loco cit.).
In this procedure, blood donors were screened for SGPT level and cut off value is fixed. The donors
whose SGPT level exceeds the cut off value are rejected and below the level are accepted. This is an
indirect screening programme of donors to find out the hepatocellular damage that could be due to
hepatitis serum hepatitis may be caused by HbsAg is an important disease that can betransmitted through
transfusion. The present methodology of detection of HbsAg in donors by Elisa test claimed full proof
are done after collection of blood and reactive units are discarded. Elisa tests are costly and donors are
lost in follow up procedure. Such donors who are active carrier may spread the infection further :in the
community. But if screening is done before collection, we have the advantage of cost benefit in terms of
man, money and material. In health aspect, 'we prevent the spread of disease by.adequate cautioning of
the donor on the basis of cut off value and request him to carry on further investigation for confirmation,
treatment etc. which the donor usually does with the hope ofgetting cured. Abriefoutline ofthe procedure
is given below.
Test
(a) Establish the mean and standard deviation from normal distribution curve ofdiseased individuals
for SGPT. This is expressed in log SGPT units.
(b) Establish the mean and standard deviation for SGPT in healthy individual from the normal
distribution curve for healthy individuals expressed log (base 10) SGPT units.
(c) Determine the cut off value at 95% confidence limit. This means that donors blood SGPT
exceeding the cut off value will be rejected and below the cut off value will be accepted. And
accordingly find out the SGPT value that cuts off upper 5% of the distribution of the areas from

[[p.30]]
454
Blood Banking and Transfusion
the table of one tail distribution of standard normal curve (consult the table in standard text
book statistics, .050 in the table corresponds to 1.65.Actually in table, it is .049, but we take an
approximate value for .050 =1.65).
(d) Since 1.65is equal to cut off value for upper 5%distribution so what will be the cut off value for
(a) and (b) above. Let us consider:
a =y ± SO =y ± e1
b =m± Sd = m j e,
So set 1.65=(x - y) e t
So we for x, :. x=(1.65)/e1 + y, thus the value of 'x' which can be obtained by substitution of
numbers will give the upper specification limit of accepting 95% blood from healthy donors"
(e) If the upper cut off value is 1.65/e1 + y, then what is (-) z, determination of z will enable us to
know the total area in the distribution curve is formed by (+) Z + (-) z around the mean. So the
area below (1.65)/e] + y or (-) z.
(-) z=(1.65/e} + y - m)/e2 = n
Read the value of (-) n from standard one tailed table.
(f) Thus, we reject peoples (healthy donors) above (1.65) et + y or those whose log SGPT has
exceeded the upper specification limit suspecting that they have the disease.
And again the proportion of normal curve that falls below (1.65) I e1 + y and is accepted as (-)
n.
y
I
I
Scalefor Healthy
o
1.65/e1+y
donors
I
I
950/0 confidencelimit
(-)n
0
or [(1.65)/e1 + y -
m] 92
(Not as per scale)
m
Scalefor unhealthy
(Diseased)Donors
Fig. 15.8.
So from the diagram (Fig. 15.8) it is thus clear that there is some error in our determination:
(i) We are rejecting 5% healthy donors on suspicion of disease false positive.
(ii) We are accepting some diseased donors allowed by area covered under (-)n (not as per scale)
false negative. But an attempt to minimise one of the error will be at the cost of increasing the
other.
Whatever we have described in terms of percentage which gives us only a vague idea. For easier
understanding let us consider a population group of 500 population of whom 15% are diseased, i.e., 75
individuals are diseased and healthy individual are 425 only confidence limit for 95% we accept the
blood of 425 x 95/100 =403.75 of healthy donor and on the other hand, we reject 5% of healthy donor

[[p.31]]
Mathematical Consideration,
455
blood exceeding the upper specification limit on suspicion that they might have the disease and it amounts
425 x 5 / 100 = 21.25. Let us consider the diseased population of 75 individual according to statement
given earlier, we are accepting (-) n of diseased blood from the diseased donor and rejecting 100- n
blood from diseases population. If we 'put it in a tabulated form as below.
Total donor population is 500 '
Healthy
425
Diseased
75
425
= 403.75 (units accepted) + 21.25 (units rejected)
75 = 75 [(-) n] units accepted + 75 (100 - n) (units rejected)
Total
= [403.75 + 75 (-n)] + [21.25 + 75 (100 - n)]
Inference
(a) We accept only 75 (-n) units of diseased blood in a total collection of 403.75 + 75 (-0).
(b) Totalblood rejected is 21.25 + 75 (100' - n).
B'oth the inferences above has to be analysed in terms ofcost benefit and decision taken accordingly
on the result of screening programme.
Correlati'on and' regression
This is not a new concept but useful and important in statistical application in laboratory statistical
procedure, where we want to know ofany linear relation between two variables. These relations, may be
positive, negative or useless and are expressed respectively as upto (+) 1, down to (-) .. and '0' i.e.,
useless or no relation, The linear relation could be applied to know whether there is any significant
relation between height and weight, smoking and lung cancer compare laboratory data obtained' from
quantitative analysis or test performed by two different methods.
The two variables are denoted by 'x' and 'y' and "coefficient or correlation" by r and is defmed as :
L(X -'x)(y - y)
r = ~"L(x
_ X)2 "L(y_ y)2
As stated earlier this r lies between (+) 1 to (-) 1. And (+) 1 means strong positive association, a
value nearer to (-) 1 indicates negative association, and '0' indicates no association or relation. In positive
association the values of 'x' and 'y' are directly related whereas in negative association they are inversely
related.
We can also attribute 'p' value to 'r' to see whether this correlation are due to chance or not. Smaller
the 'p' value, more is the difference expected between 'r' and '0'.
,
This measure does not prove any causation but may be coincidental or it may be related to any
associate variable common to both 'x' and 'yr.
Regression
Both regression and correlation are statistical approach essential to' determine the inter-relationship
between two variable, the unit of measurement being the same in each.

[[p.32]]
456
Blood Banking and Transfusion
Regression does not practice symmetry in inter-relationship between two variables. In determining
the regression coefficient one studies the mean value of one variable (dependent-y) while the other
changes. The changing variable is denoted as independent variate or 'x',
Its chief application in laboratory is "dose response" relationship in experimental animal. Here
since the dose is changed to observe the response and dose is therefore independent variate and respons~
is dependent variate respectively 'X' and Y. Regression coefficient is denoted as 'b' and is defined as
follows:
y = y + b (x - x)
L(X - x)(y - y)
b =
L(X _ X)2
Sum of cross product of deviation about mean
b =
Sum of squares of deviation about mean
So 'b' is regression coefficient of 'y' upon 'x',
where
x = Independent variate (fixed)
y = Dependent variate (subject to change)
We can similarly find the value of 'x' i.e, regression coefficient of 'x' upon 'y' denoted as bl .
b., by application of same formulae and replacing 'b' by 'b)', we can get:
x = x + b, (y - y)
where
y is the meanof yl' Y2' Y3' Y4 etc.
x is the mean of Xl' "2' X3 etc.
Sum of cross pr9duct of deviation about mean
b, =
Sum of squares of deviation about mean
L(X - x)(y - y)
b. =
L(Y _ y)2
Apart from the ones described there are few other items in statistics though not very much known
for its application in laboratory practice and transfusion medicine but in certain occasions they are
required to be used, e.g., distribution of non-continuous events forming a small fraction of the total. A
concise information on few such items are described here in below.
We ought to know little about uniform distribution and poisson distribution. They are types of
probability distribution like the binomial distribution, f(~)
which denotes percentage distribution of 'x' is
not the same for all. In uniform distribution 'x' ranges from '0' to '9' in binomial the 'x' ranges from '0' to
'n' and in poisson distribution 'x' ranges from '0' to 'a'. the mean respectively are 4.5 nrc, A. (lambda).
Standard deviation for each of the probability distribution can thus be :
SD for 4.5 = 2.87
SD for A. =-VA
SD for n1t= -Vn1t (1 - 1t)

[[p.33]]
Mathernatlcal Consideration
457
In uniform distribution 'x' is random variable. The probability of occurring each 'x' in uniform
distribution is equal and uniform i.e., each of the numbers 0, 1to 9 are likely to occur equally and can be
expressed as follows :
f(x) =I;9 (1/10) =10(1/10) =1
where,
x = 0.1-9
In poisson distribution, there is distribution of non-continuous events forming a small fraction of
total, e.g. let us consider counting of blood cells. When we count blood cells in a suspension and we
count them in a unit volume which is likely to vary in each count. The total number of cells, represented
by 'x' and Arepresents cells present in per unit volume. This can be mathematically expressed as :
f(x) =~a
e-A. AXlxl =c-AeA=1 .
where,
x =0 and upto a '
Substituting by numbers
Total no. of cells, say 2000
Randomly distributed in 50 units
Thus the rate of distribution is 2000150 =40 per unit volume.
So the mean of A, 40 and SO =-VA =-V40 = 6.35
We can thus find out the probability by substituting Aby number in formulae
f(x) = e-A. AX I XI
Similarly in case of binomial distribution the formulae is :
f(x) nex XX (1 - 1t)n - X
where nex =combination of n objects taken 'x' at a time
n1t= mean
~
=standard deviation
n1t(l- x)
However, detailed discussion on the above beyond the scope of this book and interested readers
may consult textbook of statistics.
Bayes theorem
It involves the additive and multiplicative laws and the conditional probability to develop Baye's theorem.
Additive laws are mutually exdusive events where addition is in operation, e.g. pr (A) + pr (B)
where 'A' and 'B' are mutually exclusive events and 'pr' is the probability, e.g. n = 8 mutually exclusive
outcome so each outcome has the chance of 1/8which are equally likely but the possibility of not having
clearly equal outcome is also likely so the best way to calculate the probability is to find out how many
of such pairs makes the total,of 5 or 3 and they are respectively (1 + 4, 2 + 3, 3 + 2, 4 + 1) giving a
probability 4/8, (1 + 2, 2 + 1) giving a probability of 2/8. Now, if 5 stands for 'A' and 3 stands for 'B', pr
(A or B) =pr (A) + pr (B) = 4/8 + 2/8 =6/8, i.e, chance of 'A' or 'B' is 6/8.
'
Multiplicative law: Here the events are independent and multiplication is in operation, i.e, pr (A and
B) =pr (A), pr (B), e.g. tossing of an unbiased coin and outcome is independent of each other. Referring
back to tossing of coin in previous section, we found the possibility is 1/2, therefore multiplication
probability in each toss is 1/2 and the possibility of head in 'n' toss is (t/2)1t. We can express as follows:

[[p.34]]
458
Blood Banking and Transfusion
Pr A (head) and pr B (head) = pr (A) pr (B)
= Y2XY2
= 14
= 0.25
This means the chance of head in both toss is 1/4th.
Conditionu! pro.bal>jlity
The chance that.a particular event happens is dependent on the outcome of another event. Let us consider
that chance of diseased patient surviving upto next year or less. This is dependent on the condition
(favourable or unfavourable) and hence the former is dependent on the later, the probability is, thus
conditional and is denoted as :
B / A : It indicates probability of event 'B' occurring when event 'A' has already occurred.
If A), A2, A3 are mutually disjoint events with probability Aith,
Pr (Ai) -:t 0 (i =1,2, ..... n) then for any arbitrary event 'B' which is subject of unAi such that P (B) >
0, we have:
P (Ai) P (BI Ai)
Pr (Ai / B):::
nLP (Ai) P(BI Ai)
i=l
Let us consider a population from red light area who comes forward for voluntary blood. donation,
you suspect them of having AIDS.
By application of Baye's theorem :
Al = Diseased population
= 15%
A2 = Healthy population
= 85%
BfA'l' = Blood is accepted when individual is diseased
= (-)z (false' negative)
BfA2 = Blood is accepted when the individual is healthy = 95%
The probability question that is to be taken into consideration "the blood, coming from diseased
individual, subjected to screening and passed thereafter.
To find out this, we are required to decide about the confidence limit by setting the upper cut off
value in such a way that accepts, say 95%of blood from healthy donor, using the table, the upper 5% in
standard one tailed distribution, the cut off value is 1.65. From this, we deduce the value of 'x' (refer back
to earlierdiscussion on detection ofhepatitis in SGPT), thereby achieve the specification limit for accepting
the 95% of healthy blood. Establish the mean for healthy blood. Establish the mean for healthy and
diseased individuals by serological determination of reactive donors. Find out SD for respective
observation.
We can compute' the 'Z' for both healthy and AIDS infected person (reactive units confirmed by
western blot) thereby determining (+) Z and (-) Z by consulting the standard one tailed table. The
amount of false positive (above upper specification limit) donor who are rejected. The false negative

[[p.35]]
Mathematical Consideration
459
donors thoughactually infected passesthe screenand are accepted, (-) Z are to be evaluated. Once this
is done, we can directly put them .into the operation for determining the question under consideration
and develop Baye's theorem.
Pr(AI and B)
Pr (AI) B =
Pr (B)
Pr(AI and B)
= Pr (AI and B) Pr (A2 and B)
Pc (B/AI)Pc(AI)
= Pr (BI AI') Pr (AI) + Pr (BI A2)Pr (A2)
Substituting by numbers, we can find out the percentage of screened but -diseased donors from
whom we are accepting the blood. Subjecting our finding to cost-benefit analysis, we can come to
conclusion, whether it is really worth to consider a voluntary donation programme in red light area and
if at all we consider then wha~
shoul,dbethe strategy that strikes balance betwe~n
cost and benefit.
Baye's theorem can thus be applied and developed to solve such type of problem.
Applied properly, the arithmetic and statistics can solve many a problems in blood banking and
transfusion medicine. Only few examples have been described in this chapter. But investigative studies
in blood banking and allied field requires one to applyarithmetic and statistics in arriving at an inference
that is quantitatively valid and scientifically explainable, the brief description given above deals with
fundamentals only and may not be sufficient for advanced scientific work. And readers who has an
aptitude for higher scientific research work which needs a better understanding ofthe subject for accuracy
and precision in quantitative analytical work may refer to larger textbooks of arithmetic and statistics.
Books recommended for further reading
1. A Handbook ofMedical Laboratory Technology; V.H. Talib, 1988, Sponsored by WHO.
2. Clinical Diagnosis and Management by Laboratory Methods, Todd, Sanford. Davidshon, edited by John
Bernard Henry, Seventh Asian Edition, AITBS.
3. Fundamental PhysicsofRadiology,Meredith, ~J.
andMassey, J.B., Third Asian Edition,1977, K.M. Varghese
Company.
4. Statistics in Medicine,.Theodore Colton, First Edition, 1974, Little Brown and Company.
5. Statistics for Medical Students, Robert Hoffman, 1963, First Edition, Charles C. Tho11UlS.
6. Textbook of Blood Banking Science, Zmijewski Chester; M. and Haesler Waltere, 1982 Edition, Appleton-
Century (Rofts).
.
.
7. Transfusion Medicine TechnicalManual, Saran, R.K. and Makroo, R.N., 1991 Edition, Published by DGHS,
Sponsored by WHO.
·S. Textbook of preventive and Social Medicine, Park, J.E. and Park, K., Tenth Edition, 1985, Banarsidas
Bhanot.