Biostatistics

Biostatistics
Dr. Khushbu Mishra

CONTENTS
• Introduction
• Definition
• Common statistical terms
• Sources and collection of Data
• Presentation of Data
• Analysis and interpretation
 Statistical averages
 Measures of Dispersion

Sampling and sampling methods
Sampling errors
Tests of significance
Correlation and regression
limitations

Introduction
• We, medical and dental students during period of our study,
learn best methods of diagnosis and therapy.
• After graduation, we go through research papers presented at
conferences and in current journals to know
new methods of therapy,
improvement in diagnosis and surgical techniques.
• It must be admitted that essence of papers contributed to
medical journals is largely statistical.

Training in statistics has been recognized as “indispensible” for
students of medical science.
for eg.
if we want to establish cause and effect relationship, we need
statistics.
if we want to measure state of health and also burden of
disease in community, we need statistics.

• statistics are widely used in epidemiology,
clinical trial of drug vaccine
program planning
community medicine
health management
health information system etc..
• The knowledge of medical statistics enables one to develop a
self- confidence & this will enable us to become a good
clinician, good medical research worker, knowledgable in
statistical thinking.

• Everything in medicine, be it research, diagnosis or treatment
depends on counting or measurment.
• According to Lord Kelvin,
when you can measure what you are speaking about and
express it in numbers, you know something about it but when
you can not measure, when you can not express it in numbers,
your knowledge is of meagre and unsatisfactory kind.

Bio-Statistics in Various areas
Health Statistics
Medical Statistics
Vital Statistics

• In Public Health or Community Health, it is called Health
Statistics.
• In Medicine, it is called Medical Statistics. In this we study the
defect, injury, disease, efficacy of drug, Serum and Line of
treatment, etc.,
• In population related study it is called Vital Statistics. e.g.
study of vital events like births, marriages and deaths.

• Application and uses of Biostatistics as a science..
in Physiology,
a.
to define what is normal/healthy in a population
b.
to find limits of normality
c.
to find difference between means and proportions of
normal at two places or in different periods.
d.
to find the correlation between two variables X and Y such
as in height or weight..
for eg. Weight increases or decreases proportionately with
height and if so by how much has to be found.

•
a.
b.
c.
•
a.
b.
c.

In Pharmacology,
To find action of drug
To compare action of two different drugs
To find relative potency of a new drug with respect to a
standard drug.
In Medicine,
To compare efficacy of particular drug, operation or line of
treatment.
To find association between two attributes eg. Oral cancer and
smoking
To identify signs and symptoms of disease/ syndrome.

Common statistical terms
• Variable:- A characteristic that takes on different values in
different persons, places/ things.
• Constant:- Quantities that donot vary such as π = 3.141
e = 2.718
these donot require statistical study.
In Biostatistics, mean, standard deviation, standard error,
correlation coefficient and proportion of a particular
population are considered constant.
• Observation:- An event and its measurment.
for eg.. BP and its measurment..

• Observational unit:- the “sources” that gives observation for
eg. Object, person etc.
in medical statistics:- terms like individuals, subjects etc are
used more often.
• Data :- A set of values recorded on one or more observational
units.
• Population:- It is an entire group of people or study elementspersons, things or measurments for which we
have an intrest at particular time.
• Sampling unit:- Each member of a population.
• Sample:- It may be defined as a part of a population.

• Parameter:- It is summary value or constant of a variable, that
describes the sample such as its mean,
standard deviation
standard error
correlation coefficient etc..
• Parametric tests:- It is one in which population constants
such as described above are used :- mean,
variances etc..
data tend to follow one assumed or established distribution
such as normal, binomial, poisson etc..
• Non- parametric tests:- Tests such as CHI- SQUARE test, in
which no constant of population is used.
Data donot follow any specific distribution and no assumptions
are made in non- parametric tests.eg ..good, better and best..

DEFINITION
American Heritage Dictionary® defines statistics as: "The
mathematics of the collection, organization, and interpretation of
numerical data, especially the analysis of population
characteristics by inference from sampling.”

The Merriam-Webster’s Collegiate Dictionary® definition is:
"A branch of mathematics dealing with the collection, analysis,
interpretation, and presentation of masses of numerical data."

A Simple but Concise definition by Croxton
and Cowden:
“Statistics is defined as the Collection, Presentation,
Analysis and Interpretation of numerical data.”

In the line of the definition of Croxton and Cowden, a
comprehensive definition of Statistics can be:

“Statistics defined as the science of
Collection,
Organisation,
presentation,
analysis and
interpretation of numerical data.”

• STATISTIC/ DATUM:- measured/ counted fact or piece of
information
such as height of person,
birth weight of baby…
• STATISTICS/ DATA:- plural of the same
such as height of 2 persons,
birth weight of 5 babies
plaque score of 3 persons…
• BIOSTATISTICS:- term used when tools of statistics are
applied to the data that is derived from biological sciences
such as medicine.

Types of Data

Qualitative Data

Nominal

Ordinal

Quantitative Data

Discrete

Continuous

Interval

Ratio

COLLECTION OF DATA

Data can be collected through
Primary sources:- here data is obtained by the investigator
himself. This is first hand information.
Secondary sources:- The data already recorded is utilized to
serve the purpose of the objective of study eg. records of OPD
of dental clinics.

• Main sources for collection of medical statistics:1. Experiments
2. Surveys
3. Records.
• Experiments and surveys are applied to generate data needed
for specific purposes.
• While Records provide ready- made data for routine and
continuous information.

Methods of collection of data
• Method of direct observation:- clinical signs and symptoms
and prognosis are collected by direct observation.
• Method of house to house visit:- vital statistics and morbidity
statistics are usually collected by visiting house to house.
• Method of mailed questionnaire:- this method is followed in
community where literacy status of people is very high.
Prepaid postage stamp is to be attached with questionnaire.

Presentation of data
• to sort and classify data into groups or classification.
• Objective :- to make data simple,
concise,
meaningful,
intresting,
helpful for further analysis.
• 2 main methods are
i. Tabulations
ii. Charts and diagrams

• Tabulation :• Devices for presenting data simply from masses of statistical
data.
• A table can be simple or complex, depending upon the number
or measurment of a single set or multiple set of items.
• 3 types:
a. Master table:- contains all the data obtained from a survey.
b. Simple table:- oneway table which supply answers to
questions about one characteristics only.
c. Frequency distribution table:- data is first split up into
convenient groups and the number of items which occur in
each group is shown in adjacent columns.

Table 1
states

population 1st march 2011

Andhra pradesh

8,46,65,533

Madhya pradesh

7,25,97,565

Uttar pradesh

19,95,81,477

Karnataka

7,14,83,435

Rajasthan

18,23,45,998

kerela

6,43,35,772

Frequency distribution table
• The following figures are the ages of patients admitted to a
hospital with poliomyelitis..
8, 24, 18, 5, 6, 12, 14, 3, 23, 9, 18, 16, 1, 2, 3, 5, 11, 13, 15, 9,
11, 11, 7, 106, 9, 5, 16, 20, 4, 3, 3, 3, 10, 3, 2, 1, 6, 9, 3, 7, 14,
8, 1, 4, 6, 4, 15, 22, 2, 1, 4, 6, 4, 15, 22, 2, 1, 4, 7, 1, 12, 3, 23,
4, 19, 6, 2, 2, 4, 14, 2, 2, 21, 3, 2, 1, 7, 19.
Age

Number of patients

0-4

35

5-9

18

10-14

11

15-19

8

20-24

6

Charts and diagrams
Quantitative
data
1. Histogram
2. Frequency polygon
3. Frequency curve
4. Line chart or graph
5.Cumulative frequency
diagram
6. Scatter diagram

Qualitative
data
1.Bar diagram
2. Pie or sector
diagram
3.Pictogram
4.Map diagram

Cumulative frequency diagram or
Ogive

Scatter or dot or correlation diagrams

Measures of central tendency/ statistical averages
• The word “average” implies a value in the distribution, around
which other values are distributed.
• It gives a mental picture of the central value.
• Commonly used methods to measure central tendency..
a. The Arithmetic Mean
b. Median
c. Mode.

• Mean = sum of all values
total no. of values
• Median = middle value (when the data are arranged
in order.
• Mode = most common value

• For eg..
the income of 7 people per day in rupees are as follows.
5, 5, 5, 7, 10, 20, 102= (total 154)
• Mean = 154/7 = 22
• Median= 7
• Median, therefore, is a better indicator of central tendency
when more of the lowest or the highest observations are wide
apart .
• Mode is rarely used as series can have no modes, 1 mode or
multiple modes.

Measures of Dispersion
•
a.
b.
c.

Widely known measures of dispersion are ..
The Range
The Mean or Average Deviation
The Standard Deviation.

a. Range : simplest
difference between highest and lowest figures
for eg.. Diastolic BP – 83, 75, 81, 79, 71, 90, 75, 95, 77, 94
so, the range is expressed as 71 to 95
or by actual difference of 24

• Merit :- simplest.
• Demerit :not of much practical importance.
indicates nothing about the dispersion of values
between two extreme values.
• Mean deviation:average of deviation from arithmetic mean.
M.D. = Ʃ(X – X )
ɳ

• Standard Deviation :- most frequently used
“ Root Mean Square Deviation”
denoted by greek letter σ or by initials
S.D. = Square root of Ʃ(X-X )2
ɳ

•
•
•

if sample size is less than 30 in denominator, (ɳ-1)
S.D. gives us idea of the spread of dispersion .
Larger the standard deviation, greater the dispersion of values
about the mean

Normal distribution
• large number of observations of any variable characteristics.
• A frequency distribution table is prepared with narrow class
intervals.
• Some observations are below the mean and some are above the
mean.
• If they are arranged in order, deviating towards the extremes
from the mean, on plus or minus side, maximum number of
frequencies will be seen in the middle around the mean and
fewer at extremes, decreasing smoothly on both the sides.
• Normally, almost half the observations lie above and half
below the mean and all observations are symmetrically
distributed on each of the mean.

• A distribution of this nature or shape is called normal or
gaussian distribution.

standardized normal curve
• Devised to estimate easily
the area under normal curve
between any two ordinates.
•Smooth
•Bell shaped
•Perfectly symmetrical curve
•Total area of curve is 1
mean=0
standard deviation= 1
Mean, Median and Mode all
coincide.
•Probability of occurrence of
any variable can be
calculated.

Estimation of probability (example)
• The pulse of a group of normal healthy males was 72, with a
standard deviation of 2. what is the probability that a male
chosen at random would be found to have a pulse of 80 or
higher?
• The relative deviate (z) = (x-x )
σ
= 80 – 72 = 4
2
The area of normal curve corresponding to a deviate 4=
0.49997, so, probability = 0.5- .49997 = 0.00003 i.e. 3 out of
1,00,000 individuals.

Areas of the standard normal curve with mean 0 and
standard deviation 1
Relative deviate (z)= (x-x)
σ
0.00

Proportion of area from middle
of the curve of designated
deviation.
.0000

0.50

.1915

1.00

.3413

1.50

.4332

2.00

.4772

4.00

. 4999998

Sampling
• When a large proportions of individuals or units have to be
studied, we take a sample.
• It is easier
• More economical
• Important to ensure that group of people or items included in
sample are representative of whole population to be studied.
• Sampling frame: once universe has been defined
a sampling frame must be prepared.
Listing of the members of the universe from which sample is
to be drawn.

• Accuracy
&
completeness

influences quality of sample drawn from it.

• Sampling methods
i. Simple random sampling
ii. Systematic random sampling
iii. Stratified random sampling

Sampling errors
• Repeated samples from same population
• Results obtained will differ from sample to sample.
• This type of variation from one sample to another is called
sampling error.
• Factors influencing sample error are:a. Size of sample
b. Natural variability of individual readings.
• As sample sample size increases, sampling error will
decrease.

Non – sampling errors
• Errors may occur due to
i. Inadequately caliberated instruments
ii. Observer‟s variation
iii. Incomplete coverage achieved in examining the subjects.
iv. Selected and conceptual errors

Standard error
• If we take random sample (ɳ) from the population,
and similar samples over and over again we will find that
every sample will have different mean.(X).
• Make frequency distribution of all sample means.
• Distribution of mean is nearly a normal distribution.
• Mean of sample means is practically same as population
means.
• The standard deviation of the means is a measure of sample
error and given by the formula
standard error = S.D(σ)/ √n

• Since distribution of means follows the pattern of a normal
distribution, it is not difficult to visualize that 95% of sample
means follows within limits of two standard error.

• Therefore, standard error is a measure which enables us to
judge whether mean of a given sample is within the set
confidence limits.

Tests of significance
• Standard error indicates how reliable an estimate of the mean
is likely to be.
• Standard error is applied with appropriate formulae to all
statistics, i.e, mean, standard deviation.etc..

i.
ii.
iii.
iv.

Standard error of Mean
Standard error of Proportion
Standard error of difference between means
Standard error of difference between proportions

Standard error of Mean
• we take only one sample from universe, calculate Mean and
standard deviation.
• But, how accurate is mean of our sample?
• What can be said about true mean of universe.
• In order to answer these questions,
we calculate standard error of Mean and set up confidence
limits
within which the mean(μ), of the population (of which we
have only one sample) is likely to lie.

let us suppose, we obtained a random
sample of 25 males, age 20-24 years
whose mean temperature was 98.14
deg.F with a standard deviation of
0.6. what can we say of the true mean
of the universe from which the
sample was drawn?

Confidence limits on the basis of
normal curve distribution- 95%
confidence limits= 98.14+ (2 0.12)
Range= 97.90 to 98.38degree F

25

0.6

0.6

√25
0.12

Standard error of proportion
• Standard error of proportion= √pq/n

standard error of difference between two Mean
S.E. (d)
= square root of
σ 21 + σ 2 2
Between the means
n1
n2
• The actual difference between the two means should be more than
twice the standard error of difference between two means.

• Parametric Statistical Tests:
EX: Z test
t test
F test

• Non Parametric Statistical Tests:
EX: Chi- square test
sign test

Types of problems
I

Comparison of sample mean with population mean

II Comparison of two sample means
III Comparison of sample proportion with the population
proportion
IV Comparison of two sample proportions

Steps
• Finding out the type of problem and the question to
be answered.
• Stating the Null Hypothesis (Ho)
• Calculating the standard error
• Calculating the critical ratio
difference between statistics / standard error
• Comparing the value observed in the experiment with
that at the predetermined significant level given by
the table
• Making inferences. P<0.05 significant reject the Ho
P =0.05 and P>0.05 accept the Ho

Z Test
Prerequisites to apply Z-test
• The sample or the samples must be randomly selected.
• The data must be quantitative.
• The variable is assumed to follow normal distribution in
the population.
• The sample size must be larger than 30
Two types:
• one tailed Z test
• Two tailed Z test

• The z- test has 2 applications:
i. To test the significance of difference between a sample mean
and a known value of population mean.
Z = Mean – Population mean
S.E. of sample mean
ii. To test the significance of difference between 2 sample
means or between experiment sample mean and a control
sample mean.
Z = Observed difference between 2 sample means
SE of difference between 2 sample means

t - Test
Criteria for applying t-test
• Random samples
• Quantitative data
• Variable normally distributed
• Sample size less than 30

•

Unpaired t-test: applied on unpaired data of
independent observations made on individuals of
two different or separate groups or samples drawn
from two populations

• Paired t-test: applied to paired data of independent
observations from one sample only

• It was designed by WS Gosseett whose pen name was
„student‟.
• The formula used is

t = observed difference between two means of small samples
SE of difference in the same

F-test (Analysis of variance test)
• Used for comparing more than two samples mean
drawn from corresponding normal populations.
Ex: to find out whether occupation plays any part in
causation of BP. systolic BP values of 4 occupations
are given. Determine if there is significant difference in
mean BP of 4 groups in order to assess the role of
occupation in causation of BP.
F = Mean square between samples / Mean square
within the samples

Chi-square Test
Application :
1. Proportion:
a) compare the values of two binomial samples
even if <30.Ex: Incidence of diabetes in 20 obese
and 20 non obese.
b) compare the frequencies of two multinomial
samples ex: no of diabetics and non diabetics in
groups weighing 40-50, 50-60 and >60 kg
2.Association: It measures the probability of
association between two discrete attributes. It has
an added advantage that it can be applied to find
association or relationship between two discrete
attributes when there are more than two classes
or groups.

Ex:- Trial of 2 whooping cough vaccines results of
the field trial were as below
Vaccine

Attacked Not
Total
attacked

Attack
rate

A
B

22
14

68
72

90
86

24.4%
16.2%

Total

36

140

176

-

• Null hypothesis ( Ho):- there was no difference
between the effect of two vaccines.
• Calculation of the expected number (E) in each group
of the sample or the cell of table
E=(column or vertical total x Row or horizontal total) /
sample total

Vaccine
A

B

Attacked
O=22
E=36x90 / 176
=18.4
O=14
E=17.54

Not Attacked
O=68
E=71.55
O=72
E=68.37

• Applying the χ² test.
χ²= ∑(O-E)² / E
= 0.72+0.17+0.71+0.19
= 1.79
• Finding the degree of freedom.
d.f. = (c-1) (r-1) = 1.

• Probability tables.
5% level = 3.84 P >0.05 Accept the Ho
• Inference:- The vaccine B is not superior to vaccine A

Restrictions in application of χ² test:
• Will not give reliable result with one degree of freedom if
the expected value in any cell is less than 5. Apply Yates
correction
χ² = ∑ ( | O – E | - ½ ) / E
• Yates correction cannot be applied in tables larger than 2x2
• Tells the presence or absence of association but does measure
strength of association.

• Statistical finding of relationship, does not indicate the cause
and effect.

Correlation and Regression
• To find whether there is significant association or not between
two variables, we calculate coefficient of correlation, which
is represented by symbol “r”.
• r = Ʃ (x - x ) (y - y )
√ Ʃ( x-x)2 Ʃ(y-y)2
• The correlation coefficient r tends to lie between – 1.0 and
+1.0.

Types of correlation :Perfect positive correlation:
• The correlation co-efficient(r) = +1 i.e. both variables rise or
fall in the same proportion.
Perfect negative correlation:
• The correlation co-efficient(r) = -1 i.e. variables are inversely
proportional to each other, when one rises, the other falls in the
same proportions.
Moderately positive correlation:
• Correlation co-efficient value lie between 0< r< 1
Moderately negative correlation:
• Correlation coefficient value lies between -1< r< 0
Absolutely no correlation:
• r = 0, indicating that no linear relationship exits between the 2
variables.

conclusion
• Statistics is central to most medical research .
• Basic principles of statistical methods or techniques equip
medical and dental students to the extent that they may be able
to appreciate the utility and usefulness of statistics in medical
and other biosciences.
• Certain essential bits of methods in biostatistics, must be learnt
to understand their application in diagnosis, prognosis,
prescription and management of diseases in individuals and
community.

References

• PARK‟S textbook of preventive and social medicine- 22nd
edition.
• Methods in Biostatistics- 7th edition by BK Mahajan.

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