1.
K i n g s u k S a r k a r , M D
A s s t . P r o f .
D e p t . o f C o m m u n i t y M e d i c i n e , D S M C H
FUNDAMENTALS OF
BIOSTATISTICS
2.
statistics:
- It refers to the subject of scientific activity
dealing with the theories and methods of
collection, compilation, analysis and
interpretation of data.
Bio-statistics:
- An art & science of
collection, compilation, analysis and
interpretation of data.
Data(sing. Datum):
- A set of observations, usually obtained by
3.
Classification of data-
Qualitative/Attribute
Quantitative/Variable: Continuous & Discreet
Qualitative Data:
- Can not be expressed in number
- Not measurable
- Can only be categorized under different
categories & frequencies
- E.g., Religion is an attribute; can be categorized
into Hindu, Muslim, Christian
- Human Blood Group: A,B,AB or O
- Sex: M/F
4.
Quantitative Data/variable:
- In statistical language, any
character, characteristic or quality that
varies is called variable
- It has got magnitude
Continuous variable:
- It is expressed in numbers & can be
measured
- Can take up infinite no. of values in a
certain range
- E.g., weight, height, blood sugar
5.
Discreet variable:
- Countable only
- Takes only some isolated values
- E.g., numbers of a family members, no. of
workers in a factory, no. of persons suffering
from a particular disease
According to source-
Primary Data
Secondary Data
6.
Primary Data:
- Collected directly from the field of enquiry
- original in nature
- E.g., measurement of BP, weight, height, blood
sugar
Secondary Data:
- Collected previously by some other
agency/organization
- Used afterwards by another
- E.g., hospital records, census data
7.
Nominal scales
Ordinal Scales
Interval Scales
Ratio
Nominal Scales:
- Used when data are classified by major
categories or subgroups of population
- Religion can be assigned to following categories-
Muslim, Hindu, Christian
- Outcome of treatment: cured or not cured; died
or survived
8.
Ordinal Scales:
- Assign rank order to categories placed in an
order
- E.g., students rank in a class; Grades A,B,C,D;
- Literacy status: illiterate, just
literate, primary, secondary, higher
secondary, graduate, post graduate
- Disease condition: mild, moderate, severe
Interval Scale:
- Distance between two measurement is
defined, not their ratio
- E.g., intelligence score in IQ tests, temperature in
Centigrade
9.
Ratio Scale:
- Both the distance & ratio between two
measurements are defined
- E.g., length, weight, incidence of disease, no. of
children in a family
Dichotomy/ Binary Scale:
- A scale with only two categories
- E.g., disease→ present/absent; sex→male /female
Population:
- An aggregate of objects, animate or inanimate,
under study
- A group of units defined according to aims &
objective of the study
Sample:
- a finite subset of or part of population
- Every member of population should have equal
chance to be included in sample
10.
Parameter:
- constant, describes the characteristics of
population
Statistic:
- Function of observation, which describes a
sample
Statistic Parameter
Mean x (x bar) µ(Mu)
Standard Deviation s s (sigma)
No. of Subject n N
Proportion P P
11.
• Main sources for collection of medical statistics are:
1. Experiments:
- Performed in the laboratories of
physiology, biochemistry, pharmacology,, clinical pathology
- Hospital words→ for investigations & fundamental research
- Used in preparation of thesis/dissertation, scientific paper for
publication in scientific journals & books
2. Surveys:
- Carried out for epidemiological studies in the field by trained teams
to find out incidence or prevalence of health or disease situations in
a community
- Used in OR→ assessment of existing condition, how to follow a
program, to study merits of different methods adopted to control of
a disease
- Provide trends in health status, morbidity, mortality, nutritional
status, health practices, environmental hazards
- Provide feedback needed to modify policy
- Provide timely earning of public health hazards
12.
3. Records:
- Maintained as a routine in registers or books
over a long period of time
- Used for keeping vital statistics: births, deaths,
marriage, hospitalization following illness,
- Used in demography & public health practices
- Collected data are qualitative
13.
DATA INFORMATION
Statistical data is presented usually in tabular
forms through different types of tables and in
pictorial forms; diagrams, charts
Method of presentation:
A. Tabulation
B. Drawing
14.
Tabular presentation:
- A form of presenting data from a mass of
statistical data
- at first frequency distribution table is prepared
- Table can be simple or complex
• Frequency distribution table or frequency table:
- All frequencies considered together form
“frequency distribution”
- No of person in each group is called the
frequency of that group
- Frequency distribution table of most biological
variables develop normal, binomial or Poisson
distribution.
15.
• Presentation of quantitative data is more cumbersome as
- Characteristic has a measured magnitude as well as
frequency
- Table x: presentation of quantitative data of
height in markingsHeight of groups in Cm Markings Frequency of each group
160-162 //// //// 10
162-164 //// //// //// 15
164-166 //// //// //// // 17
166-168 //// //// //// //// 19
168-170 //// //// //// //// 20
170-172 //// //// //// //// //// / 26
172-174 //// //// //// //// //// //// 29
174-176 //// //// //// //// //// //// 30
176-178 //// //// //// //// // 22
178-180 //// //// // 12
Total 200
16.
- Data needs consolidation by way of
tabulation to express some meaning
- Tabulation → a process of summarizing raw
data & displaying it in a compact form for
further analysis
- Orderly management of data in columns &
rows
17.
•General Principle in designing Table:
- Table should be numbered
- Brief & self-explanatory title should be there
mentioning time, place, person
- Headings of columns & rows should be clear &
concise
- Data to be presented according to size of
importance chronologically, alphabetically,
geographically
- Data must be presented meaningfully
- Table should not be too large
- Foot notes given, if necessary
- Total no of observations ; the denominator should
be written
- Information obtained should be summarized in
the table
18.
• Frequency distribution drawings:
- After classwise or groupwise tabulation, the
frequencies of a charecteristics can be
presented by two kinds of drawings
- Graphs & Diagrams
- May be shown by either lines, dots, figures
o Presentation of quantitative data is
through graphs
o Presentation of
qualitative, discreet, counted data is
through diagrams
19.
1. Histogram
- Graphical presentation of frequency distribution
- Variable characters of different groups are
indicated in the horizontal line (x-axis) is called
abscissa
- No. of observations marked on the vertical line
(y-axis) is called ordinate
- Frequency of each group forms a triangle
20.
2. Frequency Polygon:
- An area diagram of frequency distribution
developed over a histogram
- Mid points of the class intervals at the height of
frequency are joined by straight lines
- It gives a polygon, figure with many angles
21.
3. Frequency Curve:
- If no. of observation are very large & group
interval reduced
- Frequency polygon tends to loose its
angulation
- Gives rise to a smooth curve → frequency
curve
22.
4. Line Chart or Graph:
- A frequency polygon presenting variation by lin
- Shows trend of event occurring over a period of
time
- Shows rise, fall or periodic fluctuations vertical axis
may not start from zero, but some point above
frequency
23.
5. Cumulative Frequency Diagram or “Ogive”
- Graph of the cumulative frequency distribution
- An ordinary frequency distribution table→ relative
frequency table
- Cumulative frequency: total no. of persons in
each particular range from lowest value of the
characteristic up to & including any higher group
value
24.
6. Scatter or Dot Diagram:
- Prepared after tabulation in which frequencies of
at least two variables have been cross classified
- Shows nature of correlation between two
variable character in same person(s)( e.g., height
& weight)
- Also called correlation diagram
25.
1. Bar Diagram:
- Graphically present frequencies of different categories
of qualitative data
- Vertical/ horizontal
- May be descending/ascending order
- Widths should be equal
- Spacing between bars should also be equal
i. Simple Bar Diagram:
- Each bar represents frequency of a single category with a
distinct gap from one another
26.
ii. Multiple bar diagram:-
- Used to show comparison of two or more sets of related
statistical data
iii. Component/ proportional bar diagram:
- Used to compare sizes of different component parts
among themselves
- Also shows relation between each part & the whole
27.
2. Pie/ sector Diagram:
- A circle whose area is divided into different
segments by different straight lines from cenre to
circumference
- Each segment express proportional components
of the attributes
- Angle ( ) of a sector is calculated by
Class frequency X 3.6 or
(Class frequency/total frequency)X 360
28.
3. Pictogram/ Picture Diagram:
- A popular method to denote the
frequency of the occurrence of events to
common man such as attacks, deaths,
number operated, admitted, discharged,
accidents, etc. in a population.
29.
• 4. Map diagram/ spot Map:
- These diagrams are prepared to visualize
the geographic distribution of frequency of
characteristics
- One point denotes occurrence of one
more events
30.
• When a series of observations have been
tabulated in the form of frequency distribution
→→it is felt necessary to convert a series of
observation in a single value, that describes the
characteristics of that distribution,→ called
Measure Of Central Tendency
• All data or values are clustered round it
• These values enable comparisons to be made
between one series of observations and another
• Individual values may overlap, two distributions
have different central tendency
• E.g., average incubation period of measles is 10
days and that of chicken pox is 15 days.
31.
Measures of Central tendency
Mean Mode
Median
Arithmetic Geometric Harmonic
Mean(AM) Mean(GM) Mean(HM)
32.
• Arithmetic mean:
- Sum of all observations divided by number
of observations
- Mean(x)=Sx/n; x is a variable taking
different observational values & n= no. of
observations
- Exmp.
• ESR of 7 subjects are 8,7,9,10,7,7, & 6 mm for
1st hr. Calculate mean ESR.
- Mean(x)= (8+7+9+10+7+7+6)/7=54/7=7.7
mm
33.
• Median :
when observations are arranged in ascending or
descending order of magnitude, the middle most
value is known as Median.
• Problem:
- From same example of ESR, observations are
arranged first in ascending order: 6,7,7,7,8,9,10.
- Median= {7+1}/2=8/2=4th observation I,e., 7
- When n is Odd no., Median={n+1}2 th observation
- When n is Even no., Median={n/2th + (n/2+1)th}/2
th observation
• Problem: suppose, there are 8 observations of ESR
like 5,6,7,7,7,8,9,10
• Median={8/2th +(8/2+1)th}/2={4th+5th
obs}/2=(7+7)/2=7
34.
• Mode:
- The observation, which occurs most
frquently in series
• Problem: ESR of 7 subjects are 8,7,9,10,7,7,
& 6 mm for 1st hr. Calculate the Mode.
- Mode is 7.
37.
• Geometric mean:
- Used when data contain a few extremely large or
small values
- It’s the nth root product of n observastions
• GM=ⁿ√(x₁.x₂.x₃….xn)
• Harmonic Mean:
- Reciprocal of the arithmetic mean of reciprocals of
observations
arithmetic mean of reciprocals of observations=S(⅟x)
- HM=n/S⅟x
- got limited use
- A.M>GM>HM
38.
• Measures of central tendency do not provide
information about spread or scatter values
around them
• Measures of dispersion helps us to find how
individual observations are dispersed or scattered
around the mean of a large series of data
• Different measures of Dispersion are:
i. Range
ii. Mean deviation
iii. Standard deviation
iv. Variance
v. Coefficient of variation
39.
• Range:
- Difference between highest & lowest value
- Defines normal value of a biological
characteristic
• Problem: Systolic blood pressure (mm of Hg) of 10
medical students as follows: 140/70, 120/88,
160/90, 140/80, 110/70, 90/60, 124/64, 100/62,
110/70 & 154/90
• Range of Systolic BP of medical students = highest
value- lowest value=160-90=70mm of Hg
• Range of Diastolic BP= 90-60=30 mm of Hg
40.
• Mean deviation:
- Average deviations of observations from mean
value
- Mean Deviation(S) =(x-x)/n,
where x=observation,
x=Mean
42.
• To estimate variability in population from values of a
sample, degree of freedom is used in placed of no. of
observations
• Standard deviation is calculated by following stages:
- Calculate the mean
- Calculate the difference between each observation &
mean
- Square the difference
- Sum the squared values
- Divide the sum of squares by the no. of observations(n) to
get mean square deviation or variances(s)
- Find the square root of variance to get “Root-Mean-
Square-Deviation”
• Use: sample size calculation of any study
- Summarizes deviation of a large series of observation
around mean in a single value
43.
• Coefficient of Variation:
- Used to denote the comparability of variances
of two or more different sets of observations
- Coefficient of Variation=(Sd/Mean)X100
- Coefficient of Variation indicates relative
variability
44.
NORMAL DISTRIBUTION
• Most important useful distribution in theoretical statistics
• Quantitative data can be represented by a histogram &
by joining midpoints of each rectangle in the histogram
we can get a frequency polygon
• when no. of observations become very large & class
intervals get very much reduced→ frequency polygon
loses its angulation →gives rise to a smooth curve known
as frequency curve,
• Most biological variables , e.g., height, weight, blood
cholesterol etc, follows normal distribution can be
graphically represented by “normal curve”
45.
• If a large no. of observations of any variables
such as height, weight, blood pressure, pulse rate
etc. are taken at random to make a
representative sample of the world and if a
frequency distribution table is made, it will show
following characteristics:
- Exactly half the observations will lie above & half
below the mean and all observations are
symmetrically distributed on either side of mean
- Maximum no. of frequencies will be seen in the
middle around the mean and fewer at
extremities, decreasing smoothly on both sides
47.
• Normal Curve:
- Observations of a variable, which are normally
distributed in a population, when plotted as a
frequency curve will give rise to Normal Curve
• Characteristics of a Normal Curve:
- Smooth
- Bell shaped
- Bilaterally symmetrical
- Mean, Median, Mode coincide
- Distribution of observation under normal curve
follows the same pattern of normal distribution as
already mentioned
50.
SAMPLING TECHNIQUE
Universe/population:
- Aggregate of units of observation about which certain
information is required
- Population is a set of persons (or objects) having a
common observable characteristics
- E.g., while recording pulse rate of boys in a school, all
boys in the school constitute the population/universe
Sample:
- A portion or part of total population selected in some
manner
Sapling Frame:
- A complete, non-overlapping list of all the sampling units
(persons or objects) of the population from which the
sample is to be drawn
- E.g., telephone directory acts as a frame for conducting opinion
51.
• Statistic:
- A characteristic of a sample, whereas a
• parameter
- a character of a population
Types of sampling: non-probability &
probability/random sampling
• Non-probability sampling:
- Easier, less expensive o perform
- Sampling is done by choice & not by chance
- Information collected cannot be presumed to be
representative of the whole universe
- E.g, Quota Sampling, convenience sampling,
Purposive sampling, Snowball Sampling, Case
Study
52.
• Probability/Random Sampling:
- Sample are selected from universe by
proper sampling technique
- Each member of the universe has equal
opportunity to get selected
- Composition of sample from universe
occurs only by chance
Types:
oSimple Random Sampling:
53.
oStratified Random Sampling:
oSystemic Random Sampling:
oCluster Sampling:
oMultistage sampling:
oMultiphase Sampling:
54.
• Exercise no. 1
Following are the diastolic blood pressure values (in mmHg)
of 10 male adults.
80, 60, 70, 80,65, 74, 66, 80, 70, 55
Solution:
Mode= 80
Arranging in ascending order: 55,60,65,66,70,70,74,80,80,80
Median={10/2th+(10/2+1)th}/2={5th + 6th}/2={70+70}/2=70
Mean=700/10=70
55.
Exercise No. 5.
The following table shows the number of children
per family in a village
Calculate the measure of central tendency:
No of children per family No of families
0 30
1 40
2 70
3 30
4 20
5 10
56.
Solution:
Table 1.1 showing number of children in families
• Average (x)no. of children=400/200=2
No. of children in
a family(x)
No. of families(f) Total no. of
children(fx)
0 30 0x30=0
1 40 1x40=40
2 70 2x70=140
3 30 3x30=90
4 20 4x20=80
5 10 5x10=50
Total 200 400
57.
Exercise no. 8
Marks obtained by 50 students in community medicine in
final MBBS Part-I Exam as follows:
Calculate central tendency.
Marks No. of students
41-50 5
51-60 18
61-70 15
71-80 7
81-90 5
58.
• Solution:
Average marks obtained by students=3165/50=63.3
Marks
obtained
No. of
students(f)
Mid value
of marks
group(x) of
students
Total marks
obtained
by each
group(fx)
41-50 5 45.5 227.5
51-60 18 55.5 999
61-70 15 65.5 982.5
71-80 7 75.5 528.5
81-90 5 85.5 427.5
Total 50 3165
59.
Calculation of Median:
N/2=3165/2=1582.5
Median class=60.5-70.5
Median=L+{(N/2 –cf) xh}/f
• where:
• L = lower boundary of the median class
h= class width
N = total frequency
cf = cumulative frequency of the class previous to the median
class
f = frequency in the median class
Class boundary frequency Cumulative frequency
40.5-50.5 227.5 227.5 <N/2
50.5-60.5 999 Cf=1226.5 <N/2
60.5-70.5 f=982.5 2209 >N/2
70.5-80.5 528.5 2737.5
80.5-90.5 427.5 3165
Total 3165
60.
• Median= 60.5+ (1582.5 - 1226.5)x10/982.5
= 60.5 + 3560/982.5
= 60.5 + 3.62
= 64.12
*Modal class: the class having maximum frequency
Class boundary frequency
40.5-50.5 f1=227.5
50.5-60.5 fm=999 Modal Class
60.5-70.5 f2=982.5
70.5-80.5 528.5
80.5-90.5 427.5
Total 3165
61.
• Mode=L + (fm –f1)/(2fm- f1 – f2)x h
Where, L= lower boundary of modal class
fm =Frequency of modal class
f1= frequency of pre-modal class
f2= Frequency of post-modal class
h= width of modal class
Median= 60.5 +(999 –227.5 )/(2x 999- 227.5- 982.5 )x10
=60.5 -771.5/(1998-1210)x10
=60.5 – 771.5/788x10
=60.5 – 9.79
=50.71
62.
• Exercise no. 11
Calculate measures of dispersion from following data:
15,17,19,25,30,35,48
Solution:
Range=48- 15= 33
Mean deviation= Σ(x- x)/n
Observation(x) Mean(x) (x-x)
15 X=Σx/n=189/7=27 -12
17 -10
19 -8
25 -2
30 3
35 8
48 11
Σx=189 Σ(x-x)=54, ignoring- or +
signs
64.
• Coefficient of variation=(SD/Mean)x 100
=√50.6/27 x 100
=
65.
• Exercise no. 20
In the following data A & B are given below:
Calculate mean deviation & standard deviation.
A-item B-frequency
10-20 4
20-30 8
30-40 8
40-50 16
50-60 12
60-70 6
70-80 4
66.
• Solution:
a=assumed mean
SD=√{(sumfd1)2 – (sum fd1)/N}2/√(N-1) x h
• x= sumfd1 x h + a
Data A -
Class
interval
Data B-
frequency
(f)
Mid value
(x)
d1=(x-a)/h
fd1
fd1
2
10-20 4 15 (15-35)/10=-
2
-8 64
20-30 8 25 -1 -8 64
30-40 8 a=35 0 0 0
40-50 16 45 1 16 256
50-60 12 55 2 24 576
60-70 6 65 3 18 324
total 54 Σfd1=74 Σfd1
2=1284
67.
• SD=√{1284- 74/54}/√(54-1) x 10
= √{1284- 1.37}/√53 x 10
= √( 1282.63/53) x 10
= √24.2 x 10
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