BIOSTATISTICS (MPT) 11 (1).pptx

E . J E E V A
L E C T U R E R I N S T A T I S T I C S
F A C U L T Y O F A L L I E D H E A L T H S C I E N C E .
BIOSTATISTICS

UNIT - 1
 1. Introduction to statistics:
 ( Definition, types and Application of Biostatistics in
physiotherapy).
 (Data – Types, Presentation, collection methods.
Computing in Biostatistics).
 2. Exploratory tools for univariate data:
 (Types of variables: quantitative and qualitative
variables)

 (Simple plots for continuous variable- dot plots, stem
and leaf plots, histograms, interpreting plots.)
 (Numerical summarises for continuous variables –
Mean, Mode, Standard deviations, Quartiles,
Percentiles interquartiles range).
 Frequency table:
 Various types of graphs,

3.PROBABILITIES AND DISTRIBUTION CURVE
 1. Introduction to probability and proportions:
 2. Normal distribution curve (properties, importance
etc)
 3.Discrete random variables (Binomial distribution)
 4. Continuous random variables ( Normal
distribution, 2 score, obtaining normal distribution
probabilities from tabular and statistical software's)

4. SAMPLING DISTRIBUTION OF ESTIMATES:
 Differentiate between sample and population.
 Parameters and estimates
 Sampling distribution of sample proportions
 Standard errors of differences
 Student’s t-distribution.

Sir Francis Galton
is considered as
the Father of
Biostatistics.
Sir Ronald Fisher is
considered as the
Father of Statistics.

Statistics(1)
 Statistics is a branch of mathematics that deals with data
collection, organization, analysis, and interpretation and
presentation of numerical data.
 In other words Statistics is concerned with scientific
methods for
 Collecting,
 Organizing,
 Summarizing,
 Presenting and analyzing data
 As well as deriving valid conclusions and making
reasonable decisions on the basis of this analysis.

Statistics(2)
 Is the art and science of data
 It deals with :
 Planning research
 Collecting data
 Describing data
 Presenting data
 Analyzing data
 Interpreting result
 Reaching decision

Statistics(3)
 Collecting Data
 Eg: Sample, survey, Observe, Simulate
 Characterizing data
 Eg: Organize,classify,count, summarize.
 Presenting data
 Eg:Tables, charts, Statements.
 Interpreting Results
 Eg: Inference, Conclude, Specify confidence.

Uses of statistics:
 Descriptive information for any population
 Prove association between variables
 Prove relation between risk and factors
 Compare new rates with old ones (comparing 2021 and 2022
data).
 Prioritization of problems
 Evaluate health programs and services
 Compare local results with foreign ones
 (for eg: Comparing Hospital patients)

Uses of statistics:
 Medical studies:
 No of new diseases grown in last 10 year. Increase in
no. of patients for a particular disease.
 Sports Studies:
 Used to compare run rates of to different teams. Used to
compare to different players.(Team 1 & Team 2)
 Education:
 Money spend on girls education in comparison to boys
education?
 Comparison for result for last 10 years.

Biostatistics definition
 Biostatistics can be defined as the application of the mathematical
tools used in statistics to the fields of biological sciences and
medicine.
 Biostatistics is a growing field with applications in many areas of
biology including “ Epidemiology, Medical sciences, Health
sciences, Educational research and Environmental sciences”.

Statistics vs Biostatistics:
 Statistics may be defined as
a science of collection,
presentation, analysis and
interpretation of numerical
data.
 Statistical methods applied
to health related problems.
 It is the branch of statistics
concerned with
mathematical facts and data
related to biological events
such as genetics, biology,
epidemiology, and many
other.
Statistics Biostatistics

Role of biostatistics
 Protocol development
 Study implementation
 Study monitoring
 Data analysis
 Report manuscript/ writing
 Interpretation.

Biostatistics in various areas:
 1.Health Statistics
 2. Medical Statistics
 3.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.

Statistics are widely used in epidemiology:
 Statistics are widely used in epidemiology
 1. Clinical trial of drug vaccine
 2. Program Planning
 3.Community medicine
 4.Health management
 5.Health information system etc..
 Everything in medicine, be it research, diagnosis or treatment
depends on counting or measurement.

Why should medical student learn biostatistics?
 We have to clarify the relationship between certain
factors and disease.
 Enumerate the occurrences of disease
 Explain the etiology of disease (which factors cause
disease)
 Predict number of disease occurrences
 Read understand and criticize the medical
literature.
 The planning, conduct and interpretation of much of
medical research are becoming increasingly reliant on
statistical methods.

Uses of Biostatistics in medical sector
 Documentation of medical history of disease
 Planning and conduct of clinical studies.
 Evaluating the merits of different procedures.
 In providing methods for definition of “normal” and
“abnormal”.
 To provide the magnitude of any health problems in the
community.
 To find out the basic factors underlying the ill health.
 To evaluate the health programs which was introduced in
the community (success/failure).
 To introduce and promote health legislation.

Data
 Data is any information in raw or organized form
using Alphabets, numbers or symbols that refers to
or represents preferences, ideas, objects, categories
etc.
 Data can be defined as a collection of facts or
information, set of groups and values.

Data
Qualitative
data
Quantitative
data
Discrete Continuous

Types of data:
 They are Two of data:
 Qualitative data (or) subjective data
 Quantitative data (or) Objective data.

Data Types:
Types of data
Qualitative data Quantitative data
Nominal Ordinal Discrete Continuous
Interval Ratio

Qualitative Data:
 Qualitative data:[Its Non-Numerical
Variables].
 A qualitative data, also called a categorical variable,
are variables that are not numerical. It describes data
that fits into categories.
 Variables take on values that are names or labels.
 Example:
 A) Eye colors:[ blue, green, brown, black]
 B)The color of a ball;[red, green, blue etc.]

Qualitative Data examples:
 For example:
 1. Eye colours :
 a) Black b ) Blue c) Green d) Brown.
 2. Stages of disease:
 a) Mild b) Moderate c) Severe.
 3. Are you regular exercise?
 a) Yes b)No

Quantitative Data:
 Quantitative data: [Its Numerical data].
 Variables that have are measured on a numeric or
quantitative scale.
 A quantitative data is naturally measured as a
number for which meaningful arithmetic operations
make sense.
 Example:
 Height, age, salary, Temperature, area, air pollution
index etc..

Quantitative data examples:
 For example:
 1. Age :
 a) Below 5 b ) 6-10 c) 11-15 d) above 16.
 2. Salary:
 a) below 10000 b) 10000-20000 c)21000-25000 d)
26000-30000 e) above 31000.
 3.How many books in their library?
 a) 100 b) 200 c)500 d)1000 e) above 1000.

Discrete Vs continuous data
 Discrete data (countable) is information that can only take
certain values. These values don’t have to be whole numbers
but they are fixed values – such as shoe size, number of teeth,
number of kids, etc.
 Discrete data includes discrete variables that are
finite, numeric, countable, and non-negative integers (5, 10,
15, and so on).
 continuous data
 Continuous data (measurable) is data that can take any value.
Height, weight, temperature and length are all examples of
continuous data.
 Continuous data changes over time and can have different
values at different time intervals like weight of a person.

Discrete numbers[counted],
 A set of data is said to be discrete if the values
belonging to the set are distinct and separate.
 Values are distinct and separate.
 Values are invariably whole numbers.
 For eg: Number of pages in a book.

Continuous numbers[Measured],
 A set of data is said to be continuous if the values
belonging to the set can take on any value within a
finite interval.
 For eg:
 Water temperature, Height,weight.

Scales of Measurements:
 1.Nominal
 2.Ordinal
 3.Interval
 4.Ratio

Nominal data:
 A nominal scale is a measurement scale, in which
numbers serve as “tags” or “labels” only, to identify
or classify an object.
 A nominal scale measurement normally deals only
with non-numeric (qualitative) variables or where
numbers have no values.
 Gender: 1-male, 2-Female
 Stages of disease: 1- Mild, 2-moderate, 3-Severe
 Hair colour: 1-brown, 2-black, 3-Gray, 4-Other.

Ordinal data:
 Ordinal scale is the 2nd level of measurement that reports the ranking and ordering
of the data without actually establishing the degree of variation.
 “Ordinal” indicates “order”. Ordinal data is quantitative data which have naturally
occurring orders and the difference between is unknown. It can be named, grouped
and also ranked.
 Ordinal data (Ranking scale):
 Characteristics can be put into ordered categories.
 Eg: Socio-economic status (Low/Medium/High).
 For example: 1.How satisfied are you with our products:
 1-Totally satisfied
 2-Satisfied
 3- Neutral
 4-Dissatisfied
 5-totally Dissatisfied

Interval data:
 The interval scale is defined as a quantitative
measurement scale where he difference between
two variables.
 Interval scale is the 3rd level of measure scale.
 Interval scale they don’t have a “True Zero”
 Interval Estimation in statistics, the evaluation of a
parameter.
 For eg: The difference between 60 and 50 degrees is
a measurable is 10 degrees.

Ratio:
 Ratio data is defined as a quantitative data, having
the same properties as interval data.
 It is absolute “Zero”
 For example: Weight machine, Calculator, Height
machine.

Primary Data Vs Secondary Data
Primary Data:
Primary data is the data that is collected for the first
time through personal experiences or evidence,
particularly for research.
It is also described as raw data or first-hand
information.
The mode of assembling the information is costly.
The data is mostly collected through observations,
physical testing, mailed questionnaires, surveys,
personal interviews, telephonic interviews, case
studies, and focus groups, etc.

Primary Data Vs Secondary Data
Secondary Data:
 Secondary data is a second-hand data that is already collected and
recorded by some researchers for their purpose, and not for the
current research problem.
 It is accessible in the form of data collected from different sources such
as government publications, censuses, internal records of the
organisation, books, journal articles, websites and reports, etc.
 This method of gathering data is affordable, readily available, and
saves cost and time.
 However, the one disadvantage is that the information assembled is for
some other purpose and may not meet the present research purpose or
may not be accurate.

DATA COLLECTION
Sources of data
Primary Sources Secondary
Sources
INTERVIEW
QUESTIONAIR
E
INVESTIGA
TION
PUBLISHED
UNPUBLISHED

PRIMARY DATA
 Primary data is a type of information that is obtained directly
from first-hand sources by means of surveys, observation or
experimentation.
 Direct personal interview: Data is personally collected by the
interviewer.
 Telephonic Interviews: Data is collected through an interview over
the telephone with the interviewer.
 Indirect Oral investigation: Data is collected from third parties
who have information about subject of enquiry.
 Information from correspondents: Data is collected from
agents appointed in the area of investigation.
 Mailed questionnaire: Data is collected through questionnaire
mailed to the informant.
 Questionnaire filled by enumerators: Data is collected by
trained enumerators who fill questionnaires.

How to collect primary data:
 1. Sampling: It is a process through which we
choose a smaller group to collect data that can be the
best representative of the population.
 2. survey: It can be done in face to face
mode(interviews) or indirect mode (Telephone,
internet etc.)
 3.Census: It is method in which data is collected
from every unit of population.

secondary data:
 Secondary data are those which have already been
collected by someone else and which have through some
statistical analysis.
 Sources of data:
 Publications of central, state, local government.
 Technical and trade journals
 Books, magazines, Newspaper.
 Reports & publications of industry, bank, stock exchange.
 Reports by research scholars, Universities, economist.
 Public records.

Sources of secondary data:
 1.Published source:
 Government publications, semi-government
publication etc..
 2. Unpublished source:
 Census of India, National sample survey
organization ( They collected by the organizations for
their own record).

Primary data Vs Secondary data
Primary data Secondary data
Original and New Re-used and old
Primary sources Secondary sources
Less economical More Economical
High on reliability Low on reliability
Interviews, surveys, fieldwork and
internet communications via email
Original research
Historical and legal documents Articles in newspaper
Collected real Collected from the past

BIOSTATISTICS (MPT) 11 (1).pptx

Methods of data presentation:
 1) Informal: Text and Semi tabular.
 2) Formal: Tables, graphs and measures of
variation.

Graphical presentation:
Graphic representation is another way of analyzing numerical data. A graph is
a sort of chart through which statistical data are represented in the form of
lines or curves.
 Charts and diagrams are useful methods of presenting simple statistical data.
 There are several methods of presenting data-tables, charts, diagrams, graphs,
pictures, and special curves.
 Bar chart
 Pie diagram
 Histogram
 Line diagram
 Scatter correlation plot
 Frequency polygon
 Pictogram.

Use the right type of graphic
 Charts and graphs:
 Bar chart: comparisons, categories of data
 Line graph: display trends over time
 Pie chart: show percentages or proportional share

Measures of frequency:
 Measure of Frequency [Count, Percent, Frequency]
 For eg: 1,1,2,1,2,1,3,4,1,5,1,4,5,3,3,5,6,3,2,1,2,1.
Count Frequency Percentage
1 8 36.36%
2 4 18.18%
3 4 18.18%
4 2 9.09%
5 3 13.63%
6 1 4.54%
Total 22 100%

Bar chart:
 Bar charts are mainly used for illustrating qualitative
(discrete)variables in a data series.
 Bar charts a way of presenting a set of numbers by the
length of a bar- is proportional to the magnitude to be
represented.
 Bar charts are a popular media of presenting statistical
data because they are easy to prepare, and enable values
to be compared.
 Simple bar chart
 Multiple bar chart
 Component bar chart

Bar chart types:
Simple bar charts sort data into simple
categories.
 Multiple (or compound) bar charts divide data
into groups within each category and show
comparisons between individual groups as
well as between categories.
 Component (or stacked) bar charts, which like
grouped bar charts, use grouped data within
categories.

Bar chart:
 Length of the bars, drawn vertical or horizontal,
indicates the frequency of a character.
 Bar chart or diagram is a popular and easy method
adopted for visual comparison of the magnitude of
different frequencies in discrete data.
 The frequency may be shown on Y-axis (vertical
bars) or on X-axis (horizontal bars).

Simple Bar diagram
Year 2010 2011 2012 2013
Profit 70 50 30 80

subject Test 1 Test 2
English 89 91
Hindi 95 92
Mathematic
s
97 95
Science 90 97
Social
science
89 89
Computer 89 92
For example(vertical multiple bar chart).
89
95
97
90
89 89
91
92
95
97
89
92
84
86
88
90
92
94
96
98
Test 1
Test 2

Multiple Bar diagram
Year I Division II Division III Division Fail
2014 50 150 250 150
2015 60 200 300 140
2016 50 250 350 150

Multiple and component bar chart:
Multiple Bar chart
Component Bar chart

Vertical bar chart:
19 24
18
16
22
11
0
5
10
15
20
25
mon tue wed thu fri sat
major surgeries
Day of
the week
No of
patients
19 24 18 23 25 11

Horizontal bar chart:
15
15
18
16
22
11
0 5 10 15 20 25
mon
tue
wed
thu
fri
sat
major surgeries
major surgeries

Year science Arts Computer Total
2015 240 560 220 1020
2016 280 610 280 1170

Multiple and component bar diagram
Day of
the week
Major
surgeries
15 15 18 16 22 11
Minor
surgeries
18 22 20 23 25 14
15 15
18
16
22
11
18
22
20
23
25
14
0
5
10
15
20
25
30
major
surgeries
minor
surgeries
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
15 15
18
16 22 11
18 22
20
23 25 14
minor surgeries
major surgeries

Line diagram example:
 The month-wise number of outpatients in two
general hospitals in 2007 is given in table below.
Draw a line chart to represent the data.
Mont
h
jan feb mar april may june july aug sep oct nov dec
Hos
A
9 10 8 9 11 15 17 19 14 11 9 8
Hos
B
7 9 6 7 9 13 15 16 12 9 7 6

9 10
8 9
11
15
17
19
14
11
9 8
7
9
6
7
9
13
15
16
12
9
7
6
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10 11 12
Line diagram
Hospital B
hospital A

Steps a pie diagram:
 Step 1: Total all the values in the data to be charted
and convert each value in the data to a percentage.
 Step 2: The pie chart represents 100% of the data
and there are 360 degrees in a circle. The data in
percentage is converted to degrees using the
following formula
 Number of degrees= Percentage /100*360o
 Step 3: Draw a pie chart manually , calculating the
number of degrees.

 Draw a pie chart from the following hypothetical
data. The number of beneficiaries of health program
for a given year in four districts is
 District A(38,400) District B (30,720) District
C(11,520), and District D(15,360).
 Using the formulas:
 Number of degrees= Percentage /100*360o

Pie chart example:
District Frequency Percentage
A 38400 38400/frequency
total*100= 40%
B 30720 32%
C 11520 12%
D 15360 16%
Total 96000 100%

Accounti
ng
percenta
ge
Rent 17%
Food 30%
Utilities 22%
Clothes 22%
Phone 9%
For example:
17%
30%
22%
22%
9%
Accounting
Rent
Food
Utilities
Clothes
Phone

Histogram
 A histogram is a graphical display of data using bars of different
heights. In a histogram, each bar groups numbers into ranges. Taller
bars show that more data falls in that range. A histogram displays the
shape and spread of continuous sample data

Box Plotting
 Box plots (also called box-and-whisker
plots or box-whisker plots) give a good
graphical image of the concentration of the data.
 They also show how far the extreme values are
from most of the data.
 A box plot is constructed from five values: the
minimum value, the first quartile, the median, the
third quartile, and the maximum value.

Box Plotting
The image above is a boxplot. A boxplot is a standardized way of displaying the
distribution of data based on a five number summary (“minimum”, first quartile
(Q1), median, third quartile (Q3), and “maximum”). It can tell you about your
outliers and what their values are. It can also tell you if your data is symmetrical,
how tightly your data is grouped, and if and how your data is skewed.

Statistical concepts of classification of
Data
 Classification is the process of arranging data into
homogeneous (similar) groups according to their common
characteristics.
 Raw data cannot be easily understood, and it is not fit for
further analysis and interpretation. Arrangement of data
helps users in comparison and analysis. It is also important
for statistical sampling.

Histogram:
 Histograms are the most commonly used to graphically
represent quantitative data (that are measured on an
interval scale) as grouped frequency distribution of both
continuous and discontinuous types.
 It is a pictorial diagram of frequency distribution. It
consists of a series of blocks. The class intervals are
given along the horizontal axis and the frequencies along
the vertical axis.
 The area of each block or rectangle is proportional to the
frequency in the histogram of the frequency distribution.

Histogram eg:
 Draw a histogram to graphically represent the
frequency distribution .
Age
groups
20-30 30-40 40-50 50-60 60-70 70-80 80-90
No of
patients
28 41 82 98 76 33 12

Cumulative frequency diagram or Ogives:
 Ogive is a graph of the cumulative frequency distribution.
 To draw this, an ordinary frequency distribution table in
a quantitative data has to be converted into a relative
cumulative frequency table.
 Cumulative frequency is the total number of persons in
each particular range from lowest value of the
characteristic up to and including any higher group
value.
 The cumulative frequencies are plotted corresponding to
the group limits of the characteristic. On joining the
points by a smooth free hand curve, the diagram made is
called ogive.

Cumulative frequency table:
 In a frequency table you keep count of the number of
times a data item occurs by keeping a Tally. The
number of times the item occurs is called the
frequency of that item.
 In a frequency table you can also find a “Running
total of frequencies” This is called cumulative
frequencies

Ogive curve:
 The Ogive is a Cumulative frequency curve.
 It is a free hand graph showing the curve of a
cumulative frequency.
 The ogive is constructed by plotting the upper class
limit on the X axis and the corresponding cumulative
frequency on the Y axis.
 Ogive is best used when the total frequency at any
given time is to be displayed.

Classification of Data
There are four types of classification. They are:
 Geographical classification
When data are classified on the basis of location or areas, it is called
geographical classification
 Chronological classification
Chronological classification means classification on the basis of time, like
months, years etc.
 Qualitative classification
In Qualitative classification, data are classified on the basis of some attributes
or quality such as gender, colour of hair, literacy and religion. In this type of
classification, the attribute under study cannot be measured. It can only be
found out whether it is present or absent in the units of study.
 Quantitative classification
Quantitative classification refers to the classification of data according to some
characteristics, which can be measured such as height, weight, income, profits
etc.

Quantitative classification
There are two types of quantitative classification of data:
Discrete frequency distribution and Continuous frequency
distribution.
In this type of classification there are two elements
 variable
Variable refers to the characteristic that varies in magnitude
or quantity. E.g. weight of the students. A variable may be
discrete or continuous.
 Frequency
Frequency refers to the number of times each variable gets
repeated. For example there are 50 students having weight
of 60 kgs. Here 50 students is the frequency.

Types of statistics:
 There are basically two types of statistics
 Descriptive statistics
 Inferential Statistics

Descriptive statistics:
 Measures of central tendency are statistics that
summarize a distribution of scores by reporting the
most typical or representative value of the
distribution.
 Measures of dispersion are statistics that indicate the
amount of variety or heterogeneity in a distribution
of scores.

Descriptive statistics:
 Descriptive statistics are used to describe the basic
features of data in a study.
 Measures of Frequency [Count, Percent, Frequency]
 Measures of Central Tendency [Mean, Median,
Mode]
 Measures of Dispersion[ Range, Variance, Standard
Deviation]
 Measures of Variation [percentiles, Deciles,
Quartiles]

Inferential Statistics:
 Inferential statistics makes inferences and
predictions about a population based on a sample of
data taken from the population.
 The sample is a set of data taken from the population
to represent the population.
 Hypothesis testing,
 Probability distribution,
 correlation testing, and Regression Analysis under
the category of inferential statistics.

 Frequency distribution is a table that displays the
frequency of various outcomes in a sample.
 Each entry in the table contains the frequency or
count of the occurrences of values within a particular
group or interval, and in this way, the table
summarize the distribution of values in the sample.
Frequency distribution:

Measures of frequency:
 Measures of Frequency [Count, Percent, Frequency]
 For eg: n = 1,1,2,1,2,1,3,4,1,5,1,4,5,3,3,5,6,3,2,1,2,1.
Count Frequency Percentage
1 8 36.36% 8/22*100=
2 4 18.18% 4/22*100
3 4 18.18% 4/22*100
4 2 9.09% 2/22*100
5 3 13.63% 3/22*100
6 1 4.54% 1/22*100
Total 22 100%

Frequency distribution:
 Marks 0btained by 20 students in a subject are
 15,18,25,26,18,32,15,25,25,22,25,25,2,32,22,10,24,18
,20,26. n=20.
 Present the data in the form of a frequency
distribution table.
Class interval Tally marks Frequency
1-10 II 2
11-20 IIII 6
21-30 IIII IIII 10
31 above II 2

Class (Marks) Tally Frequency Cumulative
Frequency
11-15 II 2 2
16-20 III 3 2+3=5
21-25 III 3 5+3=8
26-30 IIII 5 8+5=13
31-35 IIII I 6 13+6=19
36-40 IIII I 6 19+6=25
41-45 III 3 25+3=28
46-50 II 2 28+2=30
Total 30

Measures of central tendency:
 Measures of Central Tendency[Mean, Median, Mode].
 Mean:
 The Arithmetic mean is widely used in statistical
calculation. It is sometimes simply called Mean.
 To obtain the mean, the individual observations are first
added together, and then divided by the number of
observations.
 The operation of adding together is called “Summation”
and is denoted by the sign Σ or S.
 The mean is denoted by the sign X (called “X bar”).

Average or Mean (merits)
 Merits:
 Easy to understand and compute
 Based on the value of every item in the series.
 Limitations:
 Affected by extreme values.
 Not useful for the study of qualities like
intelligence, honesty and character.

 For example:
 N =8,2,3,4,5,6,7,1. find the mean?
 Mean= Sum of all the number of observations/ total no
of data.
 Step 2===Mean (x)= 8+2+3+4+5+6+7+1/8
 Step3 = 36/8
 Mean= X=4.5.

 1) The weight (in kg) of ten persons is as follows:
 62,52,71,56,76,53,62,67,58 and 73. n=10
 Calculate the arithmetic mean.
 2) The calculation of arithmetic mean is shown in
table.
 X: 5,7,9,11,13,6,12,17,10,5
Find the Mean:

Median:
 The median is an average of a different kind, which does
not depend upon the total and number of items. To
obtain the median, the data is first arranged in an
ascending to descending order of magnitude, and
then value of the middle observation for odd
numbers is located, which is called the median.
 If there are 10 values instead of 9, the median is worked
out by taking the average of the two middle values.
That is, If the number of items or values is even,
the practice is to take the average of the two middle
values.

 Arrange the observations in the series in ascending to
descending order .The central observations of the
arranged series gives the median, (n) is the number of
observations.
 For odd numbers: Middle value of the data. (1,3,5 etc)
 For eg: 1,2,5,8,7. n=5
 1,2,5,7,8
 Median=5
 For even numbers: Sum of two middle values/2.
(2,4,6,8,10 etc)
 For eg: x=1,2,5,8,7,1. n=6
 1,1,2,5,7,8 = 2+5/2 =7/2 median=3.5

Mode:
 The mode is the commonly occurring value in a
distribution of data. It is the most frequent item or
the “most frequently” value in a series of
observations.
 They are three types of mode
 Unimodal
 Bimodal
 Multimodal (or) Trimodal.

 For eg: X=1,2,3,4,1,6,7,8,1,2,2,6,6.
 The value”1” which occurs for the maximum number of
times, is the modal value.
 Mode=1,2,6(trimodal).
 For eg:
 1) x= 2,5,9,3,5,4,7. find the mode? Mode=5(unimodal)
 2) x= 2,5,2,3,5,4,7. find the mode? Mode=2,5 (bimodal)
 X= 1,5,6,8,9,4,3 Find the mode? Mode= No mode.
 Find the mode.
Mode eg:

Measures of dispersion:
 Measure of Dispersion [ Range, Variance, Standard
Deviation]
 Range:
 The range is by far the simplest measures of
dispersion. It is defined as the difference between the
highest and lowest in a given sample or data.
 Mean deviation:
 It is the average of the deviations from the arithmetic
mean.

 In statistics, the range of a set of data is the
difference between the largest and smallest values.
 The range is the difference between the highest and
lowest values within a set of numbers.
 Range= Largest value-smallest value.(L-S)
Range:

 1) x= 2,5,9,3,5,4,7. find the range?
 Range=Highest value – Lowest value
 Range= 9-2
 = 7.
 The range value is “7”.
Range:

Standard Deviation:
 The standard deviation is the most frequently used
measures of dispersion. In simple terms, it is defined
as “Root-Means-Square-Deviation.”
 It is denoted by the Greek letter sigma σ or by this
initials S.D.
 The standard deviation is a statistic that measures
the dispersion of a dataset relative to its mean and is
calculated as the square root of the variance.
 If the data points are further from the mean, there is
a higher deviation within the data set.

 The standard deviation of a population is defined by
the following formula;
 σ= sqrt [Σ(x-x)2/N-1].
 Where σ is the population standard deviation,
 x is the population mean,
 xi is the ith element from the population,
 and N-1 is the number of elements in the
population.

 Variance (σ2) is a measurement of the spread
between numbers in a data set.
 It measures how far each number in the set is from
the mean and is calculated by taking the differences
between each number in the set and the mean,
squaring the differences and dividing the sum of the
squares by the number of values in the set.
Variance:

 The formula for variance is,
 Variance σ2= Σ(x-x )2/n-1
 Where
 Xi= the ith data point
 X = the mean of all data points
 n= the number of data points.
Mean deviation

Coefficient of variation
 Formula for Coefficient of Variation
 Mathematically, the standard formula for the
variation is expressed in the following way:
 Coefficient of variation=S.D/µ *100.
 Where
 S.D is a standard deviation
 µ is a population mean.

Measures of Variation:
(Quartiles, deciles, Percentiles)
 Quartiles divide a rank-ordered data set into four
equal parts. The values that divide each part are called
the first, second, and third quartiles; and they are
denoted by Q1,Q2 and Q3.
 Q1 is known as first or lower quartile, covering 25%
items.
 The second quartile or Q2 is the same as Median
of the series.(50%)
 Q3 is called third or upper quartile, covering 75%
items.
 Quartiles are great for reporting on a set of data and for
making box and whisker plots.

Deciles:
 Deciles divides a series into 10 equal parts,
 For any series, there are 9 deciles denoted by
D1,D2………D9.
 D1= size of n+1/10th item
 D9= Size of 9(n+1)/10th item.
 A deciles is used to categorize large data sets from
highest to lowest values.

Percentiles:
 Percentiles divide a series into 100 equal parts. For
any series, there are 99 percentiles denoted by
p1,p2,p3………p99.
 P1=size of n+1/100th item
 P99=size of 99(n+1)/100th item.
 Percentiles are used to understand and interpret data.
 In everyday life, percentiles are used to understand
values such as test scores, health indicators, and other
measurements.

Deciles:
 Deciles divides a series into ten equal parts, For any
series, there are nine deciles denoted by D1,D2,….D9.
There are called as 1st deciles , second decile so on.
 The values which divide an array into ten equal parts
are called deciles. The first , second,…ninth deciles
by respectively. The fifth deciles (corresponds
median. The second , fourth, sixth and eighth deciles
which collectively divide the data five equal parts are
called deciles.
 D1=size of n+1/10th item
 D9=size of 9(n+1)/10th item.

E . J E E V A
S T A T I S T I C I A N
A . C . S M E D I C A L C O L L E G E .
Correlation coefficient

History of correlation
 Karl pearson (1857-1936) a British statistician,
developed the method of expressing the
relationship between two variables.
 Correlation is used in describing the strength of
the relationship between two variables.
 To find the linear relationship between two variables.
 Eg: X and Y variables.

Correlation definition
 The relationship or association between two
quantitatively measured or continuous variables is
called correlation.
 The extent or degree of relationship between two
sets of figures is measured in terms of another
parameter called correlation coefficient.
 It is denoted by letter “r”.

 When two variables characters in the same series or
individuals are measureable in quantitative units
such as
 For example:
 1.Height and weight
 2.Temperature and pulse rate
 3. Age and vital capacity
 4. Smokers and Non smokers etc..

Types of correlation
 1. Positive correlation(r=+1)
 2.Negative correlation(r=-1)
 3. No correlation(r=0)

Positive correlation
 A positive correlation is a relationship between
two variables where if one variable increases, the
other one also increases.
 A positive correlation also exists in one decreases
and the other also decreases.
 The “r” values is (Positively numbers).

Negative correlation
 Negative correlation is a relationship between
two variables in which one variable increases as the
other decreases, and vice versa.
 In statistics, a perfect negative correlation is
represented by the value -1, a 0 indicates
no correlation, and a +1 indicates a perfect
positive correlation

No correlation
 If there is no correlation between x and y, that just
means that there's no relationship, connection, or
interdependence between the two variables. You
could think of it as meaning that x and y have
nothing to do with each other.
 The “r” values equal to “Zero”

Correlation types
 Rank correlation method:
 Spearman’s rank correlation:
 Karl Pearson's method: Karl Pearson's coefficient of
correlation can be calculated for simple data,
ungrouped frequency(or discrete ) data, and grouped
frequency (or continuous)data.
 Karl Pearson's coefficient of correlation “r” can be
calculated using a variety of formulae.

Correlation
 When to use it?
 When you want to know about the association or relationship
between two continuous variables
 Ex) food intake and weight; drug dosage and blood pressure; air temperature
and metabolic rate, etc.
 What does it tell you?
 If a linear relationship exists between two variables, and how strong that
relationship is
 What do the results look like?
 The correlation coefficient = Pearson’s r
 Ranges from -1 to +1
 See next slide for examples of correlation results

Correlation
 How do you interpret it?
 If r is positive, high values of one variable are associated with high values of the
other variable (both go in SAME direction - ↑↑ OR ↓↓)
 Ex) Diastolic blood pressure tends to rise with age, thus the two variables are
positively correlated
 If r is negative, low values of one variable are associated with high values of the
other variable (opposite direction - ↑↓ OR ↓ ↑)
 Ex) Heart rate tends to be lower in persons who exercise frequently, the two
variables correlate negatively
 Correlation of 0 indicates NO linear relationship
 How do you report it?
 “Diastolic blood pressure was positively correlated with age (r = .75, p < . 05).”
Tip: Correlation does NOT equal causation!!! Just because two variables are highly correlated, this does
NOT mean that one CAUSES the other!!!

Correlation
Guide for interpreting
strength of correlations:
 0 – 0.25 = Little or no
relationship
 0.25 – 0.50 = Fair degree
of relationship
 0.50 - 0.75 = Moderate
degree of relationship
 0.75 – 1.0 = Strong
relationship
 1.0 = perfect correlation

For example
Si.no Temperature
(X)
Pulse rate
(Y)
1 12 21
2 15 22
3 32 21
4 14 14
5 22 18

Concept of Skewness
A distribution is said to be skewed-when the mean,
median and mode fall at different position in the
distribution and the balance (or center of gravity) is
shifted to one side or the other i.e. to the left or to the
right.
Therefore, the concept of skewness helps us to
understand the relationship between three measures-
• Mean.
• Median.
• Mode.

Symmetrical Distribution
 A frequency distribution is said to be symmetrical if the
frequencies are equally distributed on both the sides of
central value.
 A symmetrical distribution may be either bell – shaped or
U shaped.
 In symmetrical distribution, the values of mean, median
and mode are equal i.e. Mean=Median=Mode

Skewed Distribution
 A frequency distribution is said to be skewed if the
frequencies are not equally distributed on both the sides
of the central value.
 A skewed distribution maybe-
• Positively Skewed
• Negatively Skewed

Graphical Measures of Skewness
 Measures of skewness help us to know to what degree and in which
direction (positive or negative) the frequency distribution has a departure
from symmetry.
 Positive or negative skewness can be detected graphically (as below)
depending on whether the right tail or the left tail is longer but, we don’t
get idea of the magnitude
 Hence some statistical measures are required to find the magnitude of
lack of symmetry Mean=Media
n=Mode
Mean<Media
n<Mode
Mean> Median>
Mode
Symmetrical Skewed to the
Left
Skewed to the
Right

Kurtosis
Kurtosis is another measure of the shape of a frequency
curve. It is a Greek word, which means bulginess.
While skewness signifies the extent of asymmetry, kurtosis
measures the degree of peakedness of a frequency distribution.
Karl Pearson classified curves into three types on the basis of
the shape of their peaks. These are:-
Leptokurtic
Mesokurtic
Platykurtic

Kurtosis
• When the peak of a curve
becomes relatively high then
that curve is called
Leptokurtic.
• When the curve is flat-topped,
then it is called Platykurtic.
• Since normal curve is neither
very peaked nor very flat
topped, so it is taken as a
basis for comparison.
• This normal curve is called
Mesokurtic.

BIOSTATISTICS (MPT) 11 (1).pptx

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