PG Student :- Dr. Vinay Dange Dr. Amol Askar PG Teacher :- Dr. S. V. Akarte

Dr. D. Nandanwar

STANDARD DEVIATION & VARIANCE

THINK OF THESE…

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• Crime rate • Unemployment

figures• 2010 BAR Passing

rate• Mortality rates• Net Reproduction

Rate• Proportion of voters

favoring a candidate• Enrolment trend• Drop-out rate

• Number of Accident per year

• Annual growth rate• Monthly income• Doctor population

ratio• Prevalence of

disease• Average life span • Registered vehicles

annually • Ratio of male

teachers to the female

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Statistics• Statistics is a branch of mathematics that deals with

the methods of collection, compilation, analysis, presentation, and interpretation of data.

Biostatistics

• Is defined as application of statistical methods to medical, biological and public health related problems.

APPLICATIONS OF BIOSTATISTICS• Assess community needs

• Understand socio-economic determinants of health

• Plan experiment in health research

• Analyse their results

• Study diagnosis and prognosis of the disease for taking effective action

• Scientifically test the efficacy of new medicines and methods of treatment.

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Descriptive Statistics

Inferential Statistics

GENERAL CATEGORIES OF STATISTICS

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It is concerned with the gathering, classification, and presentation of data and summarizing the values to describe the group characteristic.

DESCRIPTIVE STATISTICS

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It pertains to the methods dealing with making of inference, estimate or prediction about a large set of data (population) using the information gathered from a sample.

INFERENTIAL STATISTICS

Spot the difference! 05/02/2023

Choose a sample…

Study the sample…

Describe the sample.

Descriptive Statistics

Choose a sample…

Study the sample…

Describe the sample…

Use such estimates( CONCLUSIONS) to describe the population from where the sample was drawn.

Inferential Statistics

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•Population refers to entire group of people or study elements ---

animals, subjects, measurements, things of any form for which we have an interest at a particular time.

•Samples are elements of the population selected through a process. They have of the same characteristics with the population.

POPULATION AND SAMPLE

• PARAMETERThis is the value that describes population or universe

• STATISTIC It is a measure that derived from sample, such as sample

mean, sample standard deviation

This summary describes sample

POPULATION

SAMPLE

ILLUSTRATION 1

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DATA

•Qualitative Vs Quantitative data•Grouped Data Vs Ungrouped data•Primary data Vs Secondary Data •Discrete Vs Continuous data•Nominal Vs Ordinal

Data are any bits or collection of information, ideas, figures or concepts.

RAW DATA – THOSE DATA IN THEIR ORIGINAL FORM AND STRUCTURE

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When you ask 1st year residents about their age, date of birth, ethnic group, religion, birth order, occupation of his father, occupation of her mother, educational background of his parents, place of birth, ambition, favorite subject, most liked Grade school teacher and hobbies – any such information given by them will be

RAW DATA.

Grouped Data – those data placed in tabular

form characterized by category or class intervals with the corresponding frequency

Religion Groups FrequencyHindu 101Muslim 27Christian 20Sikh 15Total 163

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Primary Data – data are measured and gathered by the researcher

You submit a statistical data to your Professor regarding the educational profile of the teachers in your college which you yourself had gathered through interview.

education PercentageMBBS 10%MBBS & Diploma 25%MBBS & MD 45%MBBS, MD, FRCP, etc 20%

Total 100%

Table . Educational Profile of Teachers in medical college

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Comparison of continuous and discrete data

•Continuous data is more precise than discrete

•Continuous data is more informative than discrete

•Continuous data can remove estimation and rounding of measurements

•Continuous data is often more time consuming to obtain

•Discrete should also be converted to continuous data when possible as to obtain a higher level of information and detail

• For each orange tree, the number of oranges is measured.

TEST 1

–Quantitative

• For a particular day, the number of cars entering a college campus is measured.

• Time until a light bulb burns out (4 months)

–Quantitative

–Quantitative

• blue/green color, gold frame• smells old and musty• texture shows brush strokes of oil

paint• peaceful scene of the country• masterful brush strokes

• picture is 10" by 14" • with frame 14" by 18"• weighs 8.5 pounds• surface area of painting is

140 sq. in.• cost $300

Quantitative data

Qualitative data

TEST 2 – Oil Painting

• Students• Girls• Smart/Intelligent• Hard working

• 32 students • 10 A grades• 92% students Muslim by

religion• 15 students good in

mathematics

Qualitative data

Quantitative data

TEST 3 -- Class

TEST 4 – conversion of quantitative data to qualitative data

Haemoglobin level in Gm%

Hypo, Normo or hypertensiveBlood pressure in mm of Hg

Tall or ShortHeight in cm

Anaemic or Non anaemic

IQ scores Idiot, Genius, Normal

Qualitative dataQuantitative data

Classify each set of data as discrete or continuous.

• 1) The number of suitcases lost by an airline.

• 2) The height of corn plants.

• 3) The distance of your house to gym.

• 4) The time it takes for a car battery to die.

• 5) The production of tomatoes by weight.

TEST 5

TEST 6 -- conversion of discrete to continuous data

• Religion Qualitative , Nominal data

• Disability Ordinal data

• Main food corps

• military rank (General, colonel, major, etc.),

• Anxiety level

TEST 7

Ordinal data

Ordinal data

Nominal data

• IQ Interval data

• Stethoscope units sold Ratio data

Example: Two ways of asking about Smoking behavior. Which is better, A or B? & why?

a) Do you smoke? Yes No b) How many cigarettes did you smoke in the last 3

days (72 hours)?

(a) Is nominal, so the best we can get from this data are frequencies.

(b) is ratio, so we can compute: mean, median, mode, frequencies.

TEST 8

Method of determining class intervals

Size of Class Interval

Mean

Median

Mode

Measures of Central Tendency

Measures of dispersion / variabilityRange Interquartile rangeMean deviationStandard deviationCoefficient of variation

STANDARD DEVIATION is a special form of average deviation from the mean. is the positive square root of the arithmetic mean of

the squared deviations from the mean of the distribution.

is considered as the most reliable measure of variability.

is affected by the individual values or items in the distribution.

STANDARD DEVIATION

Standard Deviation shows the variation in data. If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large.

Standard Deviation is often denoted by the lowercase Greek letter sigma, .

The bell curve which represents a normal distribution of data shows what standard deviation represents.

One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.

STANDARD DEVIATION FORMULA

The standard deviation formula can be represented using Sigma Notation:

2( )xn

STANDARD DEVIATION 1) Find the mean of the data.

2) Subtract the mean from each observation.

3) Square each deviation of the mean.

4) Find the sum of the squares.

5) Divide the total by the number of items.

6) Take the square root of the value.

VARIANCE

Variance is the average squared deviation from the mean of a set of data. It is used to find the standard deviation.

VARIANCE FORMULA

2( )xn

The variance formula includes the Sigma Notation, , which represents the sum of all the items to the right of Sigma.

Mean is represented by and n is the number of items.

VARIANCE

1. Find the mean of the data.

5. Divide the total by the number of items.

4. Find the sum of the squares.

3. Square each deviation of the mean.

2. Subtract the mean from each value – the result is called the deviation from the mean.

353535353535

210Mean= 35

731149351527

210Mean= 35

x x-ẋ (x- )ẋ 2

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

∑(x- )ẋ 2 0

FIND THE STANDARD DEVIATIONx x-ẋ (x- )ẋ 2

73 38 1444

11 -24 576

49 14 196

35 0 0

15 -20 400

27 -8 64

∑(x- )ẋ 2 2680

FIND THE STANDARD DEVIATION

Class Limits(1)

F(2)

28-29 4

26-27 9

24-25 12

22-23 10

20-21 17

18-19 20

16-17 14

14-15 9

12-13 5

N= 100

GROUPED DATA

VARIANCE FOR GROUPED DATA

Variance for Ungrouped Data

_f( Mp-X)2

279.56

364.05

228.12

55.70

2.21

53.80

185.50

286.29

291.85

∑ f (Mp-X)2= 1,747.08

_(Mp-X)2

69.89

40.45

19.01

5.57

0.13

2.69

13.25

31.81

58.37

_Mp – X

8.36

6.36

4.36

2.36

0.36

-1.64

-3.64

-5.64

-7.64

_X

20.14

20.14

20.14

20.14

20.14

20.14

20.14

20.14

20.14

FMp(4)

114.0

238.5

294.0

225.0

348.5

370.0

231.0

130.5

62.5

∑fMp= 2,014.0

Midpoint(3)

28.5

26.5

24.5

22.5

20.5

18.5

16.5

14.5

12.5

Class Limits

(1)

F(2)

28-29 4

26-27 9

24-25 12

22-23 10

20-21 17

18-19 20

16-17 14

14-15 9

12-13 5

N= 100

FIND THE STANDARD DEVIATION

APPLICATIONS OF SD• A SD is universally accepted unit of dispersion of values from

mean value

• SD summarises the variation of large distribution and defines normal limits of variation.

• SD measures position or distance of observation from mean

• SD indicates whether the variation of difference of an individual from mean is by chance.

• SD is used to calculate standard error of mean and SE of difference between 2 means

• SD helps to find the size of sample

• SD is used to calulate relative deviate or Z score

• SD is used in calcualtion of coefficient of variation

Merits of SD It is rigidly defined It is based on all observations It is not much affected by sampling fluctuations.

Demerits of SD It is difficult to understand and calculate It can not be calculated for qualitative data It is unduly affected by extreme deviations

FIND THE VARIANCE AND STANDARD DEVIATION

The math test scores of five students are: 92,88,80,68 and 52.

1) Find the mean: (92+88+80+68+52)/5 = 76.

2) Find the deviation from the mean: 92-76=16 88-76=12 80-76=4 68-76= -8 52-76= -24

3) Square the deviation from the mean:

2( 8) 64

2(16) 2562(12) 1442(4) 16

2( 24) 576

The math test scores of five students are: 92,88,80,68 and 52.

The math test scores of five students are: 92,88,80,68 and 52.

4) Find the sum of the squares of the deviation from the mean:

256+144+16+64+576= 1056

5) Divide by the number of data items to find the variance:

1056/5 = 211.2

The math test scores of five students are: 92,88,80,68 and 52.

6) Find the square root of the variance:

211.2 14.53

Thus the standard deviation of the test scores is 14.53.

DISCRETE AND CONTINUOUS DATA• There are two types of Quantitative Data:• 1. Discrete (in whole numbers)• Exp: Number of Questions in Exam 5, 7, 14

• Number of cars,

• Number of students 3000

• 2. Continuous (in decimal points) Exp: Temperature of Yanbu on Sunday 26.5 degrees

• Your Height 5.3”

• Your Weight 120.5 lbs

• Shoe size 7.5

•THANK YOU