IGNOU STATISTICS2

7/31/2019 IGNOU STATISTICS2

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Satheesh

Lecturer in Education

M.C.T. Training CollegeMalappuram

www.sathitech.blogspot.comwww.mctinfotech.blog.com

[email protected] : 09562253564


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Statistics - Definition

statistics may be defined

as the collection,

presentation, analysis

and interpretation of

numerical data

- Croxten & Cowden


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Statistics

The term Statisticsseems

to have derived from the

Latin word statusor

Italian word statistaor

the German word

statistik. Each of which

means Political state


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Why Statistics in Education ?

Data

Collection

Presentation

(Tabulation)

Analysis

Interpretation


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NATURE OF DATA

Continuous discrete

HeightWeighttemperature

Family sizeEnrolmentof children


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SCORING & TABULATION OF SCORES

Frequency DistributionFrequency distribution is

an important method of condensing

and presenting data.

This representation is alsocalled Frequency Table

Continuous (grouped)frequency distribution

Discretefrequency distribution


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Discrete frequency distribution

It is a frequency distribution

in which we make an array

by listing all the values

occurring in the series and

noting the number of times

each value occurs.


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The marks obtained by 25 students of a class inMathematics, out of 10 marks are as follows-

construct a Discrete frequency distribution

1, 7, 6, 5, 9

10, 5, 6, 8, 2

7, 8, 3, 8, 3

1, 4, 4, 5,6

4, 3, 2, 6, 7

MARKS TALLY No. OF STUDENTS

1

2

3

4

5

6

78

9

10

2

2

3

3

3

4

33

11

TOTAL 25


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Continuous (Grouped) Frequency

Distribution

Continuous (Grouped) Frequency

Distribution is a table in which the dataare grouped into different classes and

the number of observations falls in each

class are noted.


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Construct a Continuous frequency distribution

for the following set of observations

MARKS TALLY No. OF STUDENTS

20 29

30 3940 49

50 -59

60 69

70 - 79

70, 45, 33, 64, 50

25, 65, 75, 30, 20

55, 60, 65, 58, 52

36, 45, 42, 35, 40

51, 47, 39, 61, 53

59, 49, 41, 20, 55

42, 53, 78, 65, 45

49, 64, 52, 48, 46

III

IIII IIIIII

IIII IIII

IIII

III

IIII

II

3

512

10

7

3

TOTAL 40


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Upper limitNOT

Included

Lower limit

Included

Upper limit

Included

Exclusive Classes0 10

10 20

20 30

30 40

40 50

Inclusive Classes0 9

10 19

20 29

30 39

40 49

Lower limit

Included

TYPES OF CLASSES


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CUMULATIVE FREQUENCY DISTRIBUTION

Cumulative frequency Distribution is a table

which gives how many observations are lying

below or above a particular value


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CUMULATIVE FREQUENCYDISTRIBUTION

LESS THANCUMULATIVEFREQUENCY

DISTRIBUTION

GREATER THANCUMULATIVEFREQUENCY

DISTRIBUTION


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LESS THAN CUMULATIVE

FREQUENCY DISTRIBUTION

Less than cumulative frequency distribution

is a table which gives the number of

observations falling below the upper limit of

a class


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Construct Less than Cumulative

Frequency Distribution

Class Frequency

0 5 4

5 10 710 15 12

15 20 5

20 25 2


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Class Frequency


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Greater than Cumulativefrequency distribution

greater than Cumulative frequencydistribution is table which gives the number

of observations lying above the lower limit of

the class


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Construct Greater than Cumulative


Class Frequency

0 5 4

5 10 710 15 12

15 20 5

20 25 2


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Answer

Class Frequency >CF

0 5 4 (4+7+12+5+2) 30

5 10 7 (7+12+5+2) 26

10 15 12 (12+5+2) 19

15 20 5 (5+2) 7

20 25 2 2Greater than Cumulative Frequency

Distribution


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Rule for determining the number

of classes

We have a rule for determining the number of

classes known as Sturgesrule, It is given by

k = 1 + 3.22 log N,

k - number of classes

N is the total observations


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Graphical and Diagrammatic

representation of data

The following are commonly used graphs and

Diagrams.

Histogram

Frequency Polygon

Frequency Curve

Cumulative Frequency Curve (Ogive)

Less than Cumulative Frequency Curve (Less than Ogive)

Greater than Cumulative Frequency Curve (Greater than Ogive)

Pie Diagram (Sector Diagram)

Bar Diagram


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Histogram Graphical representation of continuous

(Grouped) frequency distribution It is a graph including vertical rectangles with

no space between the rectangles.

The class interval taken along the horizontalaxis (Xaxis) and the respective class

frequencies are taken on the vertical axis (Y

axis) using suitable scales of each classes.


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For each class a rectangle is drawn with base as

width of the class and height as proportional tothe class frequency.

The area of each rectangle will be proportionalto or equal to respective frequencies of the

class

The total area of the histogram will be

proportional or equal to the total frequency of

the distribution.


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Histogram

10 20 30 40 50 60

Class Frequency

0 10 4

10 20 10

20

30 21

30 40 9

40 50 4

50

60 2

Total 50


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Bar Diagram It is graphical representation of the data which

can be divided into different categories. These diagrams are generally drawn in the

shape of horizontal or vertical bars.

The bars should be of equal breadth and theheight of the bars should be proportional to

the magnitude of each quantity.

Leave equal space between the bars.


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Category No. ofStudents

Distinction 20

First class 40

Second class 50

Third class 45

Failure 25

Total 180

Draw simple bar diagram

No

.ofStude

nts


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Frequency Polygon

It is a graphical representation of continuous

frequency distribution

It can be constructed by drawing Histogram

or directly plotting the points

To draw Frequency Polygon by drawing

Histogram, join the mid-points of the top of

the rectangles of the Histogram using straight

lines


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Frequency Polygon can also drawn by joining the

consecutive points, plotted by taking the mid-points

of the classes on X-axis and corresponding

frequencies on Y-axis. The end points are extended at each end and to join

the X-axis.

the total area under the Frequency Polygon is equal

to or proportional to (numerically) the total

frequency of the given distribution.


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Construct Frequency Polygon for the

following frequency distribution

Class Frequency

0 10 4

10 20 1020 30 21

30 40 9

40 50 450 60 2

Total 50


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First Method


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Second Method


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Third Method


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Frequency Curve

It is a graphical representation of continuous

frequency distribution

It can be constructed by drawing Histogram or

directly plotting the points To draw Frequency curve by drawing Histogram,

join the mid-points of the top of the rectangles of

the Histogram using smooth curve by free hand

method


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Frequency curve can also drawn by joining the

consecutive points, plotted by taking the mid-points

of the classes on X-axis and corresponding

frequencies on Y-axis. The end points are extended at each end and to join

the X-axis.

The total area under the Frequency Curve is equal

to or proportional to (numerically) the total

frequency of the given distribution.


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Construct Frequency Curve for the

following frequency distribution

Class Frequency

0 10 4

10 20 1020 30 21

30 40 9

40 50 4

50 60 2

Total 50


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First Method


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Second Method


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Third Method


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Cumulative Frequency Curve (Ogive)

It is the graphical representation of

cumulative Frequency Distribution

Two types

a). Less than Cumulative Frequency Curve (Less

than Ogive)

b). Greater than Cumulative Frequency Curve

(Greater than Ogive)

h l i


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Less than Cumulative Frequency

Curve (Less than Ogive) It is the graphical representation of Less than

Cumulative Frequency distribution.

Less than Cumulative Frequency Curve is drawn byjoining smoothly the points obtained by plotting the

upper limit of the actual classes against their Less

than cumulative Frequencies.

Construct Less than Cumulative Frequency


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Construct Less than Cumulative Frequency

Curve for the following frequency

distribution

Class Frequency


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Less than Cumulative Frequency Curve

Greater than Cumulative Frequency


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Greater than Cumulative Frequency

Curve (Greater than Ogive)

It is the graphical representation of Greater than

Cumulative Frequency distribution.

Greater than Cumulative Frequency Curve is drawnby joining smoothly the points obtained by plotting

the Lower limit of the actual classes against their

Greater than cumulative Frequencies.

C G h C l i F


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Construct Greater than Cumulative Frequency

Curve for the following frequency distribution

Class Frequency >CF

0 10 5 120

10 20 12 115

20 30 28 103

30 40 40 75

40 50 21 35

50 60 10 14

60 - 70 4 4


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Greater than Cumulative Frequency Curve

Pie Diagram


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Pie Diagram

Pie diagram consist of circle whose area

proportional to the magnitude of the variable theypresent

The component part of the variable represented by

means of sectors of the circle

The area of the sector proportional to the

frequencies of the component parts of the variable.

If A1 and A2 are the total magnitude of the two

variables, to represent the data by means of Piediagram, draw two circles with radius r1 and r2 given

by


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Draw Pie Diagram for the following

data

Category No. of Students

Distinction 20

First class 40

Second class 50

Third class 45

Failure 25


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CategoryNo. of

StudentsAngle of the Sector

Distinction 20

First class 40

Second class 50

Third class 45

Failure 25

Total 180 360

500


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Assignment

Diagrammatic and

Graphic representation of

Data - Merits andLimitations

Last Date: 12.12.2011


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Analysis &

Interpretationof Data


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MEASURES OF CENTRAL TENDENCY

When we collected data from a sample of

study, the majority of scores in that collected

data always show a tendency to be closer the

central value. This phenomenon is calledcentral tendency.

The value of the point around which scores

tend to cluster is called Measures of CentralTendency.

M f C t l

T d


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MODE

Measures of Central Tendency

Arithmetic Mean

MedianMode

ARITHMETIC MEAN


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ARITHMETIC MEAN Case I: Ungrouped Data (Discrete data)

Letx1, x2, x3, ..xn are N observations

Then A.M (X) =

=

A.M=

Sum of the observations

Total No. of observations

x1+x2+x3+xn

N

x

Case II Ungro ped Freq enc


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Case II: Ungrouped Frequency

Distribution

Ifx1, x2, x3, .xn areobservations and

f1, f2, f3, ..fn then A.M is given by

f1x1+f2x2+f3x3+fnxn

f1+f2+f3+fnA.M =

fxf

A.M =


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Case III: Grouped Frequency


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Case III: Grouped Frequency

Distribution

Direct Method

A.M =

x -Mid-value of classes

f -Frequency

N -Total frequency


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Home work

Class f

0 - 9 3

10 19 1020 - 29 13

30 - 39 9

40 - 49 5

TOTAL 40


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Calculate A.M

Class f

0 - 10 3

10 20 12

20 - 30 20

30 - 40 10

40 - 50 5

TOTAL 50

Answer


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Class fmid-value

(x)d f d

0 - 10 3 5 -2 -6

10 20 12 15 -1 -12

20 - 30 20 25 - A 0 0

30 - 40 10 35 1 10

40 - 50 5 45 2 10

N=50 = 2

A.M (X) =A+ = 25+ = 25.4

Answer


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Arithmetic Mean Merits

It is rigidly defined

AM is easy to understand

Simple to calculate Based on all observations

It is capable for further algebraic treatment.

Used for group comparison


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Case II: N is even

Median =Average of observation and

observation when the data are arranged

in ascending or descending order of

magnitude.

Median =


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Calculate Median:


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Calculate Median:

30, 26, 42, 28, 35, 20, 32, 50

Data in Ascending order of magnitude:

20, 26, 28, 30, 32, 35, 42, 50

Here N = 8

Median =

=

= = 31

Median : Grouped (Contiguous)


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Median : Grouped (Contiguous)


Median =lm + ( ) c

lm Actual lower limit of Median Class(Median Class Class in which (observation falls

N Total Frequency

cfm

Cumulative frequency Up to MedianClass

fm frequency of Median Classc Class interval


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Graphical Determination of Median


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Method : 1

Steps:Draw Less than or Greater than

Ogive.

Locate N/2 on the Y Axis

At N/2 draw a perpendicular tothe Y Axis and extent it to

meet the Ogive

From that point of intersection

draw a perpendicular to the XAxis

The point at which the

perpendicular meets the X-

Axis will be the Median.

N/2

N

Median



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Steps:

Draw Less than and

Greater than Ogive

simultaneously

Draw perpendicular

from the point of

intersection to the X -

Axis

The point at which theperpendicular meets the

X- Axis will be the

Median.


Method : 2

Median

di i


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Median Merits

It is rigidly defined

It is easy to understand

Simple to calculate

It can be located by mere inspection It is not affected by extreme values

It can be calculated for a distribution having open

end classes It can be determined graphically.

M di d i


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Median demerits

It is not based on all observations

Median is a non-algebric measure and hence not

suitable for further algebric treatment

It is cant be used for computing other statisticalmeasures such as Standard Deviation, Coefficient of

correlation etc.

When there are wide variations between the values

of different scores, a Median may not be

representative of the distribution.

MODE


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MODE

Mode is the value of the variable whichoccurs most frequently.

In certain cases there may be Two or Three

Modes in a distribution. When there are Two Modes we call it Bi-Modal

Distribution

If there are Three Modes, we call itTri-Modal

Distribution.


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C ti Di t ib ti


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Continuous Distribution

Mode =lm + ( ) c

lm Actual lower limit of Modal Class(Modal Class Class having

maximum frequency

f1 Frequency of the class just below theModal Class

f2

Frequency of the class just above the

Modal Class

c Class interval

C l l t M d


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Calculate Mode

Class Frequency

80 84 4

75 79 8

70 74 8 f265 69 12

60 64 9 f1

55 59 7

50 54 5

45 40 3

Modal

ClassMode = lm + ( ) c

=64.5 + ( )5

= 66.9

Herelm = 64.5

f1 = 9

f2 = 8

C= 5

M d M it


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Mode Merits

Easy to locate

Not affected by extreme values

Can calculate the Mode for the distribution

having open-end classes, if open-end classes

have less frequency

It is useful in business matters.

M d d it


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Mode demerits

It is not based on all observations

It is not capable for further algebric

treatment

A slight change in the distribution may

extensively disturb the Mode

As there be 2 or 3 modal values, it becomes

impossible to set a definite value of a Mode.


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Do we Need another

Statistical

Measures?

Consider the Marks of two Groups


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p

Group 1

8, 12, 11, 12,

10, 8, 9, 11,

12, 10, 8, 10,9, 10, 12, 8,

10, 9, 10, 11

Mean = 10

Group 1

15, 2, 8, 12,

4, 17, 20, 6,

2, 18, 16, 0,3, 9, 6, 10,

15, 17, 9, 11

Mean = 10


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MEASURES OF DISPERSION


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MEASURES OF DISPERSION

The statistical measures used todetermine the extent of dispersion of

the scores from the central value

(Arithmetic Mean) of the distribution

are known as Measures of Dispersion

Measures of Dispersion measures the

spreading of observations from thecentral value of the distribution.

Commonly used Measures of


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y

Dispersion are:

Mean

Deviation

RangeQuartileDeviation

Standard

Deviation

Standard Deviation


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Standard Deviation

Standard Deviation is thesquare root of the average

of the squares of the

deviations of the scorestaken from the mean.

SD denoted by the symbol

(sigma).

Calculation of Standard Deviation


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Steps

Find the Arithmetic Mean of the given data.

Find the deviations from Arithmetic Mean of

scores.

Find the average of squares of deviations

taken from the Mean.

Find the square root of the average of

squares of deviations.

Calculation of SD - Discrete Series


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Calculation of SD Discrete Series

Letx1, x2, x3, ..xnareNobservations

Case I: Discrete Data


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Calculate Standard Deviation: 35, 49, 32, 45, 39

S.D


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Ungrouped Distribution


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Ungrouped Distribution

Calculate Standard Deviation


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Calculate Standard Deviation

Score Frequency22 5

27 10

32 25

37 30

42 2047 10

N=100

Answer


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Answer


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S.D-Continuous Frequency


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Distribution

Calculate SD


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Calculate SD

Score Frequency20 24 5

25 29 10

30 34 25

35 39 30

40

44 2045 - 49 10

N=100

S.D =

Answer


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Answer


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For a large distribution, Short-cut method

(A d M M th d) b d t


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(Assumed Mean Method) can be used to

calculate Standard Deviation


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Calculation of MEAN DEVIATION


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Calculation of MEAN DEVIATIONDiscrete Data


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Answer


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= 15

Score (x)

8 7

10 5

12 314 1

16 1

18 3

20 7

22 8

Discrete Distribution


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Calculate Mean Deviation


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Score (x) f

22 5

27 10

32 25

37 30

42 20

47 10

Answer


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Score (x) f fx

22 5 110 14 70

27 10 270 19 90

32 25 800 4 100

37 30 1110 1 30

42 20 840 6 12047 10 470 11 110

N=100 fx=3600=520

AM =

= 3600/100

= 36



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Score (x) f

20 - 24 5

25 29 1030 34 25

35 39 30

40 44 2045 - 49 10

Score

Answer


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ClassScore

(x)f fx

20 - 24 22 5 110 14 7025 29 27 10 270 19 90

30 34 32 25 800 4 100

35 39 37 30 1110 1 30

40 44 42 20 840 6 120

45 - 49 47 10 470 11 110

N=100fx

=3600 =520

AM =

= 3600/100

= 36

QUARTILE DEVIATION

(SEMI INTER QUARTILE RANGE)


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(SEMI INTER QUARTILE RANGE)

The quartile deviation is half the differencebetween the upper and lower quartiles in a

distribution.

Quartile:Any of three points that divide an ordered

distribution into four parts each containing one

quarter of the scores.


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Class Frequency

30 35 10

35 40 16

40 45 1845 50 27

50 55 18

55 60 8

60 65 3

Answer


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Class Frequency


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Range is the difference between the highest and

lowest scores in a Distribution

find Range 53, 51, 70, 45, 60, 62, 40, 53, 71, 55

Range (R) = H L

= 71 40

=31

Range (R) = H LH Highest Value

L Lowest Value



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Observation frequency

5 3

6 8

7 12

8 10

9 8Total 41

Range (R) = H L

= 9 - 5=4

continuous distribution


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In a continuous distribution, Range is the differencebetween the upper limit of the highest class and

lower limit of the lowest class

Class Frequency

10 20 12

20 - 30 2030 - 40 10

40 - 50 5

Range (R) = H L

= 50 - 10

=40


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Negative correlation: When the first variableincrease or decrease, the other variable decrease or

increases respectively, then the relationship

between this two variables are said to be in

Negative correlation.

Eg: Time spend to practice and Number oftyping error


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Zero correlation: if there is no relationshipbetween two variables, then the relationship

between this variable are said to be in Zero

correlation.

Eg: Body weight and Intelligent


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It indicates the nature of the relationship betweentwo variables.

It predicts the value of one variable given the valueof another related variable.

It helps to ascertain the traits and capacities of

pupils.

Use of Coefficient of Correlation


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It helps to determine the validity of a test.

It helps to determine the reliability of a test.

It can be used to ascertain the degree of the

objectivity of a test.

It can answer the validity arguments for or against a

statement.

Properties of Correlation


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For a perfect positive correlation, the Coefficient of

Correlation is +1 and for a perfect Negative correlation, theCoefficient of Correlation will be -1.

Perfect positive or Negative correlation is possible only in

Physical Science.

In a Social Science like Education, the correlation between

two variables will lie within the limit +1 and -1

Positive correlation varies from 0 to +1 and Negative

correlation varies from 0 to -1

Zero correlation indicates that there is no consistent

relationship between two variables.

Calculation of Correlation Coefficient


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There are two important techniques forcalculating Correlation coefficient

Rank Correlation

Product Moment Correlation

Rank Correlation


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Spearman who for the first time measuresthe extent of correlation between two set of

scores by the method of Rank Difference

Find Rank Correlation Coefficient


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Name of

Students

Score in

Maths

Score in

Physics

Nikhil 45 68

Santhosh 53 76

John 67 70

Jenna 40 64

Gopal 35 54

Mohammed 50 66


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Normal Probability Curve


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The normal probability curve is curve that graphically

represents a Normal Distribution.

In a Normal Distribution, when the scores are arranged in

the order of magnitude, those at the centre will have the

maximum frequency.

The frequencies will gradually go on decreasing towards theright and left of the score at the centre. Because of this

property, the curve representing a normal distribution will

show symmetry on either side of its central axis. Hence it

will be in bell-shaped


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These special features of the Normal Distribution

will be seen in the dispersion of scores regarding


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will be seen in the dispersion of scores regarding

natural phenomena as intelligence, height, weight

etc. in a population.

This characteristic of Normal Distribution is found

to be true to a great extent with regard to

achievement scores of a well conductedexamination, if the number taking the examination

is sufficiently large.

Hence properties of Normal Distribution and

Normal Distribution curve are of great importance

in the study of group and their characteristics with

respect to given variables.


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All the three Measures of Central Tendency, viz Mean,

Median, and Mode of a normal curve coincide, that is, they


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are all equal.

The first and third quartiles are equidistant from the median. The ordinate at the mean is the highest. The height of other

ordinates at various sigma distances from the mean are also in

fixed relationship with the height of the mean ordinate.

The curve will gradually go on the nearer to the base line, butit will never meat the base line. For practical purpose, the

curve may be taken to end at points -3 to +3 distance from

the mean, because this region will cover almost 100% of the

cases.

Between -1 and -1, there are 68.26% of the frequencies




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I can prove

anything by

Statistics exceptthe truth

-George Canning

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