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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
Frequency Distribution
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)
Frequency Distribution
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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Graphical Determination of Median
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
Graphical Determination of 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.
Graphical Determination of 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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Discrete Distribution
Calculate Mean Deviation
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Calculate Mean Deviation
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
Continuous Distribution
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Continuous Distribution
Calculate Mean Deviation
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Calculate Mean Deviation
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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Continuous Distribution
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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
Discrete Distribution
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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
Between -2 and -2, there are 95.44% of the frequencies
Between -1 and -1, there are 99.73% of the frequencies
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I can prove
anything by
Statistics exceptthe truth
-George Canning