B.Ed.104 unit 4.1-statistics

MAEER’s
MIT SaintDnyaneshwarB.Ed.College,
Alandi(Devachi),Pune.
B.Ed. 104
ASSESSMENT AND EVALUATION FOR
LEARNING
Unit 4- INTERPRETING TEST SCORES
(Statistics)
By.
Asst.Prof. Gangotri V. Rokade

UNIT4 INTERPRETINGTESTSCORES
(1 CREDIT)
Statistical measures to interpret the test scores (Meaning,
Characteristics, and Uses)
4.1 Measures of Central Tendency : Mean, Median, Mode
4.2 Measures of Variability : Quartile Deviation, Standard
Deviation
4.3 Percentile and Percentile Rank
4.4 Co-efficient of correlation by Spearman’s Rank Difference
method
4.5 Standard Scores: Z and T (Concept Only)
4.6 Graphical representation of data : Histogram, Frequency
polygon
4.7 Normal Probability Curve : Properties, Uses
4.8 Skewness and Kurtosis

UNIT 4 INTERPRETING TEST SCORES
Origin of d word Statistics?
1. Derived from an Italian word: ‘Statista’
2. Derived from a Latin word: ‘Status’
Both ‘statista’ and ‘status’ means political state.
Values in statistics denotes collection of data of any political state
regarding its population, income and expenditure, industrial
production, housing, education, births, agricultural products etc.
3. Derived from a Greek word: ‘Statizein’ : means arrange
systematically

UNIT 4 INTERPRETING TESTSCORES
Important concepts of statistics:
1. Scores- the figures, which denotes educational achievement
of an individual or student in quantitative form. For e.g.
80,45,68
2. Series:
a. discrete series- A series in which constituent scores cannot
be divided up to micro-level. E.g. no. of students, runs scored by d
batsman in cricket.
b. continuous series- A series in which constituent scores can
be divided up to any micro-level. E.g. time – hr, min, hr

3. Class interval- groups of scores are formed. Each group of scores is
known as class interval. It is denoted by C.I. for e.g. if there are scores
45,46,47,42 and 48 …then instead of expressing them separately, these
are denoted by a single class interval 42-48.
3. Length of class interval- ( i) -the actual difference between the upper
and lower limit of the given class. For e.g.
In class interval …..actual lower limit in d frequency
distribution is considered to start before by 0.5 and actual
upper limit ends after by 0.5 in reference with its face value.
So… in the class interval 42-48,
Length of the class interval = Actual upper limit- actual lower limit
= 48.5- 41.5
=7

5. Unclassified scores- scores which are written randomly/not
classified. For e.g. 22,56,34,24
6. Classified scores- division of scores into classes on the basis of
some criterion. For e.g. 41-50, 51-60,61-70
7. Frequency distribution- if the scores are written systematically
and in the form of class interval, then these types of scores are
known as classified scores and distribution is called as frequency
distribution.
8. frequency- it is a figure which indicates how many times a
specific score repeats itself in a given class interval. For e.g.
11,16,45,45,11,16,45

9. Range: Difference between lower limit and upper limit of the
end point of the class interval is known as range.
It is denoted by ‘R’
e.g. 1. unclassified scores 6,12,14,16,20
R = H- L =20-6 = 14
e.g. 2. classified data 21-30,31-40,41-50
R= (H-L)+1 = (50-21)+1= 29+1=30

4.1 Measures of Central Tendency : Mean, Median, Mode
Central tendency: In any given series, most of the scores are
concentrated around the mean or centre of the series. This
tendency of scores is known as central tendency.
Measures of central tendency: The measures or units, which
are used for the measurement of central tendency.
There are 3 measures or units of central tendency.
1. Mean
2. Median
3. Mode

Measures of central tendency:
1. Mean: (average): M
e.g. Unclassified data: e.g. 1) 10,15,20,25,30
∑X Total of all scores
M= Mean= ----------- = ------------------------
N No. of scores
e.g. 2) 30,32,28,10,25,30

2. Median: Mdn
e.g. Unclassified data: e.g. 1) 10,8,12,15,6,25,5
Arrange d data in asc/desc order:
5,6,8,10,12,15,25
10+ 12 22
Mdn=Median = --------- = ------ = 11
2 2
e.g. 2) 65, 57, 23, 11, 40, 35, 52

3. Mode : Mo.
A score which appears the maximum number of times in a series.
e.g. 10,22,34,22,35,22,10
Mo.= 22

1. Computation of Mean: (average): M
from unclassified data.
M= Mean= ----------- = ------------------------
N No. of scores
e.g.1
X (scores) f ( frequency)
40 5
38 7
26 10
19 8
14 6
10 4

X (scores) f ( frequency) f x X
40 5 200
38 7 266
26 10 260
19 8 152
14 6 84
10 4 40
N= 40 ∑ fX = 1002
∑ f x X 1002
M= Mean= -------------- = ---------- = 25.05
N 40
Mean of the given frequency distribution is = 25.05

1. Computation of Mean: (average): M
from unclassified data.
M= Mean= ----------- = ------------------------
N No. of scores
e.g.2
Mean=?
X (scores) f ( frequency)
30 1
26 2
22 3
18 3
15 1

Computation of Mean from classified scores
(using long method) e.g.1
C.I f
40-44 5
35-39 6
30-34 9
25-29 15
20-24 7
15-19 5
10-14 3

Computation of Mean from classified scores:
( using long method) e.g.1
C.I Xm f f x Xm
40-44 (40+44)/2= 42 5 42 x 5 = 210
35-39 (35+39)/2= 37 6 222
30-34 32 9 288
25-29 27 15 405
20-24 22 7 154
15-19 17 5 85
10-14 12 3 36
N= 50 ∑fXm=1400
∑ f x Xm 1400
M= Mean= -------------- = ----------
N 50
M = 28
Mean is = 28

( using long method) e.g.2
C.I f
90-99 3
80-89 4
70-79 7
60-69 8
50-59 13
40-49 17
30-39 11
20-29 9
10-19 7
0-09 3

Computation of Mean from classifiedscores: e.g.2
C.I Xm f f x Xm
90-99 3
80-89 4
70-79 7
60-69 8
50-59 13
40-49 17
30-39 11
20-29 9
10-19 7
0-09 3
N= ∑fXm=
∑ f x Xm
M= Mean= -------------- = ----------
N
Mean is =

( using a Short method/ deviation method)
e.g.1
C.I f
40-44 5
35-39 6
30-34 9
25-29 15
20-24 7
15-19 5
10-14 3

Computation of Mean from classified scores:
( using a short method) e.g.1
C.I f d f x d
40-44 5 +3 +15
35-39 6 +2 + 12 +36
30-34 9 +1 +9
25-29 (AM=27) 15 0 0
20-24 7 -1 -7
15-19 5 -2 -10 - 26
10-14 3 -3 -9
N= 50 ∑fd=10
∑ f x d 10
M= AM + -------------- x i = 27 + ---------- x 5 = 27+1 = 28
N 50
AM= (25+29)/2 = 27
Mean is = 28

Computation of Mean from classifiedscores using short method/
deviationmethod e.g.2
C.I f
90-99 3
80-89 4
70-79 7
60-69 8
50-59 13
40-49 17
30-39 11
20-29 9
10-19 7
0-09 3

ComputationofMeanfromclassifiedscores: (usingashortmethod)e.g.2
C.I f d f x d
90-99 3
80-89 4
70-79 7
60-69 8
50-59 13
40-49 17
30-39 11
20-29 9
10-19 7
0-09 3
N= ∑fd=
∑ f x d
M= AM + -------------- x i = + ---------- x = =
N
AM= Mean is =

Median: Computation of Median (Mdn.) from
frequency distribution: classified scores
e.g.1
C.I f
40-44 4
35-39 6
30-34 9
25-29 10
20-24 6
15-19 3
10-14 2

Computation of Median (Mdn.) from frequency
distribution: classified scores
e.g.1
C.I f F ( cumulative frequency)
40-44 4 36+4= 40
35-39 6 30+6= 36
30-34 9 21+9= 30
(l) 24.5 25-29 10 (fm) 11+10= 21
20-24 6 5+6= 11 (F)
15-19 3 2+3= 05
10-14 2 02
N= 40
N/2-F (40/2)-11
Mdn= l + ---------- x i = 24.5+ ----------------- x 5 = 24.5 + 4.5= 29
fm 10
Median = 29

Computation of Median(Mdn) from classifiedscores using short
method/ deviationmethod e.g.2
C.I f
90-99 2
80-89 3
70-79 10
60-69 14
50-59 20
40-49 12
30-39 8
20-29 6
10-19 5

e.g.2
C.I f F
90-99 2
80-89 3
70-79 10
60-69 14
50-59 20
40-49 12
30-39 8
20-29 6
10-19 5
N=
N/2-F ( )-
Mdn= l + ---------- x i = + ----------------- x = + =
fm
Median =

e.g.3 C.I f F
75-79 1
70-74 2
65-69 6
60-64 14
55-59 3
50-54 12
45-49 8
40-44 2
N=
N/2-F ( )-
Mdn= l + ---------- x i = + ----------------- x = + =
fm
Median =

Mode: Computation of Mode (Mo.) from frequency
distribution: classified scores
Mode = 3 Mdn – 2 M
e.g.1 C.I f
80-86 2
73-79 8
66-72 11
59-65 15
52-58 12
45-51 7
38-44 5

Let us find the Mean first:-
e.g.1 C.I f d fd
80-86 2
73-79 8
66-72 11
59-65 15
52-58 12
45-51 7
38-44 5
N= ∑fd=
∑ f x d
M= AM + -------------- x i = + ---------- x = + =
N
AM= ( + )/ =
Mean is =

Let us find the Median:-
e.g.1
C.I f F
80-86 2
73-79 8
66-72 11
59-65 15
52-58 12
45-51 7
38-44 5
N=
N/2-F ( ) -
Mdn= l + ---------- x i = + ----------------- x = + =
fm
Median =

= 3 ( ) - 2 ( )
=
Mode of the given frequency distribution is=
**************************

B.Ed.104 unit 4.1-statistics

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