Educational Statistics

1. SKEWNESS PROBITY CURVE
The skewness in statistics is a measure of
asymmetry or the deviation of a given random
variables distribution from a symmetric distribution
( like normal distribution )
In the normal curve model the median and the
mode all coincide and there is perfect balance
between the right and lift halves of the figure . A
distribution is said to be “skewed” when the
distribution and the balance is shifted to one side or
the other –to left or right

2. DEFINITIONS :-
According to Umerji .R.R “ skewness means a lack
of symmetry or asymmetry”
According to Garret H.E “ A distribution is said to
be skewed when the mean . Median and mode fall at
different points and the balanced is shifted to right
or left side of the distribution”.

3. TYPES OF SKEWNESS
Skewness in statistics can be divided into tow
(2) categories . They are
 Positive Skewness
 Negative Skewness
1. Positive Skewness:
Distributions are skewed positively or the right
when scores are massed at the low (or left) end
of the scale and are spread out gradually towards
the high or right end as shown in the figure

5. FREQUENCY DISTRIBUTION TABLE OF TEST SCORES
CLASS-INTERVAL
(CI)
FREQUENCY
(f)
CUMULATIVE
FREQUANCY (F)
MID-POINT
(x)
fx
50-54 02 300 52 2704
45-49 05 298 47 2209
40-44 15 293 24 1764
35-39 28 278 37 1369
30-34 30 250 32 1024
25-29 35 220 27 729
20-24 42 185 22 484
15-19 48 143 17 289
10-14 55 95 12 144
5-9 40 40 7 49
N=300 Ʃfx=6590

6. Q1 . Mean = Ʃfx
N
6590
300
21.96
Q2 . Median =
l + N/2 - F
fm

7. = 9.5 + 150-40 x 5
55
= 9.5 + 110 x 5
55
= 9.5 + 2 x 5
= 9.5 x 10
= 19.5
Q3. Mode = 3Median – 2 Mean
= 3 x 19.5 – 2 x 21.96
= 58.5 - 43.92
= 14.58

8. The positive skewness distribution
mean > median > mode
21.96 > 19.5 > 14.58
Characteristics of positive skewness
1. The value of mean is more then the value of median
and mode
2. The scores are massed at the low or left end of the
scale
3. Curve spread out gradually towards the high or
right end

9. 2) NEGATIVE SKEWNESS
Distribution are said to be skewed negatively or to
the left when scores are massed at the high end of
the scale ( the right end ) and are spread out more
gradually towards the low end (or left ) as shown in
figure
In a negatively skewed distribution the mean lies to the
left of the median

11. FREQUENCY DISTRIBUTION TABLE OF TEST SCORE
CLASS – INTERVAL
( C I )
FREQUENCY
(F)
CUMULATIVE
FREQUANCY
(F)
MID-
POINT
(x)
fx
90-99 50 300 94.5 4725.0
80-89 65 250 84.5 5492.5
70-79 55 185 74.5 4097.5
60-69 45 130 64.5 2902.5
50-59 25 85 54.5 1362.5
40-49 25 60 44.5 1112.5
3039 18 35 34.5 621.5
20-29 10 17 24.5 245.0
10-19 5 7 14.5 72.5
0-9 2 2 4.5 9
N=300 Ʃfx=20,640

12. Q1 . Mean = Ʃfx
N
= 20640
300
= 68.8
Q2 . Median = l + N/2 – F x 10
fm
= 79.5 + 150 – 185 x 10
65

13. = 79.5 + 35 x 10
65
= 79.5 + 0.538 x 10
= 79.5 + 5.38
= 84.88
Q3 . Mode = 3 Median – 2 Mean
= 3 x 84.88 – 2 x 68.8
= 254.64 - 137.6
= 117.2

14. The negative skewness distribution
Mean < Median < Mode
68.8 < 84.88 < 117.2
Charateristics of negative skewness
1. The value orf mean is less than the value of median
and mode
2. The score are massed at the high or right end of the
scale
3. Curve spread out gradually towards the low or left
end

15. MEASUREMENT OF SKEWNESS
There are two formula to calculate the measurement of
skewness
Pearson’s coefficient of skewness = Mean – Mode
Standard deviation
Pearson’s co efficient of skewness = 3(Mean – Median)
Standard deviation

16. Frequency Distribution Table
C I f F x fx x2 fx2
171-175 7 100 173 1211 29929 209503
166-170 26 93 168 4368 28224 733824
161-165 42 67 163 6846 26569 1115898
156-160 20 25 158 3160 24964 499280
151-155 5 5 153 765 23409 117045
N=
100
Ʃfx =
16350
Ʃfx =
267555

17. Q1 . Mean = Ʃfx
N
= 16350
100
= 163.50
Q2 . Median = l + N/2 – f x 5
fm
= 160.5 + 50-25 x 5
42

18. = 160.5 + 25 x 5
42
= 160.5 + 0.59 x 5
= 160.5 + 2.95
= 163.45
Q3. Mode = 3 Median – 2 Mean
= 3 x 163.45 - 2 x 163.50
= 490.35 - 327
= 163.35

19. Q4 .
1 (Ʃfx)2
Standard deviation = N-1 Ʃfx2 - N
S .D(σ) = 1 ( 16350 )
100 - 1 2675550 - 100
1 267322500
= 99 2675550 - 100

20. = 0.010 (2325)
= 23.25
= 4.82
Q5 . Calculate the pearson’s coefficient of skewness
= Mean – Mode
S D
= 163.5 – 163.39
4.85
= 0.11
4.85
= 0.02

21. Q6 . The pearson’s coefficient of skewness
= 3 (mean – median )
S D
= 3 ( 163.5 – 163.48 )
4.85
= 3 ( 0.02 )
4.85
= 0.06
4.85
= 0.01

22. CONCLUSION
The skewness is a measure of symmetry or asymmetry of
symmetry of data distribution. The skewness coefficient of a
set of data points helps us determine the overall shape of the
distribution curve , wheather its positive or negative. The
coefficient number also helps us determine whether the right
tail or the left tail of the distribution is more pronounced

23. REFERENCE
 Psychology of learning and instruction . Dr H V Vamadevappa.
Shreyas Publications Davanagere
( Page No – 426-428 )
 Ferguson . A Statistical Analysis in Psychology and Education .
Newyork
 WWW.Slideshare. com