Unformatted text preview: 83.43
× c3 =
× 27 =
3
N
100
100
∑ fd ' 4
809
65529
=
= 655.29
× 81 =
× c4 =
µ4
N
100
100
Moments about mean
µ1 = 0
µ2 = µ 2 − µ 1 2
= 15.57 – (2.61)2
= 15.57 – 6.81 = 8.76
µ3 = µ 3 – 3µ 2 µ 1 + 2 µ 13
= 83.43 – 3(2.61) (15.57)+2 (2.61)3
= 83.43 – 121.9 + 35.56 = −2.91
µ4 = µ 4 – 4µ 3 µ 1 + 6µ 2 µ 12 – 3 µ 14
= 665.29 – 4 (83.43) (2.61) + 6 (15.57) (2.61)2 − 3(2.61)4
= 665.29 – 871.01 + 636.39 – 139.214
= 291.454
173
µ = fd 4 7.9
Skewness:
7.9.1 Meaning:
Skewness means ‘ lack of symmetry’ . We study skewness to
have an idea about the shape of the curve which we can draw with
the help of the given data.If in a distribution mean = median =
mode, then that distribution is known as symmetrical distribution.
If in a distribution mean ≠ median ≠ mode , then it is not a
symmetrical distribution and it is called a skewed distribution and
such a distribution could either be positively skewed or negatively
skewed.
a) Symmetrical distribution: Mean = Median = Mode
It is clear from the above diagram that in a symmetrical
distribution the values of mean, median and mode coincide. The
spread of the frequencies is the same on both sides of the center
point of the curve.
b)Positively skewed distribution: Mode Median Mean
It is clear from the above diagram, in a positively skewed
distribution, the value of the mean is maximum and that of the
mode is least, the median lies in between the two. In the positively
skewed distribution the frequencies are spread out over a greater
range of values on the right hand side than they are on the left hand
side.
174 c) Negatively skewed distribution: Mean Median Mode
It is clear from the above diagram, in a negatively skewed
distribution, the value of the mode is maximum and that of the
mean is least. The median lies in between the two. In the negatively
skewed distribution the frequencies are spread out over a greater
range of values on the left hand side than they are on the right hand
side.
7.10 Measures of skewness:
The important measures of skewness are
(i) Karl – Pearason’ s coefficient of skewness
(ii) Bowle...
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 Winter '08
 Moshiri
 Business, Statistics

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