Skewness and kurtosis explained using examples and case studies based on climatic changes to explain these concepts.

1. SKEWNESS and kurtosis
J001, J009, J015, J019
MBA (TECH), EXTC

2. 2
SKEWNESS
•
Distributions (aggregations of observations) can be spread evenly around both
sides of the central tendency, like so:
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Such distributions are considered symmetrical with no skew.
5
5
Mean =
Median =

3. 3
SKEWNESS
•
As scores are weighted and distribute unevenly around the median, the mean is
“pulled” toward the extreme outlier and it diverges away from the median.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
7
5
Mean =
Median =
2120

4. SKEWNESS
•
When the outlying scores are on the higher end of the scale the distribution
becomes positively skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
+

5. SKEWNESS
•
When the outlying scores are on the lower end of the scale the distribution
becomes negatively skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
_

6. Example: skewness

7. 7
Example: skewness

8. 8
Problem: College Men’s Heights
Height
(inches)
Class Mark, x Frequency, f
59.5–62.5 61 5
62.5–65.5 64 18
65.5–68.5 67 42
68.5–71.5 70 27
71.5–74.5 73 8
xf
305 -6.45 208.01 -1341.7
1152 -3.45 214.25 -739.15
2814 -0.45 8.51 -3.83
1890 2.55 175.57 447.7
584 5.55 246.42 1367.63
∑ 6745 n/a 852.75 −269.33
2, m3 67.45 n/a 8.5275 −2.6933
Finally, the skewness is
g1 = m3 / m2
3/2
= 2.6933 / 8.5275− 3/2
= 0.1082−

9. 9
kurtosis
•
Distributions of data and probability distributions are not all the same shape.
Some are asymmetric and skewed to the left or to the right. Many times,
there are two values that dominate the distribution of values.
Kurtosis is the measure of the peak of a distribution, and indicates how high the distribution is around the mean.

10. Types of kurtosis

Mesokurtic
A distribution
identical to the
normal distribution
 Leptokurtic
A distribution that is
more peaked
than normal
 Platykurtic
A distribution that is
less peaked than
normal

11. 11
Formulae: kurtosis
•
Moment coefficient of kurtosis
One measure of kurtosis uses the fourth moment about fourth power of
standard deviation in dimensionless form:

12. Problem: test scores
67 24.125 338742.19 677484.375
62 19.125 133784.49 267568.985
57 14.125 39806.485 119419.454
52 9.125 6933.1643 6933.16431
47 4.125 289.53149 1737.18896
42 -0.875 0.5861816 6.44799805
37 -5.875 1191.3284 9530.62695
32 -10.88 13986.758 41960.2742
27 -15.88 63511.875 127023.75
22 -20.88 189891.68 379783.36
16311447.6
Xi
65-69 2
60-64 2
55-59 3
50-54 1
45-49 6
40-44 11
35-39 8
30-34 3
25-29 2
20-24 2
40
C.I. fi
Where,
SD = 7.22
(leptokurtic)

14. CASE STUDY II

15. THANK YOU