STATISTICAL RESEARCH AND TRAINING CENTER
J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City
Training Course on Basic Statistics for Research
August 24-28, 2009
Measures of Skewness
Prepared by:
Josefina V. Almeda
Professor and College Secretary
School of Statistics
University of the Philippines, Diliman
August 2009
2
Symmetric Distribution
* a distribution is symmetric if it can be folded
along the vertical axis so that the two sides
coincide
* if the distribution is symmetric, the mean, the
median, and the mode are equal and are
located at the same position along the
horizontal axis
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
3
FIGURE 1a. Example of a Symmetric
Distribution
25
No. of Provinces
20
15
10
5
0
1
2
3
4
5
6
7
8
9
mean = median = mode
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
4
Skewed Distribution
* if the two sides do not coincide,
distribution is said to be asymmetric
* a distribution that is asymmetric
with respect to a vertical axis is said
to be skewed.
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
5
Two Types of Skewness
Positively Skewed or Skewed to the Right Distribution
* distribution tapers more to the right than to the left
* has a longer tail to the right compared to a much
shorter left tail
* values are more concentrated below than above the
mean
* large values in the right tail are not offset by
correspondingly low values in the left tail and
consequently the mean will be greater than
the median
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
6
FIGURE 1b. Example of a Skewed to the Right
Distribution
35
Number of Provinces
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
Mode < Median < Mean
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
7
Negatively Skewed or Skewed to the Left Distribution
* distribution tapers more to the left than to the right
* has longer tail to the left compared to a much
shorter right tail
* values are more concentrated above than below the
mean
* small values in the left tail will make the mean less
than the median
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
8
FIGURE 1c. Example of a Skewed to the Left
Distribution
35
Number of Provinces
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
mean < median < mode
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
9
Measure of skewness
*
shows the degree of asymmetry, or
departure from symmetry of a
distribution
* indicates also the direction of the
distribution
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
X
Pearson’s First and Second Coefficient of
Skewness
X - Mo
s
1.
SK 
2.
SK = 3(X - Md)
s
10
where X = mean, Mo = mode, s = standard deviation
where X  mean
Md = median, s = standard deviation
Remarks:
1. Since the mode is frequently only an approximation,
formula 2 is preferred.
2. Interpretation of the measure of skewness:
SK > 0: positively skewed since X > Md > Mo
SK < 0: negatively skewed since X < Md < Mo
Statistical Research and Training Center
Training Course on Basic Statistics for Research
August 24 - 28, 2009
Example: Given the following statistics on weights of
court justices in kilograms:
X
=74.1 Md = 75
Mo = 84
11
s = 11.25
Using the first formula,
SK =
74.1  84
  0.88
11.25
(skewed to the left)
Using the second formula,
SK =
3(74.1  75)
  0.24
11.25
Statistical Research and Training Center
(skewed to the left)
Training Course on Basic Statistics for Research
August 24 - 28, 2009
STATISTICAL RESEARCH AND TRAINING CENTER
J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City
Training Course on Basic Statistics for Research
August 24-28, 2009
Thank you.