1
SYMMETRY AND SKEWNESS
Symmetry
A distribution is said to be symmetrical about the mean if when presented diagramatically, it has as a line of
reflective symmetry the vertical axis through the mean
When a distribution is symmetrical about the mean, the mean and median coincide. The mode may also
coincide with the median on the occasion that it exits.
Thus the relationship between the mean, median and mode provides information about the manner in which
the observations are distributed among categories
Symmetrical Distributions
a) Normal Distribution
The three measures of central tendency, mean, mode or median are equal
b)Triangular Distribution
The three measures of central tendency, mean, mode or median are equal
c) Bimodal distribution
The mean and median are equal but are not good as measures of central tendency since not many values in the
distribution cluster around them. The two modes are however good as measures of central tendency
Skewness
Skewness refers to lack of symmetry. When the longer tail of the distribution extends to the right i.e a few
observations are extremely large, the distribution is positively skewed (skewed to the right).
When the longer tail of the distribution extends to the left, i.e, a few observations are extremely small, the
distribution is said to be negatively skewed (skewed to the left)
When a distribution is skewed to the right, the mode is less than the median, which in turn is less than the
mean.
When a distribution is skewed to the left, the mean is less than the median which in turn is less than the mode.
Therefore for a skewed distribution, the mean is different from the median, which in turn is different from the
mode.
Skewness tells us about the direction of spread.
a)Karl Pearson’s coefficient of skewness
For skewed distributions, the mean tends to lie on the same side of the mode as the longer tail. Thus a measure
of asymmetry is provided by their difference i.e Mean-Mode. Thus we have the formula
SKp =
X̄ − M ode
mean − mode
=
standard deviation
s
To avoid using the mode we employ the empirical formula
mean − mode = 3(mean − median)
Thus the formula become
3(mean − median)
3(X̄ − median)
=
standard deviation
s
Theoretically the value of coefficient of skewness given by these formulae lie between −3 and +3. However in
practice the value of this coefficient usually lies between −1 and +1
b)Bowley’s coefficient of skewness
It is also called quartile coefficient of skewness as it is based on quartiles and is defined by
SKp =
SKB
=
Q3 + Q1 − 2Q2
Q3 − Q1
=
Q3 + Q1 − 2M edian
Q3 − Q1
The value of this coefficient lies between −1 and +1. This coefficient of skewness is particularly useful when we
have open-ended distributions and where extreme values are present.
c)Percentile coefficient of skewness
This measure was devised by Kelly and is defined as
SKK
=
P90 + P10 − 2P50
P90 − P10
=
D9 + D1 − 2D5
D9 − D1
2
Exercise 1
The following data is the profit of 1000 companies in thousands Kenya shillings. Calculate the coefficient of
skewness and comment
Profit
100 − 119
120 − 139
140 − 159
160 − 179
180 − 199
200 − 219
220 − 239
Number of Companies
17
53
199
194
327
208
2
1000
Exercise 2
The following table gives the distribution of the monthly income in Kenya pounds of 500 workers in a factory
Monthly Income
Below 100
100 − 150
150 − 200
200 − 250
250 − 300
300 and above
Number of workers
10
25
145
220
70
30
500
i) Obtain the limits of the central 50 percent of the observed workers
ii) Calculate the coefficient of skewness