13.5 - Shapes of distributions

Histograms and box plots can be quite useful in suggesting the shape of a probability distribution. Here, we'll concern ourselves with three possible shapes: symmetric, skewed left, or skewed right.

Skewed Left
For a distribution that is skewed left, the bulk of the data values (including the median) lie to the right of the mean, and there is a long tail on the left side.
Skewed Right
For a distribution that is skewed right, the bulk of the data values (including the median) lie to the left of the mean, and there is a long tail on the right side.
Symmetric
For a distribution that is symmetric, approximately half of the data values lie to the left of the mean, and approximately half of the data values lie to the right of the mean.

The following examples probably illustrate symmetry and skewness of distributions better than any formal definitions can.

Example 13-5 Section

Consider a random sample of weights (in pounds) of 40 female college students:

135 117 137 135 133 145 129 157 113 134
144 141 132 138 133 134 132 135 152 141
140 119 138 136 156 141 116 131 138 128
120 148 130 140 121 137 121 145 145 125

Do these data suggest that the distribution of female weights is symmetric, skewed right, or skewed left?

Solution

The histogram:

Histogram of female weights

and box plot of the 40 weights:

Box plot of female weights

suggest that the distribution of female weights is symmetric.

Example 13-6 Section

Consider a random sample of 26 grades on an easy statistics exam:

100 100 99 98 97 96 95 95 95 94
93 93 92 92 91 90 90 90 89 84
80 75 68 65 50 45        

Do these data suggest that the distribution of exam scores is symmetric, skewed right, or skewed left?

Solution

The histogram:

Histogram of easy exam scores

and box plot of the 26 grades:

Box Plot of Easy Exam Scores

suggest that the distribution of easy exam scores is skewed to the left.

Example 13-7 Section

Consider the lifetimes (in years) of a random sample of 39 Energizer bunnies:

0.2 3.6 3.1 0.9 0.7 7.8 1.4 0.4 3.1 3.4
5.3 3.2 0.3 3.1 6.0 2.8 5.6 0.2 1.4 0.9
2.4 0.8 1.8 1.0 2.9 0.5 0.9 3.2 1.3 11.1
0.8 1.8 1.4 0.2 1.0 1.1 1.6 0.7 3.2  

Do these data suggest that the distribution of lifetimes of Energizer bunnies is symmetric, skewed right, or skewed left?

Solution

The histogram:

Histogram of Energizer Lifetimes

and box plot of the lifetimes of 39 Energizer bunnies:

Boxplot of energizer lifetimes

suggest that the distribution of lifetimes of Energizer bunnies is skewed to the right.