# 13.5 - Shapes of distributions

13.5 - Shapes of distributionsHistograms 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

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:

and box plot of the 40 weights:

suggest that the distribution of female weights is symmetric.

## Example 13-6

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:

and box plot of the 26 grades:

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

## Example 13-7

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:

and box plot of the lifetimes of 39 Energizer bunnies:

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