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# Histograms and Boxplots Study Guide (page 3)

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Updated on Oct 5, 2011

## Boxplots

Another graph that is extremely useful is the boxplot. To create a boxplot, we need the five-number summary. The five-number summary consists of the first quartile, the median, the third quartile, the smallest observed value, and the largest observed value. The following steps lead to a boxplot:

1. Draw a scale that extends below the smallest and above the largest values in the data set on either the horizontal or vertical axis.
2. Draw parallel line segments at the first quartile, the median, and the third quartile. Connect the ends of the three parallel line segments to form a box.
3. Extend "whiskers" from the center of the first quartile line segment and the center of the third quartile line segment to the smallest and largest observations, respectively, as long as these most extreme observations are within 1.5 IQR of the closest quartile. Otherwise, extend them to the smallest value within 1.5 IQR of the first quartile and to the largest value within 1.5 IQR of the first quartile.
4. If there are any observations beyond 1.5 IQR of the nearest quartiles (so the whiskers do not extend to these values), mark these observations with an asterisk (*).
5. The mean may also be marked using a circle, which allows an easy comparison of the relative sizes of the mean and median.

Any value that is more than 1.5 IQR units below Q1 or above Q3 is defined to be an outlier.

#### Example

Create a boxplot for the orchestra members' heights introduced in Lesson 4. Identify any outliers that might be present.

#### Solution

Referring again to our work with the orchestra members' heights in Lessons 5 and 6, we have the following five-number summary:

 First quartile (Q1): 62.5 Median (Q2): 66.85 Third quartile(Q3): 69.5 Smallest value: 53.5 Largest value: 83.8

As determined in Lesson 6, the IQR for the orchestra members' heights is 7 inches; thus, 1.5 IQR = 1.5 (7) = 10.5 inches. Q1 – 1.5 IQR = 62.5 – 10.5 =52.5 inches. Because the smallest orchestra member is 53.5 inches, which is greater than the 52.5 just calculated, the lower whisker extends only to 53.5 inches. Now, Q3 + 1.5 IQR = 69.5 + 10.5 = 80 inches. The two tallest orchestra members are 76.7 and 83.8 inches tall. The member who is 76.7 inches tall is within 1.5 IQR of the third quartile, but the tallest member is not. Therefore, the upper whisker extends from Q3 to 76.7 (the largest observation within 1.5 IQR of Q3), and a star is used to designate the 83.8 inches that is beyond the end of the whisker. Finally, the mean is denoted with a circle in the box.

Here, the orchestra member who is 83.8 inches tall is an outlier, but the shortest member who is 53.5 inches tall is not. Notice that the histogram based on 4-inch class intervals reflects which values are outliers better than the one based on 2-inch class intervals (see Figure 7.4).

## Shape of a Distribution

Unlike dotplots and stem-and-leaf plots, histograms and boxplots may be used with very large data sets.All four, but especially the histograms and boxplots, provide a visual display of the shape of the distribution. Three specific shapes will be discussed most frequently: symmetric, right skewed, and left skewed. If a vertical line can be drawn through the center of a histogram such that the area to the left of the line is a mirror image of the area to the right, the distribution is symmetric (see Figure 7.5). For boxplots, a distribution is symmetric if the shape of the box and length of the whiskers for observations that are smaller than the median is a mirror image of the shape of the box and length of the whiskers for observations that are greater than the median. The most common continuous distribution that is unimodal and symmetric is the normal distribution.

If a distribution has one mode (is unimodal) and is not symmetric, the distribution is said to be skewed. Proceeding to the right of the mode in a unimodal distribution, we move to the upper, or right, tail of the distribution. Similarly, we move to the lower, or left, tail of the distribution as we proceed to the left of the mode in a unimodal distribution. If the upper tail stretches out farther than the lower tail, the distribution is right or positively skewed (see Figure 7.5). If the lower tail stretches out farther than the upper tail, the distribution is left or negatively skewed (see Figure 7.5). These shapes may be seen in both histograms and boxplots.

## Histograms and Boxplots In Short

Numerical data can be summarized in tabular form. However, if the number of possible values of a discrete random variable is large or if the random variable is continuous, then possible values need to be grouped. Frequency or relative frequency histograms and boxplots provide visual summaries of the distributions.

Find practice problems and solutions for these concepts at Histograms and Boxplots Practice Exercises.

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