Histograms and Stem-and-Leaf Displays for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:


A histogram is a graph that displays the classes on the horizontal axis and the frequencies of the classes on the vertical axis. The frequency of each class is represented by a vertical bar whose height is equal to the frequency of the class. A histogram is similar to a bar graph. However, a histogram utilizes classes or intervals and frequencies while a bar graph utilizes categories and frequencies.

EXAMPLE 2.10 A histogram for the aspirin prices in Example 2.8 was formed using SPSS and is shown in Fig. 2-3.

A symmetric histogram is one that can be divided into two pieces such that each is the mirror image of the other. One of the most commonly occurring symmetric histograms is shown in Fig. 2-4. This type of histogram is often referred to as a mound-shaped histogram or a bell-shaped histogram. A symmetric histogram in which each class has the same frequency is called a uniform or rectangular histogram. A skewed to the right histogram has a longer tail on the right side. The histogram shown in Fig. 2-5 is skewed to the right. A skewed to the left histogram has a longer tail on the left side. The histogram shown in Fig. 2-6 is skewed to the left.

The histograms in Figures 2-4, 2-5, and 2-6 were produced by SAS.




Cumulative Frequency Distributions

A cumulative frequency distribution gives the total number of values that fall below various class boundaries of a frequency distribution.

EXAMPLE 2.11 Table 2.11 shows the frequency distribution of the contents in milliliters of a sample of 25 one-liter bottles of soda. Table 2.12 shows how to construct the cumulative frequency distribution that corresponds to the distribution in Table 2.11.

Cumulative Relative Frequency Distributions

A cumulative relative frequency is obtained by dividing a cumulative frequency by the total number of observations in the data set. The cumulative relative frequencies for the frequency distribution given in Table 2.11 are shown in Table 2.12. Cumulative percentages are obtained by multiplying cumulative relative frequencies by 100. The cumulative percentages for the distribution given in Table 2.11 are shown in Table 2.12.


An ogive is a graph in which a point is plotted above each class boundary at a height equal to the cumulative frequency corresponding to that boundary. Ogives can also be constructed for a cumulative relative frequency distribution as well as a cumulative percentage distribution.

EXAMPLE 2.12 The MINITAB-created ogive corresponding to the cumulative frequency distribution in Table 2.12 is shown in Fig. 2-7.


Stem-and-Leaf Displays

In a stem-and-leaf display each value is divided into a stem and a leaf. The leaves for each stem are shown separately. The stem-and-leaf diagram preserves the information on individual observations.

EXAMPLE 2.13 The following are the California Achievement Percentile Scores (CAT scores) for 30 seventh-grade students:

A MINITAB stem-and-leaf plot for the data is shown in Fig. 2-8. The first row represents the number 29, the second row represents the numbers 33, 35, 35, 35, and 36, etc. The first column in the MINITAB plot is a cumulative frequency that starts at both ends of the data and meets in the middle. The row that contains the median of the data is marked with parentheses around the count of observations for that row. For the rows above the median, the number in the first column is the number of items in that row plus number of items in all the rows above. Rows below the median are just the opposite.

If the stem-and-leaf is rotated 90 degrees, the shape of a histogram of the data is revealed. The numerical values that make up the data is visible in a stem-and-leaf where as they are not in a histogram.

Practice problems for these concepts can be found at:

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