### Introduction

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets.

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is more or less a number line, labeled with what the data represents, for example, distance from your home to school. The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The shape of the data refers to the shape of the distribution, whether normal, approximately normal, or skewed in some direction, whereas the center is thought of as the middle of a data set, and the spread indicates how far the values are dispersed about the center. In a skewed distribution, the mean is pulled toward the tail of the distribution.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. Remember, frequency is defined as the number of times an answer occurs. If

*f*= frequency,*n*= total number of data values (or the sum of the individual frequencies), and*RF*= relative frequency,

then

For example, if three students in Mr. Ahab’s English class of 40 students received from 90 to 100 percent, then *f* = 3, *n* = 40, and *RF* = $$ = $\frac{3}{40}$ = 0.075. Thus, 7.5 percent of the students received 90 to 100 percent. Ninety to 100 percent is a quantitative measures.

**To construct a histogram**, first decide how many **bars** or **intervals**, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The width of each bar is also referred to as the bin size, which may be calculated by dividing the range of the data values by the desired number of bins (or bars). There is not a set procedure for determining the number of bars or bar width/bin size; however, consistency is key when determining which data values to place inside each interval.

### Example 2.9

The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players; the heights are **continuous** data since height is measured:

60, 60.5, 61, 61, 61.5,

63.5, 63.5, 63.5,

64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5,

66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5,

68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5,

70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71,

72, 72, 72, 72.5, 72.5, 73, 73.5,

74

The smallest data value is 60, and the largest data value is 74. To make sure each is included in an interval, we can use 59.95 as the smallest value and 74.05 as the largest value, subtracting and adding .05 to these values, respectively. We have a small range here of 14.1 (74.05 – 59.95), so we will want a fewer number of bins; let’s say eight. So, 14.1 divided by eight bins gives a bin size (or interval size) of approximately 1.76.

### NOTE

We will round up to two and make each bar or class interval two units wide. Rounding up to two is a way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are as follows:

- 59.95
- 59.95 + 2 = 61.95
- 61.95 + 2 = 63.95
- 63.95 + 2 = 65.95
- 65.95 + 2 = 67.95
- 67.95 + 2 = 69.95
- 69.95 + 2 = 71.95
- 71.95 + 2 = 73.95
- 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the *x*-axis and relative frequency on the *y*-axis:

Interval | Frequency | Relative Frequency |
---|---|---|

59.95–61.95 | 5 | 5/100 = 0.05 |

61.95–63.95 | 3 | 3/100 = 0.03 |

63.95–65.95 | 15 | 15/100 = 0.15 |

65.95–67.95 | 40 | 40/100 = 0.40 |

67.95–69.95 | 17 | 17/100 = 0.17 |

69.95–71.95 | 12 | 12/100 = 0.12 |

71.95–73.95 | 7 | 7/100 = 0.07 |

73.95–75.95 | 1 | 1/100 = 0.01 |

Construct a histogram and calculate the width of each bar or class interval. Use six bars on the histogram. The following data are the shoe sizes of 50 male students; the sizes are continuous data since shoe size is measured:

9, 9, 9.5, 9.5, 10, 10, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5,

11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5,

12, 12, 12, 12, 12, 12, 12, 12.5, 12.5, 12.5, 12.5, 14

### Example 2.10

The following data are the number of books bought by 50 part-time college students at ABC College; the number of books is **discrete data** since books are counted:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

2, 2, 2, 2, 2, 2, 2, 2, 2, 2,

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,

4, 4, 4, 4, 4, 4,

5, 5, 5, 5, 5,

6, 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Calculate the width of each bar/bin size/interval size.

The smallest data value is 1, and the largest data value is 6. To make sure each is included in an interval, we can use 0.5 as the smallest value and 6.5 as the largest value by subtracting and adding 0.5 to these values. We have a small range here of 6 (6.5 – 0.5), so we will want a fewer number of bins; let’s say six this time. So, six divided by six bins gives a bin size (or interval size) of one.

Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. In the previous example, we added and subtracted .05, while this time, we added and subtracted .5. Given a data set, you will be able to determine what is appropriate and reasonable.

The following histogram displays the number of books on the *x*-axis and the frequency on the *y*-axis:

### Using the TI-83, 83+, 84, 84+ Calculator

Go to Appendix G. There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example 2.10.

- Press Y=. Press CLEAR to delete any equations.
- Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
- Into L1, enter 1, 2, 3, 4, 5, 6. Note that these values represent the numbers of books.
- Into L2, enter 11, 10, 16, 6, 5, 2. Note that these numbers represent the frequencies for the numbers of books.
- Press WINDOW. Set Xmin = .5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1. The window settings are chosen to accurately and completely show the data value range and the frequency range.
- Press second Y=. Start by pressing 4:Plotsoff ENTER.
- Press second Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the third picture (histogram). Press ENTER.
- Arrow down to Xlist: Enter L1 (2
^{nd}1). Arrow down to Freq. Enter L2 (second 2). - Press GRAPH.
- Use the TRACE key and the arrow keys to examine the histogram.

The following data are the number of sports played by 50 student athletes; the number of sports is discrete data since sports are counted:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,

3, 3, 3, 3, 3, 3, 3, 3

### Example 2.11

Using this data set, construct a histogram.

Number of Hours My Classmates Spent Playing Video Games on Weekends | ||||
---|---|---|---|---|

9.95 | 10 | 2.25 | 16.75 | 0 |

19.5 | 22.5 | 7.5 | 15 | 12.75 |

5.5 | 11 | 10 | 20.75 | 17.5 |

23 | 21.9 | 24 | 23.75 | 18 |

20 | 15 | 22.9 | 18.8 | 20.5 |

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram:

22, 35, 15, 26, 40, 28, 18, 20, 25, 34, 39, 42, 24, 22, 19, 27, 22, 34, 40, 20, 38, 28

### Collaborative Exercise

Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think would be appropriate. You may want to experiment with the number of intervals.