Histograms, Frequency Polygons, and Time Series Graphs

Learning objectives

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 $\text{RF}=\frac{f}{n}\text{.}$

For example, if three students in Mr. Ahab's English class of 40 students received from ninety to 100 percent, then f = 3, n = 40, and RF = $\frac{f}{n}$ = $\frac{3}{40}$ = 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.

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.

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.

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.

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.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals and resulting interval size, for both the x-axis and y-axis. The x-axis will show the lower and upper bound for each interval, containing the data values, whereas the y-axis will represent the frequencies of the values. Each data point represents the frequency for each interval. For example, if an interval has three data values in it, the frequency polygon will show a 3 at the upper endpoint of that interval. After choosing the appropriate intervals, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

A frequency polygon was constructed from the frequency table below.

Notice that each point represents frequency for a particular interval. These points are located halfway between the lower bound and upper bound. In fact, the horizontal axis, or x-axis, shows only these midpoint values. For the interval 49.5–59.5 the value 54.5 is represented by a point, showing the correct frequency of 5. For the interval occurring before 49.5–59.5, (as well as 39.5–49.5), the value of the midpoint, or 44.5, is represented by a point, showing a frequency of 0, since we do not have any values in that range. The same idea applies to the last interval of 99.5–109.5, which has a midpoint of 104.5 and correctly shows a point representing a frequency of 0. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Frequency polygons are useful for comparing distributions. This comparison is achieved by overlaying the frequency polygons drawn for different data sets.

We will construct an overlay frequency polygon comparing the scores from example4 with the students’ final numeric grades.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon, we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By using the axes in that way, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

References

Bureau of Labor Statistics, U.S. Department of Labor. (n.d.). Consumer price index. Retrieved from https://www.bls.gov/cpi/

CIA World Factbook. (n.d.). Demographics: Children under the age of 5 years underweight. Available at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en

Centers for Disease Control and Prevention. (n.d.). Overweight and obesity: Adult obesity facts. Available online http://www.cdc.gov/obesity/data/adult.html

Food and Agriculture Organization of the United Nations. (n.d.). Food security statistics. Retrieved from http://www.fao.org/economic/ess/ess-fs/en/

General Register Office for Scotland. Births time series data. (2013). Retrieved from http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html

Gunst, R., and Mason, R. (1980). Regression analysis and its application: A data-oriented approach. Boca Raton, FL: CRC Press.

Sandbox Networks. (2007). Presidents. Available online at http://www.factmonster.com/ipka/A0194030.html

Scholastic. (2013). Timeline: Guide to the U.S. presidents. Retrieved from http://www.scholastic.com/teachers/article/timeline-guide-us-presidents

World Bank Group. (2013). DataBank: CO2 emissions (kt). Retrieved from http://databank.worldbank.org/data/home.aspx

Chapter Review

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually go on the y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

Homework

Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each.

View original figure (external; image not reproduced in this text edition)

Original figure description: A histogram showing the results of a survey. Of 111 respondents, 5 own 1 t-shirt costing more than $19, 17 own 2, 23 own 3, 39 own 4, 25 own 5, 2 own 6, and no respondents own 7.

Glossary

frequency

the number of times a value of the data occurs

histogram

a graphical representation in x-y form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency; the graph consists of contiguous rectangles

relative frequency

the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes