# Descriptive Statistics

## Histograms, Frequency Polygons, and Time Series Graphs

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
1. f = frequency,
2. n = total number of data values (or the sum of the individual frequencies), and
3. RF = relative frequency,
then

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 =   =  = 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.

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

### Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When a researcher records values of the same variable over an extended period of time, it is sometimes difficult for him or her to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

### 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.

