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Graphs, Charts, and Tables—Describing Your DataHomeland Security wishes to construct a frequency distribution showing the times until the communication systems are linked. The frequency distribution is determined as follows:
s t e p 1 Group the data into classes.
The number of classes is arbitrary, but typically will be between 5 and 20, depending on the volume of data. In this example, we have n = 72 data items.
A common method of determining the number of classes is to use the 2k Ú n guideline. We get k = 7 classes, since 27 = 128 Ú 72 and 26 = 64 6 72.
s t e p 2 Determine the class width.
W = Largest value - Smallest value
Number of classes = 1,492 - 35
7 = 208.14291225 Note that we have rounded the class width up from the minimum required value of 208.1429 to the more convenient value of 225.
s t e p 3 Define the class boundaries.
0 and under 225
225 and under 450
450 and under 675
675 and under 900
900 and under 1,125
1,125 and under 1,350 1,350 and under 1,575
These classes are mutually exclusive, all-inclusive, and have equal widths.
s t e p 4 Determine the class frequency for each class.
Time to Link Systems
(in seconds) Frequency 0 and under 225 9
225 and under 450 6
450 and under 675 12
675 and under 900 13
900 and under 1,125 14 1,125 and under 1,350 11 1,350 and under 1,575 7
This frequency distribution shows that most cities took between 450 and 1,350 seconds (7.5 and 22.5 minutes) to link their communications systems.
TRY EXERCISE 2-5 (pg. 70)
2.1 Frequency Distributions and Histograms
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Chapter 2 63are spread out from the center, the data exhibit greater variation. The examples in Figure 2.3 all have the same center but are different in terms of spread.
3. We can observe the shape of the distribution. Is it reasonably flat, is it weighted to one side or the other, is it balanced around the center, or is it bell-shaped?
BUSINESS APPLICATION
Constructing Histograms
Capital Credit Union Even for applications with small amounts of data, such as the streamed movies example, constructing grouped data frequency distributions and histo- grams is a time-consuming process. Decision makers may hesitate to try different numbers of classes and different class limits because of the effort involved and because the “best” presen- tation of the data may be missed.
Fortunately, current data analysis and statistical software, including Microsoft Excel 2016, makes the construction of histograms quick and easy. Using such software frees decision mak- ers from the more tedious and time-consuming tasks associated with computing frequencies and class widths so that they can devote more time to data analysis and interpretation.
Let’s illustrate the use of Excel 2016’s Statistic Chart capability by considering Capital Credit Union (CCU) in Mobile, Alabama. CCU recently began issuing a new credit card. Man- agers at CCU are interested in how customers use the card, so they selected a random sample of 300 customers. Data on the current credit card balance (rounded to the nearest dollar) and the gender of the 300 sampled cardholders are contained in the file named Capital. To con- struct a histogram using Excel 2016, the analyst selects the data and inserts a Statistic Chart as indicated in the instructions in Figure 2.4. Excel automatically calculates the number of classes (bins) using a bin width algorithm. The resulting chart is displayed on the same sheet with the data (see Figure 2.4), but it can be moved to another sheet if desired. Excel’s histogram for- matting employs standard notation for indicating whether a value is included in a class, with brackets [], or not included in a class, with parentheses (). For example, the class (249.00, 399.00] does not include the value 249.00 but does include the value 399.00.
Although Excel 2016 automatically selects the number of classes and the class width for the histogram, analysts can adjust these two values if they desire. Selecting the horizontal axis will open a Format Axis dialog box on the right of the Excel screen. Axis options is one of the choices available and grants the decision maker the ability to select a different bin (class) width or a different number of bins (classes). Changes to these values will result in a new histogram of the data. Analysts now have the ability to quickly see the effects of different numbers of classes or class widths on the histogram without having to spend time updating frequency tables. Excel’s Design Option on the ribbon allows for alternative displays of the histogram by changing the background color or adding frequency counts to the histogram bars.
The resulting histogram in Figure 2.4 shows that the data are centered in the class ($699, $849]. The customers vary considerably in their credit card balances, but the distri- bution is quite symmetric and somewhat bell-shaped. CCU managers must decide whether the usage rate for the credit card is sufficient to warrant the cost of maintaining the credit card accounts.
Excel Tutorial
100 (a)
(b)
(c)
200 300 400 500 600 700 800
100 200 300 400 500 600 700 800
100 200 300 400 500 600 700 800
FIGURE 2.2 Histograms Showing Different Centers
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64 Chapter 2
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Graphs, Charts, and Tables—Describing Your Data100 (a)
(b)
(c)
200 300 400 500 600 700 800
100 200 300 400 500 600 700 800
100 200 300 400 500 600 700 800
FIGURE 2.3 Histograms—
Same Center, Different Spread
Excel 2016 Instructions 1. Open file: Capital.xlsx.
2. Select the Credit Card Account Balance column.
3. On the Insert tab, in the Charts group, click the Insert Statistic Chart arrow and click Histogram.
4. Select the numbers on the Horizontal Category Axis, right mouse click and select Format Axis. Change the Number Category to Currency with 0 Decimal places.
5. Add the Chart Title to the Histogram.
FIGURE 2.4 Excel 2016 Histogram of Credit Card Balances
EXAMPLE 2-4
Frequency Histograms
NBA Player Weights The weights of 505 NBA play- ers during the 2013 season are in the file NBA Player Weight. To develop the frequency histogram of player weight using Excel 2016, perform the following steps:
s t e p 1 Open the data file.
Open the file NBA Player Weight and select the column (B2:B506) containing the 505 players’ weights.
s t e p 2 Select desired chart.
On the Insert tab in the Charts group, select the Insert Statistic Chart option arrow and then click Histogram.
s t e p 3 Apply the chart tool.
With the chart still selected, on the Design tab, in the Charts Layouts group, click the Quick Layout arrow and click Layout 2. This will place the frequency at the top of each bar. Use the Design tool to also format the histogram by adding titles and axis labels.
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2.1 Frequency Distributions and Histograms
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Chapter 2 65The formatted histogram, developed with the default classes, is shown here:
s t e p 4 Format the histogram.
Make changes in the number of classes or the class width, if desired.
Select the histogram to activate the Format Data Series feature. Under Series option, use the drop-down arrow to select Horizontal Category Axis. Under Axis options, make desired changes.
Below is a modified histogram of NBA player weights with the bin width (class width) set to 20.
With this modified histogram, NBA player weights appear to be symmetrically distributed about a center in the range of just above 217 pounds to 237 pounds:
(217, 237].
TRY EXERCISE 2-17 (pg. 73)