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# Describing and Displaying Categorical Data Study Guide (page 2)

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#### Example

In the orchestra of 62 members, 36 play a string instrument, 12 play a woodwind, ten play brass, and four play percussion. Provide a tabular display of the frequency and relative frequency distributions for the type of instruments for this orchestra.

#### Solution

Note that, although the frequencies sum properly to the total number of orchestra members, the relative frequencies actually sum to 0.99, not to 1 as indicated. The reason for this is round-off error. When dividing the category frequency by the total number of members, the result was not always an exact two-decimal value.We rounded to two decimal places. It is better to use the 1.00 as the total rather than reﬂecting the rounding error in the total.

#### Example

The orchestra is going to a special awards banquet during which dinner will be served. The hosts of the banquet need to know in advance whether the orchestra members prefer steak, ﬁsh, or pasta as their main dish. Thirty-two members choose steak, 12 choose ﬁsh, and 18 choose pasta. Provide a tabular display of the frequency and relative frequency distributions of the orchestra members' main dish choices.

## Pie Charts - Visual Displays for Categorical Data

Pie charts and bar charts are common graphical approaches to displaying data. The frequencies, relative frequencies, or percentages can be presented graphically using pie charts or bar charts. For each category of chart, percent = relative frequency × 100%.

To make a pie chart, ﬁrst draw a circle to represent the entire data set. For each category, the "slice" size is the category's relative frequency times 360 (because there are 360 ° in a circle). Each slice should be labeled with the category name. The numerical value of the frequency, relative frequency, or percentage associated with each slice should also be shown on the graph. Percentages are presented most commonly in pie charts. As an example, consider the frequency distribution of the high school orchestra members' gender. Thirty-ﬁve or 56.5% were female, and 27 or 43.5% were male. For the pie chart, the slice for females is of the pie; the remaining is for the males. See Figure 3.1.

#### Example

For the relative frequency distribution of the types of instruments played by the high school orchestra members discussed earlier in the lesson, create a pie chart.

#### Solution

First, the size of each pie must be found. For example, for the woodwinds, the slice of the pie is . After performing this calculation for each instrument type, the following pie chart can be created (see Figure 3.2).

#### Example

Create a pie chart of the frequency and relative frequency distribution of the orchestra members' meal choice for the awards banquet.

#### Solution

See Figure 3.3 for an illustration of the orchestra members' meal choices.

## Bar Charts - Visual Displays for Categorical Data

Bar charts may be used to display frequencies, relative frequencies, or percentages represented by each category in a data set. If there is only a response variable and frequencies are to be presented, a bar is used for each category and the height of the bar corresponds to the number of times that category occurs in the data set. For a relative frequency bar chart, a bar is used for each category, and the height of the bar corresponds to the proportion of times that response occurs in the data set. As an illustration, consider a relative frequency bar chart of the genders of the orchestra members.

Notice that in Figure 3.4, both the x- and y-axes are labeled. Categories are displayed on the x-axis, and an appropriate scale is used on the y-axis.

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