Practice problems for these concepts can be found at:

- Organizing Data Solved Problems for Beginning Statistics
- Organizing Data Supplementary Problems for Beginning Statistics

The frequency distribution given in the Table 2.5 is composed of five *classes*.

The classes are: 80–94, 95–109, 110–124, 125–139, and 140–154. Each class has a *lower class limit* and an *upper class limit*. The lower class limits for this distribution are 80, 95, 110, 125, and 140. The upper class limits are 94, 109, 124, 139, and 154.

If the lower class limit for the second class, 95, is added to the upper class limit for the first class, 94, and the sum divided by 2, the *upper boundary* for the first class and the *lower boundary* for the second class are determined. Table 2.6 gives all the boundaries for Table 2.5.

If the lower class limit is added to the upper class limit for any class and the sum divided by 2, the *class mark* for that class is obtained. The class mark for a class is the midpoint of the class and is sometimes called the *class midpoint* rather than the class mark. The class marks for Table 2.5 are shown in Table 2.6.

The difference between the boundaries for any class gives the *class width* for a distribution. The class width for the distribution in Table 2.5 is 15.

When forming a frequency distribution, the following general guidelines should be followed:

*The number of classes should be between 5 and 15**Each data value must belong to one, and only one, class.**When possible, all classes should be of equal width.*

**EXAMPLE 2.7** Group the following weights into the classes 100 to under 125, 125 to under 150, and so forth:

The weights 111 and 120 are tallied into the class 100 to under 125. The weights 127, 129, 130, 145, and 145 are tallied into the class 125 to under 150 and so forth until the frequencies for all classes are found. The frequency distribution for these weights is given in Table 2.7

When a frequency distribution is given in this form, the class limits and class boundaries may be considered to be the same. The class marks are 112.5, 137.5, 162.5, 187.5, 212.5, and 237.5. The class width is 25.

**EXAMPLE 2.8** The price for 500 aspirin tablets is determined for each of twenty randomly selected stores as part of a larger consumer study. The prices are as follows:

Suppose we wish to group these data into seven classes. Since the maximum price is 3.15 and the minimum price is 2.50, the spread in prices is 0.65. Each class should then have a width equal to approximately 1/7 of 0.65 or .093. There is a lot of flexibility in choosing the classes while following the guidelines given above. Table 2.8 shows the results if a class width equal to 0.10 is selected and the first class begins at the minimum price.

The frequency distribution might also be given in a form such as that shown in Table 2.9. The two different ways of expressing the classes shown in Tables 2.8 and 2.9 will result in the same frequencies.

### Single-Valued Classes

If only a few unique values occur in a set of data, the classes are expressed as a single value rather than an interval of values. This typically occurs with discrete data but may also occur with continuous data because of measurement constraints.

**EXAMPLE 2.9** A quality technician selects 25 bars of soap from the daily production. The weights in ounces of the 25 bars are as follows:

Since only six unique values occur, we will use single-valued classes. The weight 4.72 occurs twice, 4.73 occurs once, 4.74 occurs six times, 4.75 occurs nine times, 4.76 occurs twice, and 4.77 occurs five times. The frequency distribution is shown in Table 2.10.

Practice problems for these concepts can be found at:

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