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So far, we have discussed frequency distribution tables for qualitative data and quantitative data that can be treated as qualitative data. In this section, we discuss frequency distribution tables for quantitative data.

Let be a set of quantitative data values. To construct a frequency distribution table for this data set, we follow the steps given below.

1 Step 1. Find the range of the data that is defined as(2.3.1)

2 Step 2. Divide the data set into an appropriate number of classes. The classes are also sometimes called categories, cells, or bins. There are no hard and fast rules to determine the number of classes. As a rule, the number of classes, say , should be somewhere between 5 and 20. However, Sturges's formula is often used, given by(2.3.2) or(2.3.3) where is the total number of data points in a given data set and log denotes the log to base 10. The result often gives a good estimate for an appropriate number of intervals. Note that since , the number of classes, should always be a whole number, the reader may have to round up or down the value of obtained when using either equation (2.3.2) or (2.3.3).

3 Step 3. Determine the width of classes as follows:(2.3.4) The class width should always be a number that is easy to work with, preferably a whole number. Furthermore, this number should be obtained only by rounding up (never by rounding down) the value obtained when using equation (2.3.4).

4 Step 4. Finally, preparing the frequency distribution table is achieved by assigning each data point to an appropriate class. While assigning these data points to a class, one must be particularly careful to ensure that each data point be assigned to one, and only one, class and that the whole set of data is included in the table. Another important point is that the class at the lowest end of the scale must begin at a number that is less than or equal to the smallest data point and that the class at the highest end of the scale must end with a number that is greater than or equal to the largest data point in the data set.

Example 2.3.4 (Rod manufacturing) The following data give the lengths (in millimeters) of 40 randomly selected rods manufactured by a company:

145	140	120	110	135	150	130	132	137	115
142	115	130	124	139	133	118	127	144	143
131	120	117	129	148	130	121	136	133	147
147	128	142	147	152	122	120	145	126	151

Prepare a frequency distribution table for these data.

Solution: Following the steps described previously, we have the following:

1 Range

2 Number of classes

3 Class width

The six classes used to prepare the frequency distribution table are as follows: 110–under 117, 117–under 124, 124–under 131, 131–under 138, 138–under 145, 145–152.

Note that in the case of quantitative data, each class is defined by two numbers. The smaller of the two numbers is called the lower limit and the larger is called the upper limit. Also note that except for the last class, the upper limit does not belong to the class. For example, the data point 117 will be assigned to class two and not class one. Thus, no two classes have any common point, which ensures that each data point will belong to one and only one class. For simplification, we will use mathematical notation to denote the classes above as

Here, the square bracket symbol “[“ implies that the beginning point belongs to the class, and the parenthesis”)” implies that the endpoint does not belong to the class. Then, the frequency distribution table for the data in this example is as shown in Table 2.3.4.

Table 2.3.4 Frequency table for the data on rod lengths.

		Frequency	Relative		Cumulative
Classes	Tally	or count	frequency	Percentage	frequency
	///	3		7.5	3
	///// //	7		17.5	10
	///// ///	8		20.0	18
	///// //	7		17.5	25
	///// /	6		15.0	31
	///// ////	9		22.5	40
Total		40	1	100

The same frequency distribution table can be obtained by using MINITAB as follows: