1.7.1. Less than method for writing classes

The classes in frequency distribution given in table 1.9 for the data on monthly-earning salaries for 50 employees were written as 301-400,

401-500, etc. Alternatively, we can write the classes in a frequency distribution table using the less than method. The technique for writing classes in previous topic is more commonly used for data sets that do not contain fractional values. The less than method is more appropriate when a data set contains fractional values.

Example:

The following data give the hourly wage rates for a sample of 30 employees selected from a population.

12.25 9.20 13.90 8.10 7.30 7.25 8.75

5.20 15.85 11.20 10.20 14.50 10.50 8.25

7.45 10.20 12.20 10.80 9.25 14.35 16.50

6.40 15.20 10.30 11.75 12.45 13.25 10.80

10.35 9.75

Construct a frequency distribution table. Find the relative frequency distribution table. Find the relative frequency and cumulative frequencies.

Solution:

The minimum value is data set is 5.20 and the maximum value is 16.50. Suppose we decide to group these data using six classes of equal width. Then

We round this number to a more convenient number, say 2.

Then we take 2 as the width of each class. If we start the first class at 5, the classes will be written as 5 to less than 7, 7 to less than 9, and so on as it shown in table 1.10.

Table 1.10

Hourly wage rate (in dollars)	f	Relative frequencies	Cumulative relative frequencies
5 but less than 7 7 but less than 9 9 but less than 11 11 but less than 13 13 but less than 15 15 but less than 17	2 6 10 5 4 3	2/30=0.067 6/30=0.2 10/30=0.333 5/30=0.167 4/30=0.133 3/30=0.1	2/30=0.067 8/30=0.267 18/30=0.6 23/30=0.767 27/30=0.9 30/30=1
	30	Sum=1.00	Sum=1

A histogram for frequencies can be drawn in the same way as for the data of table 1.10. (Fig.1.4; Fig.1.5; Fig.1.6)

Exercises

1. In survey of 20 patients who smoked, the following data were obtained. Each value represents the number of cigarettes the patient smoked per day. Construct a frequency distribution, using six classes.

10 8 6 14 22 13 17 19 11

9 18 14 13 12 15 15 5 11

16 11

Draw frequency, relative frequency and cumulative relative frequency histograms.

2. For 75 employees of a large department store, the following distribution for years of service was obtained. Construct histograms for frequency, relative frequency, and cumulative relative frequency.

Class limits	Frequency
1-5 6-10 11-15 16-20 21-25 26-30	21 25 15 0 8 6

3. In a study of 32 student grade point averages (GPA), the following data were obtained.

3.2 2.0 3.3 2.7 2.1 3.9 1.1 3.5 1.9

1.7 0.8 2.6 0.6 4.0 3.5 2.3 1.6 2.8

2.6 1.6 1.6 2.4 2.6 2.3 3.8 2.1 2.9

3.0 1.7 4.0 1.2 3.1

Construct histograms for frequency, relative frequency and cumulative relative frequency.

4. To determine their lifetimes, 80 randomly selected batteries were tested. The following frequency distribution was obtained. The data values are in hours given in table 1.11

Construct histograms for frequency, relative frequency and cumulative relative frequency.

Table 1.11

Class boundaries	Frequency
63.5-74.5 74.5-85.5 85.5-96.5 96.5-107.5 107.5-118.5 118.5-129.5	10 15 22 17 11 5