1.6. Numerical summary of grouped data

1.6.1. Mean for data with multiple-observation values

Suppose that a data set contains values occurring with frequencies, respectively.

1. For a population of observations, so that

The mean is

2. For a sample of observations, so that

The mean is

The arithmetic is most conveniently set out in tabular form.

Example:

The score for the sample of 25 students on a 5-point quiz are shown below. Find the mean.

Score	Frequency
0 1 2 3 4 5	1 2 6 12 3 1

Solution:

We must find . We need a column to display the computation of the quantity (Table 1.3):

Table 1.3

Score	Frequency
0 1 2 3 4 5	1 2 6 12 3 1	0· 1=0 1· 2=2 2· 6=12 3· 12=36 4· 3=12 5· 1=5

In the end, .

Hence the mean of the scores

Is approximately 2.7.

1.6.2. Median for data with multiple-observation values

For an ungrouped frequency distribution, find the median by examining the cumulative frequency to locate the middle value, as shown in the next example.

Example:

The number of videocassette recorders sold per month over a two-year period is recorded below. Find the median.

Solution:

As we know the median is observation.

Since then .

To find and observations we write corresponding cumulative frequency distribution (Table 1.4).

Table1.4

Class	Number of sets sold	Frequency (month)	Cumulative frequency
1 2 3 4 5 6 7	1 2 3 4 5 6 7	3 8 5 4 2 1 1	3 11 16 20 22 23 24

The and values fall in class 3.

value=3 ; value=3.

Therefore, .

1.6.3. Mode for data with multiple-observation values

As we already know, the mode is the most frequently occurring value. A similar concept can be used when the data are available in multiple-observation form.

Example:

The following data were collected on the number of blood tests a hospital conducted for a random sample of 50 days. Find the mode.

Number of tests per day	Frequency (days)
26 27 28 29 30 31 32	5 9 12 18 5 0 1

Solution:

Since 29 days were given on 18 days (the number of tests that occurs most often), the mode is 29.