Measures of Central Tendency

Measures of central tendency provide information about a “typical” observation in the data. Such measures are usually computed from sample data, rather than from population data.
STATISTIC AND PARAMETER
A statistic is a descriptive measure computed from a sample of data. A  parameter is a descriptive measure computed from an entire population of data.

The Mean and the Median

The most common measures of central tendency is the arithmetic mean, usually referred to simply as the “mean” or the “average”.

ARITHMETIC MEAN

The arithmetic mean of a set of data is the sum of the data values divided by the number of observations. If the data set is from a sample, then the sample mean,
                 

n = sample size.
Example: People across the country participate in various walks to raise funds for needed research. Recently, national and local business sponsored a Suntan G. Komen Breast Cancer Foundation 5K “Race for the Cure” in Daytona Beach, Florida. Participants could either run or walk the race. The time to complete the race appeared on a large display as each participant enthusiastically crossed the finish line. The times (rounded to the nearest minite) for a random sample of 5 participants who walked this race are:
                         45    53    45   50   48
Compute the mean times for this random sample of participants.
Solution:
The sample mean time is easily found to be

Mean for Multiple-Observation Values

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

Example: A random sample of 50 personal property insurance policies showed the following number of claims over the past two years.      

No. of claims	No. of policies
0	21
1	13
2	5
3	4
4	2
5	3
6	2
Sum	50

Calculate the mean number of claims.

Solution:

Table 1:

No. of claims (mi)	No. of policies (fi)	fi*mi
0	21	0
1	13	13
2	5	10
3	4	12
4	2	8
5	3	15
6	2	12
Sum	50	70

Hence, mean number of claims= 

Problem: Calculate arithmetic mean of data of days to maturity 40 short-term investments.
Table 2: Data of days to maturity 40 short-term investments.

Class interval	No. of investments
30—39	3
40—49	1
50—59	8
60—69	10
70—79	7
80—89	7
90—99	4
Total	40

Solution:

Table 3: Calculation of arithmetic mean of data of days to maturity 40 short-term investments.

Class interval	Midpoint (mi)	Frequency (fi)	fi*mi
30—39	34.5	3	103.5
40—49	44.5	1	44.5
50—59	54.5	8	436
60—69	64.5	10	645
70—79	74.5	7	521.5
80—89	84.5	7	591.5
90—99	94.5	4	378
Total		40	2720