# Population variance

The annual incomes of the five vice presidents of TMV Industries are: $125,000; $128,000; $122,000; $133,000; and $140,000. Consider this a population.

What is the population variance? The standard deviation? What is the range? What is the arithmetic mean income?

The annual incomes of officers of another firm similar to TMV Industries were also studied. The mean was $129,000 and the standard deviation $8,612. Compare the means and dispersions in the two firms.

Sample Variance The formula for the population mean is µ = SX/N. We just changed the symbols for the sample mean that is X = SX/n. Unfortunately, the conversion from the population variance to the sample variance is not as direct. It requires a change in the denominator. Instead of substituting n (number in the sample) for N (number in the population), the denominator is n 1. Thus the formula for the sample variance is:

2

2 E(X - X)

S = --------------------

N - 1

s2 is the sample variance.

X is the value of each observation in the sample.

X is the mean of the sample.

n is the number of observations in the sample.

Exercise 2

Determine the mean and the standard deviation of the following frequency distribution.

Class Frequency

0 up to 5 2

5 up to 10 7

10 up to 15 12

15 up to 20 6

20 up to 25 3

Exercise 3

Determine the mean, median, and the standard deviation.

Determine the coefficient of skewness using Pearson's method

Determine the coefficient of skewness using the software method

Listed below are the commissions earned ($000) last year by the sales representatives at the Furniture Patch, Inc.

$3.9 $5.7 $7.3 $10.6 $13.0 $13.6 $15.1 $15.8 $17.1

17.4 17.6 22.3 38.6 43.2 87.7

Exercise 4

A telephone number consists of seven digits, the first three representing the exchange. How many different telephone numbers are possible within the 537 exchange?

#### Solution Summary

This solution assists in determining the population variance.