7.5 The Variance and Standard Deviation

1. Variance of Probability Distribution2. Spread 3. Standard Deviation4. Unbiased Estimate5. Sample Variance and Standard Deviation6. Alternative Definitions7. Chebychev's Inequality

1

Variance of Probability Distribution

Let X be a random variable with values x1, x2,

…, xN and respective probabilities p1, p2,…, pN. The variance of the probability distribution is 2 2 2

1 1 2 2variance .N Nx p x p x p

2

Spread Roughly speaking, the variance measures

the dispersal or spread of a distribution about its mean. The probability distribution whose histogram is drawn on the left has a smaller variance than that on the right.

3

Standard Deviation of Probability Distribution

The standard deviation of probability distribution is

2 variance , where

2 2 221 1 2 2

2 2 2

1 1 2 22

,

or .

r r

r r

x p x p x p

x f x f x f

N

4

Example Variance & Standard Deviation

Compute the variance and the standard deviation for the population of scores on a five-question quiz in the table.

0 4 1 9 2 6 3 14 4 18 5 93

60

5

Example Variance & Standard Deviation (2)

2 132Variance 2.2, 2.2 1.48

60

6

Unbiased Estimate

If the average of a statistic, if that statistic were computed for each sample, equals the associated parameter for the population, then that statistic is said to be unbiased.

7

Sample Variance and Standard Deviation

The unbiased variance for a sample is

The unbiased standard deviation for a sample is

2 2 2

1 1 2 22 .1

r rx x f x x f x x fs

n

2 .s s

8

Example Variance & Standard Deviation

Compute the sample variance and standard deviation for the weekly sales of car dealership A.

9

Example Variance & Standard Deviation (2)

2 2 22 151

2 2 2

2

[ 5 7.96 2 6 7.96 2 7 7.96 13

8 7.96 20 9 7.96 10 10 7.96 4

11 7.96 1] 1.45

1.45 1.20

s

s

5 2 6 2 7 13 8 20 9 10 10 4 11 1

52 7.96

x

10

Alternative Definitions

Two alternative definitions for variance are

and, for a binomial random variable with parameters n, p, and q,

2 2 2E X

2 .npq

11

Example Alternative Definition

Find the variance when a fair coin is tossed 5 times and X is the number of heads.

2 1 1 55

2 2 4

12

Chebychev's Inequality

Chebychev's Inequality Suppose that a probability distribution with numerical outcomes has expected value and standard deviation Then the probability that a randomly chosen outcome lies between - c and + c is at least

13

.

2

1 .c

Example Chebychev's Inequality

A drug company sells bottles containing 100 capsules of penicillin. Due to bottling procedure, not every bottle contains exactly 100 capsules. Assume that the average number of capsules in a bottle is 100 and the standard deviation is 2. If the company ships 5000 bottles, estimate the number of bottles having between 95 and 105 capsules, inclusive.

14

Example Chebychev's Inequality (2)

The number of bottles containing between 100 - 5 and 100 + 5 capsules can be estimated using Chebychev's Inequality.

2 2

100

2

5

21 1 .845

c

c

15

On average, at least 84% will contain between 95 and 105 capsules. 4200 bottles.

Summary Section 7.5 - Part 1

The variance of a random variable is the sum of the products of the square of each outcome's distance from the expected value and the outcome's probability. The variance of the random variable X can also be computed as E(X 2) - [E(X)] 2.

A binomial random variable with parameters n and p has expected value np and variance np(1 - p).

16

Summary Section 7.5 - Part 2

The square root of the variance is called the standard deviation.

Chebychev's Inequality states that the probability that an outcome of an experiment is within c units of the mean is at least , where is the standard deviation.

2

1 c

17