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

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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

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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.

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Standard Deviation of Probability Distribution

The standard deviation of probability distribution is

2 variance , where

2 2 221 1 2 2

2 2 21 1 2 22

,

or .

r r

r r

x p x p x p

x f x f x fN

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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

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Example Variance & Standard Deviation (2)

2 132Variance 2.2, 2.2 1.4860

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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.

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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

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Example Variance & Standard Deviation

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

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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 11 152

7.96

x

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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

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Example Alternative Definition

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

2 1 1 552 2 4

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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

.

21 .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.

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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

1002

5

21 1 .845

c

c

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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).

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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.

21 c

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