Section 3.2 Measures of Dispersion

1. Range

2. Variance

3. Standard deviation

4. Empirical Rule for bell shaped distributions

5. Chebyshev’s Inequality for any distribution

3-1

Range

The range of a set of data is the difference between the maximum value and the minimum value.Range = (maximum value) – (minimum value)

EXAMPLE

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Find the range.

Range = 43 – 5

= 38 minutes

The population variance is the sum of squared deviations about the population mean divided by the number of observations in the population, N.

That is it is the mean of the sum of the squared deviations about the population mean.

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Variance

The population variance is symbolically represented by σ2 (lower case Greek sigma squared).

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EXAMPLE Population Variance

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Compute the population variance of this data. Recall that

17424.85714

7

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xi μ xi – μ (xi – μ)2

23 24.85714 -1.85714 3.44898

36 24.85714 11.14286 124.1633

23 24.85714 -1.85714 3.44898

18 24.85714 -6.85714 47.02041

5 24.85714 -19.8571 394.3061

26 24.85714 1.142857 1.306122

43 24.85714 18.14286 329.1633

902.8571 2

ix 2

2 902.8571

7ix

N

129.0 minutes2

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The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.

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EXAMPLE Sample Variance

For the travel time data assume we obtained the following simple random sample: 5, 36, 26.

Compute the sample variance travel time.

Travel Time, xi Sample Mean, Deviation about the Mean,

Squared Deviations about the Mean,

5 22.333 5 – 22.333

= -17.333

(-17.333)2 = 300.432889

36 22.333 13.667 186.786889

26 22.333 3.667 13.446889

xix x 2

ix x

2500.66667ix x

2

2 500.66667

1 3 1

ix xs

n

250.333 square minutes

3-9

Standard Deviation

The standard deviation of a set of sample values is a measure of variation of values about the mean.

Population standard deviation:

= square root of the population variance

Sample standard deviation: s

= square root of the sample variance, so that

2s s3-11

EXAMPLE Population Standard Deviation

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Compute the population standard deviation of this data.

Recall, from the last objective that σ2 = 129.0 minutes2. Therefore,

2 902.857111.4 minutes

7

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EXAMPLE Sample Standard Deviation

Recall the sample data 5, 26, 36 results in a sample variance of

2

2 500.66667

1 3 1

ix xs

n

250.333 square minutes

Use this result to determine the sample standard deviation.

2 500.66666715.8 minutes

3 1s s

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1.50 0.79 1.01 1.66 0.94 0.672.53 1.20 1.46 0.89 0.95 0.901.88 2.94 1.40 1.33 1.20 0.843.99 1.90 1.00 1.54 0.99 0.350.90 1.23 0.92 1.09 1.72 2.00

3.50 0.00 0.38 0.43 1.82 3.040.00 0.26 0.14 0.60 2.33 2.541.97 0.71 2.22 4.54 0.80 0.500.00 0.28 0.44 1.38 0.92 1.173.08 2.75 0.36 3.10 2.19 0.23

Wait Time at Wendy’s

Wait Time at McDonald’s

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EXAMPLE Comparing Standard Deviations

EXAMPLE Comparing Standard Deviations

Determine the standard deviation waiting time for Wendy’s and McDonald’s.

Which is the better company in terms of waiting times?

3-15

EXAMPLE Comparing Standard Deviations

Determine the standard deviation waiting time for Wendy’s and McDonald’s.

Sample standard deviation for Wendy’s:

0.738 minutes

Sample standard deviation for McDonald’s:

1.265 minutes

3-16

For many observations – especially if their histogram is bell-shaped

1. Roughly 68% of the observations in the list lie within 1 standard deviation from the average

2. And 95% of the observations lie within 2 standard deviations from the average

AverageAve-s.d. Ave+s.d.

68%

95%

Ave-2s.d. Ave+2s.d.

The empirical rule for bell shaped distributions

3-18

The Empirical Rule

The Empirical Rule

The Empirical Rule

EXAMPLE Using the Empirical RuleThe following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.

41 48 43 38 35 37 44 44 4462 75 77 58 82 39 85 55 5467 69 69 70 65 72 74 74 7460 60 60 61 62 63 64 64 6454 54 55 56 56 56 57 58 5945 47 47 48 48 50 52 52 53

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(a)Compute the population mean and standard deviation.

(b) Draw a histogram to verify the data is bell-shaped.

(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.

(d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.

(e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1

(use the raw data directly, not the empirical rule for this question. See how close the empirical rule approximation was!)

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(a) Using a TI-83 plus graphing calculator or Excel, we find

(b)

7.11 and 4.57

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22.3 34.0 45.7 57.4 69.1 80.8 92.5

(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1.

(c) According to the Empirical Rule, 99.7% of the patients that have serum HDL within 3 standard deviations of the mean.

(d) 13.5% + 34% + 34% = 81.5% of patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.

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Empirical rule for any shape distribution• Chebyshev’s Inequality

3-26

EXAMPLE Using Chebyshev’s Theorem

Using the data from the previous example, use Chebyshev’s Theorem to

(a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean.

(b) determine the actual percentage of patients that have serum HDL between 34 and 80.8.

2

11 100% 88.9%

3

2

11 100% 75%

2

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