1. Measures of Dispersion (Spread)
Lesson 3 - 2
Measures of Dispersion (Spread)

2. Objectives Compute the range of a variable from raw data
Compute the variance of a variable from raw data
Computer the standard deviation of a variable from raw data
Use the Empirical Rule to describe data that are bell shaped
Use Chebyshev’s inequality to describe any set of data

3. Vocabulary
Range – difference between the smallest and largest data values
Variance – based on the deviation about the mean (how spread out the data is)
Population Variance – ( σ2 ) computed using (∑(xi – μ)2)/N
Sample Variance – ( s2 ) computed using (∑(xi – x)2)/((n – 1)
Biased – a statistic that consistently under-estimates or over-estimates a population parameter
Degrees of Freedom – number of observations minus the number of parameters estimated in the computation
Population Standard Deviation – square root of the population variance
Sample Standard Deviation – square root of the sample variance

4. Example 1 Which of the following measures of spread are resistant?
Range
Variance
Standard Deviation
Not Resistant
Not Resistant
Not Resistant

5. Example 2
Given the following set of data: 70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62, 45, 60, 38, 73, 45, 51, 56, 51, 46, 39, 56, 32, 44, 60, 51, 44, 63, 50, 46, 69, 53, 70, 33, 54, 55, 52 What is the range? What is the variance? What is the standard deviation?
73-28 = 45
10.861

6. Empirical Rule μ ± 3σ μ ± 2σ μ ± σ 99.7% 95% 68% 34% 34% 13.5% 13.5%
0.15%
0.15%
2.35%
2.35% μ
μ - 3σ
μ - 2σ
μ - σ
μ + σ
μ + 2σ
μ + 3σ

7. within k standard deviations of the mean
Chebyshev’s Inequality
at least 88.9%
at least 75%
Nothing
At least (1 – 1/k2)*100%, k>1
within k standard deviations of the mean μ
μ - 3σ
μ - 2σ
μ - σ
μ + σ
μ + 2σ
μ + 3σ

8. Example 3 Which of the following measures of spread are resistant?
Range
Variance
Standard Deviation
Not Resistant
Not Resistant
Not Resistant

9. Example 2
Given the following set of data: 70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62, 45, 60, 38, 73, 45, 51, 56, 51, 46, 39, 56, 32, 44, 60, 51, 44, 63, 50, 46, 69, 53, 70, 33, 54, 55, 52 What is the variance? What is the standard deviation? If this was a population instead of a sample, what is the standard deviation?
10.861
10.724

10. Example 3 Compare the Empirical Rule and Chebyshev’s Inequality
Empirical Rule Chebyshev
μ ± σ
μ ± 2σ
μ ± 3σ
68%
n/a
95%
> 75%
99.7%
> 88.9%

11. Summary and Homework Summary Homework
Sample variance is found by dividing by (n – 1) to keep it an unbiased (since we estimate the population mean, μ, by using the sample mean, x‾) estimator of population variance
The larger the standard deviation, the more dispersion the distribution has
Homework
pg : 11, 14, 22, 23, 35, 39, 40, 43, 45, 51