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Lesson 3 - 2
Measures of Dispersion (Spread)
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
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 overestimates 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
Example 1
Which of the following measures of spread are
resistant?
1. Range
Not Resistant
2. Variance
Not Resistant
3. Standard Deviation
Not Resistant
Example 2
Given the following set of data:
70,
28,
56,
63,
56,
35,
51,
50,
48,
58,
46,
46,
48,
62,
39,
69,
53,
45,
56,
53,
52,
60,
32,
70,
66,
38,
44,
33,
48,
73,
60,
54,
36,
45,
51,
55,
49,
51,
44,
52
What is the range?
73-28 = 45
What is the variance?
117.958
What is the standard deviation?
10.861
Empirical Rule
μ ± 3σ
μ ± 2σ
μ±σ
99.7%
95%
68%
34%
0.15%
34%
13.5%
13.5%
2.35%
μ - 3σ
μ - 2σ
2.35%
μ-σ
μ
μ+σ
μ + 2σ
μ + 3σ
0.15%
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 - 3σ
μ - 2σ
μ-σ
μ
μ+σ
μ + 2σ
μ + 3σ
Example 3
Which of the following measures of spread are
resistant?
1. Range
Not Resistant
2. Variance
Not Resistant
3. Standard Deviation
Not Resistant
Example 2
Given the following set of data:
70,
28,
56,
63,
56,
35,
51,
50,
48,
58,
46,
46,
48,
62,
39,
69,
53,
45,
56,
53,
52,
60,
32,
70,
66,
38,
44,
33,
48,
73,
60,
54,
36,
45,
51,
55,
49,
51,
44,
52
What is the variance?
117.958
What is the standard deviation?
10.861
If this was a population instead of a
sample, what is the standard deviation?
10.724
Example 3
Compare the Empirical Rule and Chebyshev’s Inequality
Empirical Rule
Chebyshev
μ±σ
68%
n/a
μ ± 2σ
95%
> 75%
μ ± 3σ
99.7%
> 88.9%
Summary and Homework
• Summary
– 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 148-155: 11, 14, 22, 23, 35, 39, 40, 43,
45, 51
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