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Chapter 2
Descriptive Statistics
IV. Section 2-4
A. Measures of Variation
1. Range – the difference between the highest value and the lowest
value. (Maximum minus Minimum)
a. Easy to compute but only uses two numbers from a data set.
2. Deviation – The difference between the value of a data point and the
mean of the data set.
a. In a population, the deviation of x is 𝑥 − 𝜇. (Greek letter “mu”,
pronounced “moo”)
b. In a sample, the deviation of x is 𝑥 − 𝑥 (pronounced “x bar”)
c.The sum of the deviations of a set of data will always be zero.
3. Population Measures of Variance –
a. Population Variance -- The sum of the squares of the deviations,
divided by N (the number of data points in the population).
1). Find the deviations, and then square them (this makes them all
positive, so they don’t cancel each other out)
a) Add up the squared deviations, and then divide by the
number of data points.
b. Population Standard Deviation – The square root of the population
variance.
4. Sample Measures of Variance
a. Sample Variance – The sum of the squares of the deviations,
divided by n - 1 (one less than the number of data points in the
sample).
b. Sample Standard Deviation – The square root of the sample
variance.
Test Scores
88
98
84
74
23
78
66
100
52
68
4. Sample Measures of Variance
a. Sample Variance – The sum of the squares of the deviations,
divided by n - 1 (one less than the number of data points in the
sample).
b. Sample Standard Deviation – The square root of the sample
variance.
Test Scores
88
98
84
74
23
78
66
100
52
68
73.1
4. Sample Measures of Variance
a. Sample Variance – The sum of the squares of the deviations,
divided by n - 1 (one less than the number of data points in the
sample).
b. Sample Standard Deviation – The square root of the sample
variance.
Test Scores
Dev.
88
14.9
98
24.9
84
10.9
74
0.9
23
-50.1
78
4.9
66
-7.1
100
26.9
52
-21.1
68
-5.1
73.1
0
4. Sample Measures of Variance
a. Sample Variance – The sum of the squares of the deviations,
divided by n - 1 (one less than the number of data points in the
sample).
b. Sample Standard Deviation – The square root of the sample
variance.
Test Scores
Dev.
Dev.2
88
14.9
222.01
98
24.9
620.01
84
10.9
118.81
74
0.9
.81
23
-50.1
2510.01
78
4.9
24.01
66
-7.1
50.41
100
26.9
723.61
52
-21.1
445.21
68
-5.1
26.01
73.1
0
4740.9
4. Sample Measures of Variance
a. Sample Variance – The sum of the squares of the deviations,
divided by n - 1 (one less than the number of data points in the
sample).
b. Sample Standard Deviation – The square root of the sample
variance.
Test Scores
Dev.
Dev.2
88
14.9
222.01
98
24.9
620.01
84
10.9
118.81
74
0.9
.81
23
-50.1
2510.01
78
4.9
24.01
66
-7.1
50.41
100
26.9
723.61
52
-21.1
445.21
68
-5.1
26.01
73.1
0
4740.9
Variance of a population is
4740.9 divided by 10, or 474.09.
Variance of a sample is 4740.9
divided by 9, or 526.767.
Standard deviation of a
population is the square root of
474.09, or 21.7736
Standard deviation of a sample is
the square root of 526.757, or
22.9514.
Notice that STAT-Calc-1 gives you
these values!!
B. Empirical Rule
1. All symmetric bell-shaped distributions have the following
characteristics:
a. About 68% of data points will occur within one standard deviation
of the mean.
b. About 95% of data points will occur within two standard deviations
of the mean.
c. About 99.7% of data points will occur within three standard
deviations of the mean.
C. Chebychev’s Theorem
1. This applies to ANY distribution, regardless of its shape.
a. The portion of data lying with k standard deviations (k > 1) of the
1
mean is at least 1 − 2
𝑘
1) For k = 2, at least 1 – ¼ = ¾ or 75% of the data will be within 2
standard deviations of the mean.
2) For k = 3, at least 1 – 1/9 = 8/9 or 88.9% of the data will be
within 3 standard deviations of the mean.
Assignments:
Classwork:
Homework:
Pages 92-93, #1-6 All, #8-18 Evens
Problems on printed worksheet