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```Chapter
2
Descriptive Statistics
Edited by Tonya Jagoe
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The Shape of Distributions
Symmetric Distribution
• A vertical line can be drawn through the middle of
a graph of the distribution and the resulting halves
are approximately mirror images.
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The Shape of Distributions
Uniform Distribution (rectangular)
• All entries or classes in the distribution have equal
or approximately equal frequencies.
• Symmetric.
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The Shape of Distributions
Skewed Left Distribution (negatively skewed)
• The “tail” of the graph elongates more to the left.
• The mean is to the left of the median.
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The Shape of Distributions
Skewed Right Distribution (positively skewed)
• The “tail” of the graph elongates more to the right.
• The mean is to the right of the median.
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Interpreting Standard Deviation
• Standard deviation is a measure of the typical amount
an entry deviates from the mean.
• The more the entries are spread out, the greater the
standard deviation.
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
For data with a (symmetric) bell-shaped distribution, the
standard deviation has the following characteristics:
• About 68% of the data lie within one standard
deviation of the mean.
• About 95% of the data lie within two standard
deviations of the mean.
• About 99.7% of the data lie within three standard
deviations of the mean.
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
95% within 2 standard deviations
68% within 1
standard deviation
34%
34%
2.35%
2.35%
13.5%
13.5%
x s
x
xs
x  3s
x  2s
x  2s
x  3s
Notice: this just involves adding/subtracting the standard deviation to/from the
mean, just like we did for the standard deviation distribution last week.
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 68% of data lies within 1 standard deviation of the
mean… what % lies outside that interval?
100%
 68%
68% within 1
standard deviation
34%
x  3s
x  2s
x s
 32%
34%
x
xs
x  2s
x  3s
Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 32% of data lies outside 1
standard deviation of the mean…
what % lies above +1 std. dev.’s?
Below -1 std. dev.’s?
32%
 16%
2
68% within 1
standard deviation
16%
x  3s
x  2s
34%
x s
16%
34%
x
xs
x  2s
x  3s
Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 95% of data lies within 2 standard deviations of the
mean… what % lies outside that interval?
95% within 2 standard deviations
100%
 95%
 5%
x  3s
x  2s
x s
x
xs
x  2s
x  3s
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 5% of data lies outside the 2
standard deviation interval of the
mean… what % lies above +2
std. dev.’s? Below -2 std. dev.’s?
5%
 2.5%
2
95% within 2 standard deviations
2.5%
x  3s
x  2s
2.5%
x s
x
xs
x  2s
x  3s
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 99.7% of data lies within 3 standard deviations of the
mean… what % lies outside that interval?
99.7% within 3 standard deviations
100%
 99.7%
 0. 3%
x  3s
x  2s
x s
x
xs
x  2s
x  3s
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Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
If 0.3% of data lies outside the 3
standard deviation interval of the
mean… what % lies above +3
std. dev.’s? Below -3 std. dev’s?
0.3%
 0.15%
2
99.7% within 3 standard deviations
0.15%
0.15%
x  3s
x  2s
x s
x
xs
x  2s
x  3s
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Example: Using the Empirical Rule
In a survey conducted by the National Center for Health
Statistics, the sample mean height of women in the
United States (ages 20-29) was 64.3 inches, with a
sample standard deviation of 2.62 inches. Estimate the
percent of the women whose heights are between 59.06
inches and 64.3 inches.
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Solution: Using the Empirical Rule
• Because the distribution is bell-shaped (NORMAL),
you can use the Empirical Rule.
problem to
identify the
specific values
you are seeking
– circle or
highlight those.
subtract std. dev.
to find +/-1, +/-2,
& +/-3 std. dev.
values. Write all
values on diagram.
the appropriate
region.
+2.62
72.16
+2.62
69.54
+2.62
66.92
-2.62
64.3
-2.62
61.68
56.44
-2.62
59.06
%’s in the
13.5%
 34%
 47.5%
Solution: Using the Empirical Rule
64.3
59.06
• Finally, be sure to interpret your solution in the context
of the problem.
47.5% of women are between 59.06 and 64.3 inches tall.
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Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
13.5%
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
83.85%
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
81.5%
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
50%
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
6. Above 61.98?
61.68
99.85%
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
84%
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
16%
2.5%
7. Above 66.92 and below 59.06 combined?
18.5%
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
2.5%
2.5%
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
5%
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
16%
16%
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
0%
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
72.16
69.54
66.92
64.3
61.68
6. Above 61.98?
59.06
5. Below 72.16?
56.44
4. Above 64.3?
Solution: Using the Empirical Rule
What % of data falls in each interval?
1. From 59.06 to 61.68?
50%
2. From 56.44 to 66.92?
3. From 61.68 to 69.54?
7. Above 66.92 and below 59.06 combined?
8. Above 69.54 and below 59.06 combined?
9. What is the difference in the percent of data that lies above 66.92
and below 61.98?
10. What is the difference in the percent of data that lies above 59.06
and above 64.3?
47.5%