Download Slides 1-10 Descriptive Statistics

Example
Given the data below, complete the following summary statistics table. (Data are in ascending
order): 10.0, 10.5, 12.2, 13.9, 13.9, 14.1, 14.7, 14.7, 15.1, 15.3, 15.9, 17.7, 18.5
Count
Average
Median
Variance
Standard deviation
Minimum
Maximum
Range
Lower quartile
Upper quartile
Interquartile range
Sum
Variable X
13
14.3462
14.7
5.94936
2.43913
10.0
18.5
8.5
13.9
15.3
1.4
186.5
Box-and-Whisker Plot
10
12
14
16
18
20
Variable X
Lower invisible line: 11.8
Upper invisible line: 17.4
1
The Empirical Rule
99.7%
95%
68%
0.15%
2.35%
13.5%
34%
34%
13.5%
2.35%
0.15%
x  3s
  3
x  2s
xs
x
xs
x  2s
  2
 
 
  2
3
2
1

0
x  3s
  3
1
2
3
2
Example
 A set of data whose histogram is bell shaped
yields a sample mean and standard deviation
of 50 and 4, respectively. Approximately what
proportion of observations





Are between 46 and 54?
Are between 42 and 58?
Are between 38 and 62?
Are less than 46?
Are less than 58?
3
Example: The Empirical Rule
A manufacturer of automobile batteries
claims that the average length of life for
its grade A battery is 60 months with a
standard deviation of 10 months.
3 cars in your family used this brand of
batteries and none of them lasted for more
than 30 months.
What do you think about the manufacturer’s
claim?
4
What statistical lesson can we learn?
“Should we scare the opposition
by announcing our mean height
or lull them by announcing our
median height?”
5
Answer: Example
 A set of data whose histogram is bell shaped
yields a sample mean and standard deviation
of 50 and 4, respectively. Approximately what
proportion of observations





Are between 46 and 54? => 68%
Are between 42 and 58? => 95%
Are between 38 and 62? => 99.7%
Are less than 46? => 16%
Are less than 58? => 84%
6
Example: The Empirical Rule
A manufacturer of automobile batteries
claims that the average length of life for
its grade A battery is 60 months with a
standard deviation of 10 months.
3 cars in your family used this brand of
batteries and none of them lasted for more
than 30 months.
What do you think about the manufacturer’s
claim?
=> Assuming the claim was true, the chance of having 3
bad batteries is (0.15%) x (0.15%) x (0.15%), i.e., very small.
This will certainly cause you to suspect the claim.
7