Download Empirical Rule

Empirical Rule
For bell-shaped data sets:
z
Approximately 68% of the observations fall
within 1 standard deviation of the mean
z
Approximately 95% of the observations fall
within 2 standard deviations of the mean
z
Approximately 100% of the observations fall
within 3 standard deviations of the mean
Agresti/Franklin Statistics, 1 of 14
Example: IQ Score
z
z
IQ scores of normal adults on the Weschler
test have a bell-shaped distribution with mean
100 and a standard deviation of 15. What
percentage of adults have IQ between 70 and
130?
Empirical Rule shows that 95% of adults have
IQ between two standard deviations from the
mean, which is between 70 and 130.
Agresti/Franklin Statistics, 2 of 14
Parameter and Statistic
z
A parameter is a numerical summary of
the population (such as population mean)
z
A statistic is a numerical summary of a
sample taken from a population (such as
sample mean)
Agresti/Franklin Statistics, 3 of 14
Five summary statistics
z
z
z
z
z
z
Minimum =1
1st quartile = 3
Median =10
3rd quartile=12
Maximum =15
Boxplot is graphical display of fivesummary statistics
Agresti/Franklin Statistics, 4 of 14
Boxplot
Agresti/Franklin Statistics, 5 of 14
Boxplot of SUGARg
16
max
14
Q3
12
Q2=median
SUGARg
10
8
mean
6
4
2
Q1
min
0
Agresti/Franklin Statistics, 6 of 14
Comparison using boxplots
Example: Your company makes plastic
pipes, and you are concerned about the
consistency of their diameters. You
measure ten pipes a week for three
weeks. Create a boxplot to examine the
distributions.
Agresti/Franklin Statistics, 7 of 14
Minitab output
Boxplot of Week 1, Week 2, Week 3
9
8
Data
7
6
5
4
Week 1
Week 2
Week 3
Agresti/Franklin Statistics, 8 of 14
Skewed to the right
Symmetric
Skewed to the left
Agresti/Franklin Statistics, 9 of 14
Interpreting the results
z
z
z
z
Tip To see precise information for Q1, median, Q3, interquartile
range, whiskers, and N, hover your cursor over any part of the
boxplot. The boxplot shows:
Week 1 median is 4.985, and the interquartile range is 4.4525
to 5.5575.
Week 2 median is 5.275, and the interquartile range is 5.08 to
5.6775. An outlier appears at 7.0.
Week 3 median is 5.43, and the interquartile range is 4.99 to
6.975. The data are positively skewed.
Conclusion: The medians for the three weeks are similar.
However, during Week 2, an abnormally wide pipe was created,
and during Week 3, several abnormally wide pipes were
created.
Agresti/Franklin Statistics, 10 of 14
Z-Score
z
The z-score for an observation measures how far
an observation is from the mean in standard
deviation units
observatio n - mean
z=
standard deviation
z
An observation in a bell-shaped distribution is a
potential outlier if its z-score < -3 or > +3
Agresti/Franklin Statistics, 11 of 14
Example: Converting to z-score
z
z
Scores on a test have a mean of 75 and
a standard deviation of 10. Bob has a
score of 90. Convert Bob’ score to a zscore.
Bob’s z-score=(90-75)/10=1.5 which
means that Bob’s score is 1.5
standard deviation higher than the
mean.
Agresti/Franklin Statistics, 12 of 14
Inverse problem
z
z
z
If Bob’s score is 1.5 standard deviation
higher than the mean, what is Bob’s
score for the previous problem.
Denote Bob’s score=x,
then 1.5=(x-75)/10
so x=1.5(10)+75=90.
Inverse formula: x=z(s)+mean
Agresti/Franklin Statistics, 13 of 14
2.6 How are descriptive
summaries misused? (read)
z
z
Figure 2.18, page 75
HW4:
• read section 3.2
• problems 2.57, 2.62, 2.63, 2.65, 2.67, 2.68,
2.69, 2.71, 2.72
Agresti/Franklin Statistics, 14 of 14