Download Empirical Rule

Empirical Rule
For bell-shaped data sets:
Approximately 68% of the observations fall
within 1 standard deviation of the mean
Approximately 95% of the observations fall
within 2 standard deviations of the mean
Approximately 100% of the observations fall
within 3 standard deviations of the mean
Example: IQ Score
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, 1 of 14
Parameter and Statistic
Agresti/Franklin Statistics, 2 of 14
Five summary statistics
A parameter is a numerical summary of
the population (such as population mean)
A statistic is a numerical summary of a
sample taken from a population (such as
sample mean)
Minimum =1
1st quartile = 3
Median =10
3rd quartile=12
Maximum =15
Boxplot is graphical display of fivesummary statistics
Agresti/Franklin Statistics, 3 of 14
Agresti/Franklin Statistics, 4 of 14
B oxplot of SUGA Rg
16
Boxplot
max
14
Q3
12
10
g
R
A 8
G
U
S
Q2=median
mean
6
4
2
Q1
min
0
Agresti/Franklin Statistics, 5 of 14
Agresti/Franklin Statistics, 6 of 14
1
Comparison using boxplots
Minitab output
B o xp l ot of We e k 1 , W e ek 2, We e k 3
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.
9
8
7
a
t
a
D
6
5
4
Week 1
Agresti/Franklin Statistics, 7 of 14
Week 2
Week 3
Agresti/Franklin Statistics, 8 of 14
Interpreting the results
Skewed to the right
Symmetric
Skewed to the left
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, 9 of 14
Z-Score
The z-score for an observation measures how far
an observation is from the mean in standard
deviation units
z=
observatio n - mean
standard deviation
An observation in a bell-shaped distribution is a
potential outlier if its z-score < -3 or > +3
Agresti/Franklin Statistics, 11 of 14
Agresti/Franklin Statistics, 10 of 14
Example: Converting to z-score
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
2
Inverse problem
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)
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
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