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Transcript
Review – Using Standard
Deviation
Here are eight test scores from a previous Stats 201
class:
35, 59, 70, 73, 75, 81, 84, 86.
The mean and standard deviation are 70.4 and 16.7,
respectively. Work out which data points are within
a) one standard deviation from the mean i.e.
59, 70, 73, 75, 81, 84, 86
b) two standard deviations from the mean i.e.
59, 70, 73, 75, 81, 84, 86
c) three standard deviations from the mean i.e.
35, 59, 70, 73, 75, 81, 84, 86
1
Idea!
• The example suggests that there may be a
general rule which allows us to estimate the
fraction of data points which are within a
given number of standard deviations of the
mean.
2
Distribution curves
If the number of data points n is
small, one uses a small number of
class intervals and obtains a typical
histogram
3
Histogram (small n)
4
Distribution curves
If the number of data points n is
lager, one can use more subintervals
in producing a histogram for the data
set
5
Histogram (Medium n)
6
Distribution curves
If the number of data points n is very large, one
can use a corresponding large number of class
intervals and obtain a histogram which cn be
seen to approach a curve in the limit as n
increases.
In this case, it is convenient to choose the scale
on the vertical axis so that the area of each
vertical bar corresponds to the fraction of data
poins in the corresponding subinterval.
NOTE: The total area under the graph of the
histogram will therefore be 1.
7
Histogram (large n)
8
In the limit as the size of the
population increases, one
obtains a smooth curve.
• This curve was called a probability
density function in Math 112
• One obtains different curves
corresponding to different
population means and variances.
9
Normal Distributions
Shape of this curve is determined by µ and σ
– µ it’s centered, σ is how far it’s spread out.
Interpreting the Standard
Deviation
Chebyshev’s Theorem
The proportion (or fraction) of any data set lying
within K standard deviations of the mean is always
at least 1-1/K2, where K is any positive number
greater than 1.
For K=2 we obtain, at least 3/4 (75 %) of all scores
will fall within 2 standard deviations of the mean,
i.e. 75% of the data will fall between
x  2s and x  2s
11
Interpreting the Standard
Deviation
Chebyshev’s Theorem
The proportion (or fraction) of any data set lying
within K standard deviations of the mean is always
at least 1-1/K2, where K is any positive number
greater than 1.
For K=3 we obtain, at least 8/9 (89 %) of all scores
will fall within 3 standard deviations of the mean,
i.e. 89% of the data will fall between
x  3s and x  3s
12
Exercise 1
• Data collected daily at an intersection
giving the number of cars passing through.
• Mean = 375 with standard deviation 25
• Estimate the fraction of days that more than
425 cars used the intersection.
13
Exercise 1
• Data collected daily at an intersection
giving the number of cars passing through.
• Mean = 375 with standard deviation 25
• Estimate the fraction of days that more than
425 cars used the intersection.
• ANS: At least 75% lie in the interval (325,
425). Therefore, at most 25% lie outside
this interval.
14
Exercise 1
• NOTE: Assuming the data is symmetric
and therefore evenly distributed about the
mean, we can conclude that the 25% which
lie outside the interval (325,425) are evenly
distributed into 12.5% lying above 425 and
12.5% lying below 3.25. 5, 425). Therefore,
on roughly 12.5% of the days there will be
more than 425 cars using the intersection.
15
If the data is known to have a
histogram which is symmetric
about the mean and “bell
shaped”, one can improve upon
Chebyshev’s Rule
16
This Data is Symmetric, Bell
Shaped (or Normal Data)
x M
Relative
Frequency
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
17
This Data is Symmetric, Bell
Shaped (or Normal Data)
Relative
Frequency
0.5
0.4
x M
0.3
0.2
0.1
0
1
2
3
4
5
18
This Data is Symmetric, Bell
Shaped (or Normal Data)
Relative
Frequency
0.5
0.4
x M
0.3
0.2
0.1
0
1
2
3
4
5
6
19 7
The Empirical Rule
The Empirical Rule states that for bell shaped
(normal) data:
68% of all data points are within 1 standard deviations of the mean
95% of all data points are within 2 standard deviations of the mean
99.7% of all data points are within 3 standard deviations of the mean
20
The Empirical Rule
The Empirical Rule states that for bell shaped
(normal) data, approximately:
68% of all data points are within 1 standard deviations of the mean
95% of all data points are within 2 standard deviations of the mean
99.7% of all data points are within 3 standard deviations of the mean
21
Exercise 2
• Data collected daily at an intersection
giving the number of cars passing through.
• Mean = 375 with standard deviation 25
• Estimate the fraction of days that more than
425 cars used the intersection assuming the
data is bell-shaped.
22
Exercise 2
• Data collected daily at an intersection
giving the number of cars passing through.
• Mean = 375 with standard deviation 25
• Estimate the fraction of days that more than
425 cars used the intersection assuming the
data is bell-shaped.
• ANS: 2.5% (Check this!)
23
Z-Score
To calculate the number of standard
deviations a particular point is away from the
standard deviation we use the following
formula.
24
Z-Score
To calculate the number of standard
deviations a particular point is away from the
standard deviation we use the following
formula.
z
x

or
xx
z
s
The number we calculate is called the z-score
of the measurement x.
25
Example – Z-score
Here are eight test scores from a previous
Stats 201 class:
35, 59, 70, 73, 75, 81, 84, 86.
The mean and standard deviation are 70.4 and
16.7, respectively.
a) Find the z-score of the data point 35.
b) Find the z-score of the data point 73.
26
Example – Z-score
Here are eight test scores from a previous
Stats 201 class:
35, 59, 70, 73, 75, 81, 84, 86.
The mean and standard deviation are 70.4 and
16.7, respectively.
a) Find the z-score of the data point 35.
z = -2.11
b) Find the z-score of the data point 73.
z = 0.16
27
Interpreting Z-scores
The further away the z-score is from zero the
more exceptional the original score.
Values of z less than -2 or greater than +2 can
be considered exceptional or unusual (“a
suspected outlier”).
Values of z less than -3 or greater than +3 are
often exceptional or unusual (“a highly
suspected outlier”).
28
Example: Aptitude tests
Before being accepted into a manufacturing job, one
must complete two aptitude tests. Your score on the
tests will decide whether you will be in management
or whether you will work on the factory floor. One
test is a manual dexterity test, the other is a statistics
test. The manual dexterity test (out of 10) has a mean
of 6 and a standard deviation of 1. The statistics test
(out of 50) has a mean of 25 with a standard deviation
of 3. Your score is 7/10 on the manual dexterity test,
and a 34/50 on the statistics test. In which test were
29
you exceptional?
Example: Aptitude tests
The problem with comparing the two test scores
stems from the fact that the tests are on two
different scales.
If we are going to do meaningful comparisons,
then we must somehow, standardize the scores.
30
Answer
Calculate the z-score for the two tests.
– Z-score of Man. Dex.
= (7-6)/1 = 1
– Z-score of Stats.
= (34-25)/3 = 3
Your score on the stats test was exceptionally
high (3 standard deviations above the mean.
31