Download IQ Scores and the Empirical Rule

IQ Scores and the Empirical Rule
The Empirical Rule is a guide to understanding the standard
deviation and the shapes of distributions.
Suppose that a distribution is roughly symmetrical.
• Then the mean is a good measure of the location of its center.
• And the standard deviation is a good measure of its spread.
Also suppose that the distribution has a single "hump" in the center
and moderate sized tails on both sides, with no extreme outliers.
Then the Empirical Rule is a rough guide to how the observations
fall. It says that:
• About 68% of the observations fall within one standard
deviation on either side of the mean: within x ± s .
• About 95% of the observations fall within two standard
deviations on either side of the mean: within x ± 2s .
• All or almost all of the observations fall within three standard
deviations on either side of the mean: within x ± 3s .
To be specific consider 1000 IQ scores that have approximately
mean 100 and standard deviation 15. It is known that IQ scores
come close to meeting the requirements above. Then...
• x ± s means the interval 100 ± 15 or (85, 115).
This interval should contain about 680 IQ scores.
• x ± 2s means the interval 100 ± 30 or (70, 130).
This interval should contain about 950 IQ scores.
• x ± 3s means the interval 100 ± 45 or (55, 145).
This interval should contain very nearly 1000 IQ scores.
Here is a Minitab description of 1000 IQ scores:
Descriptive Statistics: IQ
Variable
IQ
N
1000
Variable
IQ
Minimum
46.00
Mean
100.37
Median
101.00
Maximum
145.00
TrMean
100.49
Q1
91.00
StDev
14.84
SE Mean
0.47
Q3
110.00
Notice that the mean is very nearly 100 and the standard deviation
is very nearly 15 as is typical of IQ scores.
Also, as we see below, the distribution of the IQ scores is
approximately symmetrical, so it is no surprise that the mean and
the median are approximately equal.
The distribution table is as follows:
Interval
Frequency
Percent
-------------------------------Below 55
3
0.3
55- 69
24
2.4
|
70- 84
123
12.3
|
|
85- 99
341
34.1 | 69.6 | 95.9 | 99.7
100-114
355
35.5 |
|
|
115-129
130
13.0
|
|
130-144
15
1.5
|
145 and Above
0
0.0
Here
• 69.6% of the observations fall within (85, 114)
• 95.5% of the observations fall within (70, 129)
• 99.7% of the observations fall within (55, 155)
These percentages are in very good agreement with the
Empirical Rule.
Here is a histogram of these 1000 IQ scores.
Histogram of 1000 IQ Scores
400
Frequency
300
200
100
0
40
55
70
85
100
115
130
145
160
IQ
The two center bars lie within one standard deviation of the mean
and contain 69.6% (close to the Rule's 68%) of the observations.
The four center bars lie within two standard deviations of the mean
and contain 95.5% (close to the Rule's 95%) of the observations.
The six center bars lie within three standard deviations of the mean
and contain 99.7% (close to the Rule's "all or almost all") of the
observations.
Now here is a quiz. What is the approximate mean and standard
deviation of the distributions represented by each of the following
histograms? The first is based on n = 1000, the second on n = 100.
400
Frequency
300
200
100
0
280
320
360
400
440
480
520
60
70
80
x
40
Frequency
30
20
10
0
20
30
40
50
y