Download 3.4 – Use of Standard Deviation

3.4 – Use of Standard Deviation
Empirical Rule
For a bell-shaped distribution
-about 68% of all values fall within 1 standard deviation of the mean
-about 95% of all values fall within 2 standard deviations of the mean
-about 99.7% of all values fall within 3 standard deviations of the mean
ex.
The prices of all college textbooks follow a bell-shaped distribution with a mean of $105
and a standard deviation of $20.
(a)
Using the empirical rule, find the interval that contains the prices of 99.7% of college
textbooks.
(b)
Using the empirical rule, find the percentage of all college textbooks with their prices
between $85 and $125.
(c)
Using the empirical rule, find the percentage of all college textbooks with their prices
between $65 and $145.
Chebyshev’s Theorem
1 

At least 1 − 2  of the data values lie within k standard deviations of the mean, for any k>1.
 k 
Note that the distribution does NOT have to be bell-shaped in order to use this theorem.
ex.
Use above formula for k=2, and interpret the result.
ex.
Use above formula for k=3, and interpret the result.
ex.
Use above formula for k=1.5, and interpret the result.
ex.
Suppose the average credit card debt for households is $9,500 with a standard deviation
of $2,600.
(a) Using Chebyshevs theorem, find at least what percentage of current credit card debts
for all households are between $3,000 and $16,000.
(b) Using Chebyshevs theorem, find the interval that contains credit card debts of at least
89% of all the households.
Range Rule of Thumb
Most values fall within 2 standard deviations of the mean. Values that fall outside of this interval
would be considered unusual.
From this we could also say that SD ≈
ex.
range
(unless we have extreme outliers)
4
The Wechsler Adult Intelligence Scale involves an IQ test designed so that the mean
score is 100 and the standard deviation is 15. Use the range rule of thumb to find the
minimum and maximum "usual" IQ scores. Then determine whether an IQ score of 135
would be considered "unusual."