Measures of Variation

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Section 2-5

• Measures of Variation

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WAITING TIMES ATDIFFERENT BANKS
All the measures of center are equal for both
banks. Mean 7.15 Median 7.20 Mode
7.7 Midrange 7.10
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RANGE
The range of a set of data is the difference
between the highest value and the lowest
value. range (highest value) - (lowest value)
EXAMPLE Jefferson Valley Bank range 7.7 - 6.5
1.2 min Bank of Providence range 10.0 - 4.2
5.8 min
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STANDARD DEVIATIONFOR A SAMPLE
The standard deviation of a set of sample values
is a measure of variation of values about the
mean. It is a type of average deviation of
values from the mean.
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STANDARD DEVIATION FORMULAS
Sample Standard Deviation
Shortcut Formula for Sample Standard Deviation
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EXAMPLE
Use both the regular formula and shortcut formula
to find the standard deviation of the
following. 3 7 4 2
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STANDARD DEVIATIONOF A POPULATION
The standard deviation for a population is
denoted by s and is given by the formula
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FINDING THE STANDARD DEVIATION ON THE TI-81/84

• Press STAT select 1Edit.
• Enter your data values in L1. (You may enter the
  values in any of the lists.)
• Press 2ND, MODE (for QUIT).
• Press STAT arrow over to CALC. Select 11-Var
  Stats.
• Enter L1 by pressing 2ND, 1.
• Press ENTER.
• The sample standard deviation is given by Sx.
  The population standard deviation is given by sx.

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SYMBOLS FORSTANDARD DEVIATION
Sample
s Sx
textbook TI-83/84 Calculators
Population
textbook TI-83/84 Calculators
s sx
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EXAMPLE
Use your calculator to find the standard
deviation for waiting times at the Jefferson
Valley Bank and the Bank of Providence.
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VARIANCE
The variance of set of values is a measure of
variation equal to the square of the standard
deviation. sample variance s2 population
variance s2
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ROUND-OFF RULE FOR MEASURES OF VARIATON
Carry one more decimal place than is present in
the original data set.
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STANDARD DEVIATION FROM A FREQUECNY DISTRIBUTION
Use the class midpoints as the x values.
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EXAMPLE
Find the standard deviation of the following.
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RANGE RULE OF THUMB
To roughly estimate the standard deviation, use
the range rule of thumb where range (highest
value) - (lowest value) If the standard deviation
s is known, use it to find rough estimates of the
minimum and maximum usual sample values by using
minimum usual value maximum usual value
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EXAMPLES

• The shortest home-run hit by Mark McGwire was 340
  ft and the longest was 550 ft. Use the range
  rule of thumb to estimate the standard deviation.
• Heights of men have a mean of 69.0 in and a
  standard deviation of 2.8 in. Use the range rule
  of thumb to estimate the minimum and maximum
  usual heights of men. In this context, is it
  unusual for a man to be 6 ft, 6 in tall?

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THE EMPIRICAL(OR 68-95-99.7) RULE
The empirical (or 68-95-99.7) rule states that
for data sets having a distribution that is
approximately bell-shaped, the following
properties apply.

• 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.

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99.7 of data are within 3 standard deviations of
the mean
95 within 2 standard deviations
68 within 1 standard deviation
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2.4
2.4
0.1
13.5
13.5
x - 3s
x - 2s
x - s
x
x 2s
x 3s
x s
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EXAMPLE
A random sample of 50 gas stations in Cook
County, Illinois, resulted in a mean price per
gallon of 1.60 and a standard deviation of
0.07. A histogram indicated that the data
follow a bell-shaped distribution. (a) Use the
Empirical Rule to determine the percentage of
gas stations that have prices within three
standard deviations of the mean. What are these
gas prices? (b) Determine the percentage of gas
stations with prices between 1.46 and 1.74,
according to the empirical rule.
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CHEBYSHEVS THEOREM
The proportion (or fraction) of any set of data
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 and K
3, we get the following statements

• At least 3/4 (or 75) of all values lie within 2
  standard deviations of the mean.
• At least 8/9 (or 89) of all values lie within 3
  three standard deviations of the mean.

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EXAMPLE
Using the weights of regular Coke listed in Data
Set 17 from Appendix B, we find that the mean is
0.81682 lb and the standard deviation is
0.00751 lb. What can you conclude from
Chebyshevs theorem about the percentage of cans
of regular Coke with weights between 0.79429 lb
and 0.83935 lb?