Formulas for Deviation
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Transcript of Formulas for Deviation
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Section 3-3
Measures of Variation
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WAITINGTIMES AT
DIFFERE
NTBANK
SJefferson Valley Bank
(single waiting line)6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7
Bank of Providence
(multiple waiting lines)4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
All the measures of center are equal for both banks.
Mean = 7.15 min
Median = 7.20 min
Mode = 7.7 min
Midrange = 7.10 min
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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 minBank of Providence range = 10.0 4.2 = 5.8 min
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STANDARDDEVIATION
FORA
SA
MPLEThe 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 valuesfrom the mean.
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STANDARDDEVIATION
FORMUL
AS
1
)( 2
!
n
xxs
Sample Standard
Deviation
Shortcut FormulaforSample
Standard Deviation
)1(
)( 22
!
nn
xxns
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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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STANDARDDEVIATION
OFA
PO
PULATION
The standard deviation for apopulation is
denoted by and is given by the formula
.)( 2
N
x !
QW
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FINDINGTHE STANDARD
DEVIATION
ON
TH
ETI
-81/841. Press STAT; select 1:Edit.2. Enter your data values in L1. (You may enter the
values in any of the lists.)
3. Press 2ND, MODE (forQUIT).4. Press STAT; arrow over to CALC. Select 1:1-Var
Stats.
5. Enter L1 by pressing 2ND, 1.
6. Press ENTER.
7. The sample standard deviation is given by Sx. Thepopulation standard deviation is given by x.
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SYMBOLS FOR
STANDA
RD
D
EVIATION
Sample
textbook
TI-83/84 Calculators
Population
s
Sx
x
textbook
TI-83/84 Calculators
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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.
Jefferson Valley Bank
(single waiting line)6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7
Bank of Providence
(multiple waiting lines)4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
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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 = 2
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ROUND-OFF RULE FOR
MEA
SURESOF
VA
RIATON
Carry one more decimal place
than is present in the
original data set.
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STANDARDDEVIATIONFROM A
FREQUE
NCYDI
ST
RIBU
TION
? A ? A
)1(
)()(22
!
nn
xfxfns
Use the class midpoints as the x values.
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EXAMPLE
Find the standard deviation of the following.
CLASS FREQUENCY
1-3 4
4-6 9
7-9 6
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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 maximumusual sample values by using
RANGE RULE OFTHUMB
4
range}s
minimum usual value
maximum usual value sx
sx
2
2
}
}
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EXAMPLES
1. 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.
2. 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 andmaximum 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
(O
R 68-95-99.7) RULEThe empirical (or68-95-99.7) 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 2standard deviations of the mean.
About 99.7% of all values fall within 3
standard deviations of the mean.
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x - 3s x - 2s x - s x x + 2s x + 3sx + s
68% within
1 standard deviation
34% 34%
95% within2 standard deviations
99.7% of data are within 3 standard deviations of the mean
0.1% 0.1%
2.4% 2.4%
13.5% 13.5%
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EXAMPLEA 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) ofany 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. ForK= 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?