formulas for deviation

8/8/2019 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?

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