week1-2 chap 3 descri data

8/8/2019 Week1-2 Chap 3 Descri Data

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Describing Data: Numerical MeasuresDescribing Data: Numerical Measures

GOALS

When you have completed this chapter, you will be able to:

ONE

Calculate the arithmetic mean, median, mode, weighted

mean, and the geometric mean.

TWO

Explain the characteristics, uses, advantages, and

disadvantages of each measure of location.THREE

Identify the position of the arithmetic mean, median,

and mode for both a symmetrical and a skewed

distribution.

Goals

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FOUR

Compute and interpret the range, the mean deviation, the

variance, and the standard deviation of ungrouped data.

Describing Data: Numerical MeasuresDescribing Data: Numerical Measures

FIVE

Explain the characteristics, uses, advantages, and

disadvantages of each measure of dispersion.

SIX

Understand Chebyshevs theorem and the Empirical Rule as

they relate to a set of observations.

Goals

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Characteristics of the Mean

It is calculated bysumming the values

and dividing by thenumber of values.

It requires the interval scale.

All values are used.It is unique.

The sum of the deviations from the mean is 0.

The Arithmetic MeanArithmetic Meanis the most widely used

measure of location andshows the central value of the

data.

The major characteristics of the mean are: A era eJoe

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Population Mean

N

X!Q

where

is the population mean

Nis the total number of observations.

Xis a particular value.

7 indicates the operation of adding.

For ungrouped data, the

Population MeanPopulation Mean is thesum of all the populationvalues divided by the total

number of population

values:

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Example 1

500484

0007300056!

!!

N

XQ

Find the mean mileage for the cars.

A ParameterParameter is a measurable characteristic of apopulation.

The Kiers

family owns

four cars. The

following is thecurrent mileage

on each of the

four cars.

56,000

23,000

42,000

73,000

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Sample Mean

nXX 7!

where n is the total number ofvalues in the sample.

For ungrouped data, the sample mean is

the sum of all the sample values dividedby the number of sample values:

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Example 2

.1.1....1

!!

!7

!

n

XX

A statisticstatistic is a measurable characteristic of a sample.

A sample offive

executives

received the

following

bonus last

year ($ ):

1 .0,1 .0,

15.0,15.0,

17.0,17.0,16.0,16.0,

15.015.0

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Properties of the Arithmetic Mean

Every set of interval-level and ratio-level data has amean.

All the values are included in computing the mean.

A set of data has a unique mean.

The mean is affected by unusually large or small

data values.

The arithmetic mean is the only measure of locationwhere the sum of the deviations of each value from

the mean is zero.

PropertiesProperties of the Arithmetic Mean3- 8


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Example

? A!!7 XX

Consi er t e set of al es: , , and .

T e meanmean is . Ill strating t e fiftropert

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Weighted Mean

n

nnw

www

www

!

The Weighted MeanWeighted Mean of a set of

numbers X1, X2, ..., Xn, withcorresponding weights w1, w2,

...,wn, is computed from the

following formula:

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Example

9.$.$

111)1.1($1)9.($1).($1).($

!!

!wX

During a one hour period on a

hot Saturday afternoon cabanaboy Chris served fifty drinks.

He sold five drinks for $0.50,

fifteen for $0.75, fifteen for

$0.90, and fifteen for $1.10.Compute the weighted mean of

the price of the drinks.

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The Median

There are as manyvalues above themedian as below it inthe data array.

For an even set of values, the median will be the

arithmetic average of the two middle numbers and isfound at the (n+1)/2 ranked observation.

The MedianMedian is themidpoint of the values afterthey have been ordered from

the smallest to the largest.

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The ages for a sample of five college students are:

1, 5, 19, 0, .

Arranging the data

in ascending ordergives:

19, 0, 1, , 5.

Thus the median is1.

The median (continued)

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Example

Arranging the data in

ascending order gives:

7 , 75, 76, 80

Thus the median is 75.5.

The heights of four basketball players, in inches,

are: 76, 7 , 80, 75.

The median is found

at the (n+1)/ =( +1)/ = .5th data

point.

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Properties of the Median

There is a unique median for each data set.

It is not affected by extremely large or smallvalues and is therefore a valuable measure of

location when such values occur.

It can be computed for ratio-level, interval-level, and ordinal-level data.

It can be computed for an open-endedfrequency distribution if the median does notlie in an open-ended class.

Properties of the Median

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The Mode: Example

Example 6Example 6:: The exam scores for ten students are:1, 9 , , , , , 1, , 1, . Because the score

of 1 occurs the most often, it is the mode.

Data can have more than one mode. If it has twomodes, it is referred to as bimodal, three modes,

trimodal, and the like.

The ModeMode is another measure of location and

represents the value of the observation that appearsmost frequently.

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Symmetric distributionSymmetric distribution: A distribution having thesame shape on either side of the center

Skewed distributionSkewed distribution: One whose shapes on eitherside of the center differ; a nonsymmetrical distribution.

Can be positively or negatively skewed, or bimodal

The Relative Positions of the Mean, Median, and Mode

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The Relative Positions of the Mean, Median, and Mode:

Symmetric Distribution

Zero skewness Mean

=Median

=Mode

Mode

Median

Mean

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The Relative Positions of the Mean, Median, and Mode:

Right Skewed Distribution

Positively skewed: Mean and median are to the right of the

mode.

Mean>Median>Mode

ode

edia

ea

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Negatively Skewed: Mean and Median are to the left of the Mode.

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Geometric Mean

GM X X X X nn! ( )( )( )...( )1 2 3

The geometric mean is used to

average percents, indexes, and

relatives.

The Geometric MeanGeometric Mean

(GM) of a set of n numbersis defined as the nth root

of the product of the n

numbers. The formula is:

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Example

The interest rate on three bonds were 5, 1, and

percent.

The arithmetic mean is (5+ 1+ )/ =10.0.

The geometric mean is

!!GM

The GMgives a more conservative

profit figure because it is notheavily weighted by the rate of

21percent.

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Geometric Mean continued

eriodoe inninatVa ue

eriodoendatVa ue nGM

Another use of the

geometric mean is to

determine the percent

increase in sales,

production or other

business or economicseries from one time

period to another.

Grow th in Sales 1999-2004

0

10

20

30

40

50

1999 2000 2001 2002 2003 2004

Year

SalesinMillions($)

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Example

!!GM

The total number of females enrolled in American

colleges increased from 755,000 in 199 to 8 5,000 in000. That is, the geometric mean rate of increase is

1. 7%.

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DispersionDispersion

refers to thespread or

variability in

the data.

Measures of dispersion include the following: range,range,

mean deviation, variance, and standardmean deviation, variance, and standard

deviationdeviation.

RangeRange = Largest value Smallest value

Measures of Dispersion

0

5

10

15

20

25

30

0 2 4 6 8 10 12

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The following represents the current years Return on

Equity of the 2 companies in an investors portfolio.

-8.1 3.2 5.9 8.1 12.3

-5.1 4.1 6.3 9.2 13.3

-3.1 4.6 7.9 9.5 14.0

-1.4 4.8 7.9 9.7 15.0

1.2 5.7 8.0 10.3 22.1

Example 9

Highest value: 22.1 Lowest value: - .1

Range = Highest value lowest value

= 22.1-(- .1)

= .2

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MeanMean

DeviationDeviationThe arithmetic

mean of the

absolute values

of thedeviations from

the arithmetic

mean.

The main features of the

mean deviation are: All values are used in the

calculation.

It is not unduly influenced bylarge or small values.

The absolute values are

difficult to manipulate.

Mean Deviation

MD =7 X - X

n

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The weights of a sample of crates containing booksfor the bookstore (in pounds ) are:

1 , 9 , 1 1, 1 , 1Find the mean deviation.

X = 10

The mean deviation is:

.11

11...11

!

!

!

7

! n

XX

MD

Example 1

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VarianceVariance:: the

arithmetic meanof the squared

deviations from

the mean.

Standard deviationStandard deviation: The squareroot of the variance.

ariance and standard Deviation

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Not influenced by extreme values.

The units are awkward, the square of theoriginal units.

All values are used in the calculation.

The major characteristics of the

Population VariancePopulation Variance are:

Population ariance

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Population VariancePopulation Variance formula:

7 (X - Q)2

NW =

X isthevalueofanobservationinthepopulation

m isthearithmeticmeanofthepopulation

N isthenumberofobservationsinthepopulation

W!

Population Standard DeviationPopulation Standard Deviation formula:

WVariance and standard deviation

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(-8. -6.62)2 (-5. -6.62)2 ... (22. -6.62)2

25W!

W

W!

2.227

=6. 8

In Example 9, the variance and standard deviation are:

7(X - Q)2

N

=

Example9continued

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Sample variance (sSample variance (s ))

s2 =7( - )2

n-1

Sample standard deviation (s)Sample standard deviation (s)

2ss !

Sample variance and standard deviation

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40.5

3!!

7!

n

XX

30.5

15

2.21

15

4.76...4.77

1

222

2

!

!

!

7!

n

XXs

Example 11

The hourly wages earned by a sample of five students are:

$7, $ , $11, $ , $ .

Find the sample variance and standard deviation.

.. !!! ss

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Chebyshevs theoremChebyshevs theorem:: For any set of

observations, the minimum proportion of the valuesthat lie within kstandard deviations of the mean is at

least:

where kis any constant greater than 1.

211k

Chebyshevs theorem

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Empirical RuleEmpirical Rule: For any symmetrical, bell-shaped

distribution:

About % of the observations will lie within 1s the

mean

About 9 % of the observations will lie within 2sof

the mean

Virtually all the observations will be within s of the

mean

Interpretation and Uses of the

Standard Deviation

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Bell -Shaped Curve showing the relationship between and .W Q

QW QW QW Q QW QW QW

68%

9 %

99.7%

Interpretation and Uses ofthe Standard Deviation

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The Mean of Grouped Data

n

XfX

7!

The MeanMean of a sample of data

organized in a frequencydistribution is computed by the

following formula:

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Example 12

A sample of ten

movie theaters

in a large

metropolitan

area tallied the

total number ofmovies showing

last week.

Compute the

mean number ofmovies

showing.

Movies

showing

frequency

f

class

midpoint

X

(f)(X)

1 up to 3 1 2 2

3 up to 5 2 4 8

5 up to 7 3 6 18

7 up to 9 1 8 8

9 up to

11

3 10 30

Total 10 66

6.61

66!!

7!

n

XX

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The Median of Grouped Data

)(2 if

CF

LMedia

!

where L is the lower limit of the median class, CF is the

cumulative frequency preceding the median class, fis

the frequency of the median class, and i is the median

class interval.

TheMedianMedian of a sa ple ofdata organized in a

fre encydistri tion is computed y:

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Finding t e edianClass

To determine t e median class forgrouped

data

Construct a cumulative frequency distribution.

Divide t e total number ofdata values by 2.

Determine w ic class will contain t is value. Forexample, ifn= , /2 = 2 , t endetermine w ic

class will contain t e 2 t value.

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Example 12 continued

o ie

o in

Fre uenc umulati e

Fre uenc1 up to 3 1 1

3 up to 5 2 3

5 up to 3 6

up to 1

up to 11 3 10

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Example 12 continued

.6)2(21

)(2 !

!

! i

f

CFn

LMedian

From the table, L= , n=1 , f= , i=2, CF=

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The Mode of Grouped Data

The modes in example 12 are 6 and 1

and so is bimodal.

The ModeMode for grouped data isapproximated by the midpoint of the

class with the largest class frequency.

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week1-2 chap 3 descri data

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