chapter 03 describing data: numerical measures

8/21/2019 Chapter 03 Describing Data: Numerical Measures

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Chapter

Three

McGraw-Hill/Irwin 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

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Chapter Three

Describing Data: Numerical Measures

GOALSWhen 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 Measures

FIVEExplain 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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There are two numerical ways

of describing data, namely

measure of location and

measure of dispersion.

Measure of Locationare often referred to as average.An Average shows the central value of the data.

Measure of dispersionoften called the variation or

spread in the data.


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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 Meanis the most widely used

measure of location andshows the central value of the

data.

The major characteristics of the mean are:Average

Joe

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Population Mean

N

X

where

is the population mean

N is the total number of observations.

X is a particular value.

indicates the operation of adding.

For ungrouped data, the

Population Meanis the

sum of all the populationvalues divided by the total

number of population

values:

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Example 1

500,484

000,73...000,56

N

X

Find the mean mileage for the cars.

A Parameteris 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

where nis the total number ofvalues in the sample.

For ungrouped data, the sample mean is

the sum of all the sample values divided

by the number of sample values:

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Example 2

4.155

77

5

0.15...0.14

n

XX

A statisticis a measurable characteristic of a sample.

A sample offive

executives

received the

followingbonus last

year ($000):

14.0,

15.0,

17.0,16.0,

15.0

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Properties of the Arithmetic Mean

Every set of interval-level and ratio-level data has a

mean.

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.

Propertiesof the Arithmetic Mean3- 10


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Example 3

0)54()58()53()( XX

Consider the set of values: 3, 8, and 4.

The meanis 5. Illustrating the fifthproperty

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Weighted Mean

)21

)2211

...(

...(

n

nnw

www

XwXwXwX

The Weighted Meanof a set of

numbersX1,X2, ...,Xn, withcorresponding weights w1, w2,

...,wn, is computed from the

following formula:

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Example 4

89.0$50

50.44$

1515155)15.1($15)90.0($15)75.0($15)50.0($5

wX

During a one hour period on a

hot Saturday afternoon cabana

boy 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 in

the 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 Medianis the

midpoint 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:

21, 25, 19, 20, 22.

Arranging the datain ascending ordergives:

19, 20, 21, 22, 25.

Thus the median is21.

The median (continued)

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Example 5

Arranging the data in

ascending order gives:

73, 75, 76, 80

Thus the median is 75.5.

The heights of four basketball players, in inches,

are: 76, 73, 80, 75.

The median is found

at the (n+1)/2 =(4+1)/2 =2.5thdata

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.

Properties of the Median

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The Mode: Example 6

Example 6:The exam scores for ten students are:

81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the scoreof 81 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.

TheModeis another measure of location andrepresents the value of the observation that appears

most frequently.

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

Skewed 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

Mode

Median

Mean

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

Mean


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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 Mean(GM) of a set of n numbers

is defined as the nthroot

of the product of the n

numbers. The formula is:

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Example 7

The interest rate on three bonds were 5, 21, and 4

percent.

The arithmetic mean is (5+21+4)/3 =10.0.The geometric mean is

49.7)4)(21)(5(3

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

1period)ofbeginningat(Value

period)ofendatValue( nGM

Another use of the

geometric mean is to

determine the percentincrease in sales,

production or other

business or economic

series 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 8

0127.1000,755

000,8358 GM

The total number of females enrolled in American

colleges increased from 755,000 in 1992 to 835,000 in

2000. That is, the geometric mean rate of increase is

1.27%.

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Dispersionrefers to the

spread orvariability in

the data.

Measures of dispersion include the following: range,

mean deviation, variance, and standard

deviation.

Range = Largest valueSmallest 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 25 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: -8.1

Range = Highest valuelowest value

= 22.1-(-8.1)

= 30.2

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Mean

DeviationThe 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 by

large or small values.

The absolute values are

difficult to manipulate.

Mean Deviation

MD =X - X

n

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

103, 97, 101, 106, 103Find the mean deviation.

X = 102

The mean deviation is:

4.25

14151

5

102103...102103

n

XX

MD

Example 10

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Variance: the

arithmetic meanof the squared

deviations from

the mean.

Standard deviation: The squareroot of the variance.

Variance and standard Deviation

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Population Varianceformula:

(X - )2N

=

X is the value of an observation in the populationm is the arithmetic mean of the population

N is the number of observations in the population

Population Standard Deviationformula:

2Variance and standard deviation

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(-8.1-6.62)2+ (-5.1-6.62)2 + ... + (22.1-6.62)225

= 42.227

= 6.498

In Example 9, the variance and standard deviation are:

(X - )2

N =

Example 9 continued

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Sample variance (s2)

s2=(X - X)2

n-1

Sample standard deviation (s)

2ss

Sample variance and standard deviation

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40.75

37

n

XX

30.515

2.21

15

4.76...4.77

1

222

2

n

XXs

Example 11

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

$7, $5, $11, $8, $6.

Find the sample variance and standard deviation.

30.230.52 ss

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Chebyshevs theorem:For any set of

observations, the minimum proportion of the valuesthat lie withinkstandard deviations of the mean is atleast:

where k is any constant greater than 1.

211k

Chebyshevs theorem

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Empirical Rule: For any symmetrical, bell-shapeddistribution:

About 68% of the observations will lie within 1s

the mean

About 95% of the observations will lie within 2s of

the mean

Virtually all the observations will be within 3sof

the mean

Interpretation and Uses of the

Standard Deviation

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Bell -Shaped Curve showing the relationship between and .

3

1 1

3

68%

95%

99.7%

Interpretation and Uses of the Standard Deviation

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The Mean of Grouped Data

n

Xf

X

The Meanof a sample of dataorganized in a frequency

distribution is computed by the

following formula:

X is the midpoint of each class

f is the frequency in each classnis the total number of frequencies

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Example 12

A sample of ten

movie theaters

in a largemetropolitan

area tallied the

total number of

movies showing

last week.

Compute the

mean number ofmovies

showing.

Moviesshowing

frequencyf

classmidpoint

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 89 up to

113 10 30

Total 10 66

6.610

66

n

XX

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The Median of Grouped Data

)(2 if

CFn

LMedian

whereLis the lower limit of the median class,

CFis the cumulative frequency preceding the median

class,f is the frequency of the median class,

andiis the median class interval.

The Medianof a sample of data organized in afrequency distribution is computed by:

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Finding the Median Class

To determine the median class for grouped

data

Construct a cumulative frequency distribution.

Divide the total number of data values by 2.

Determine which class will contain this value. Forexample, if n=50, 50/2 = 25, then determine which

class will contain the 25thvalue.

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Example 12 continued

Movies

showing

Frequency Cumulative

Frequency1 up to 3 1 1

3 up to 5 2 3

5 up to 7 3 6

7 up to 9 1 7

9 up to 11 3 10

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Example 12 continued

33.6)2(3

32

10

5)(2

if

CF

n

LMedian

From the table,L=5, n=10,f=3, i=2, CF=3

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The Mode of Grouped Data

The modes in example 12 are 6 and 10

and so is bimodal.

The Modefor grouped data isapproximated by the midpoint of the

class with the largest class frequency.

chapter 03 describing data: numerical measures

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