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Lecture 2: Methods for Describing data through

Numerical Measures

North South University School of BusinessSlide 1 of 76

Outline• Measures of central tendency and dispersion

• Characteristics, uses, advantages, and disadvantages ofeach measure of location and dispersion

• Chebyshev’s theorem and the Empirical Rule as theyrelate to a set of observations


• Quartiles, deciles, and percentiles

• Box plots

• Coefficient of skewness and coefficient of variation

• Scatter diagram

• Contingency table

Numerical Ways of Describing Data

• Measures of location


Measures of location

• Measures of dispersion

Parameter and Statistic

A ParameterParameter is a measurable characteristic of a population

A statisticstatistic is a measurable characteristic of a


A statisticstatistic is a measurable characteristic of a sample

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Measures of Location and Dispersion

• Measures for Population Data

• Measures for Sample Data


• Measures for Ungrouped data

• Measures for grouped data

Measures of Location

Mean (Arithmetic, Weighted, Geometric) Median Mode


Arithmetic Mean

The The Arithmetic MeanArithmetic Meanis the most widely used is the most widely used measure of location and measure of location and

shows the central value of shows the central value of the datathe data

Average Joe


the datathe data

It is calculated by summing the values and

dividing by the number of values

Population Mean

N

X

For ungrouped data, the For ungrouped data, the

Population MeanPopulation Mean is is the sum of all the the sum of all the population values population values

divided by the total divided by the total number of populationnumber of population


where µ is the population mean N is the total number of observations. X is a particular value. indicates the operation of adding.

number of population number of population values:values:

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Example 1

The Kiers family owns four cars. The following is

the current mileage on

each of the four

56,000

42,000


500,484

000,73...000,56

N

X

Find the mean mileage for the cars.

cars. 23,000

73,000

Sample Mean

For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample

values:


n

XX

where n is the total number of values in the sample.

Example 2

A sample of five

executives received the

14.0, 15.0, 17.0, 16 0


4.155

77

5

0.15...0.14

n

XX

following bonus last

year ($000):

16.0, 15.0

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 datavalues.

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

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Example 3

Consider the set of values: 3, 8, and 4. The meanmean is 5. Illustrating the fifth

property


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

property

Weighted Mean

The Weighted MeanWeighted Mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2,

...,wn, is computed from the


)21

)2211

...(

...(

n

nnw

www

XwXwXwX

n pfollowing formula:

Example 4

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


89.0$50

50.44$1515155

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

wX

for $1.10. Compute the weighted mean of the price of the drinks.

The Median

There are as many values above the

median as below it in

The MedianMedian is the midpoint of the values after they have been

ordered from the smallest


the data array.

For an even set of values, the median will be the arithmetic average of the two middle numbers and is

found at the (n+1)/2 ranked observation.

to the largest.

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The ages for a sample of five college students are:

21, 25, 19, 20, 22.

Arranging the data in ascending order

The median (cont’d)


ggives:

19, 20, 21, 22, 25.

Thus the median is 21.

Example 5

Arranging the data in ascending order

gives:

The heights of four basketball players, in inches, are: 76, 73, 80, 75.


gives:

73, 75, 76, 80

Thus the median is 75.5.

The median is found at the

(n+1)/2 = (4+1)/2 =2.5th data point.

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 oflocation when such values occur


location when such values occur.

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

The Mode

The ModeMode is another measure of location and represents the value of the observation that

appears most frequently.


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Example 6

The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of

81 occurs the most often, it is the mode.


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

trimodal, and so on.

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

The Relative Positions of the Mean, Median, and Mode


Skewed distribution: One whose shapes on either side of the center differ; a nonsymmetrical distribution.

Can be positively or negatively skewed, or bimodal

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

Zero skewness Mean

= Median

= Mode


Mode

Median

Mean

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

The Relative Positions of the Mean, Median, and Mode: Left Skewed Distribution


Mean < Median < Mode

ModeMean

Median

Geometric Mean

The Geometric Mean(GM) of a set of n positive

numbers is defined as the nthroot of the product of the nnumbers. The formula is:


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

The geometric mean is used to average percents,

indexes, and relatives.

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 GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent.

Example 8

• The return on investment earned by Atkins ConstructionCompany for four successive years was: 30%, 20%, -40%, and 200%. What is the geometric mean rate ofreturn on investment?


294.10.36.02.13.1... 421 n

nXXXGM

The average rate of return is 29.4%

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Geometric Mean (cont’d)

Another use of the geometric mean is to determine the

percent increase in sales, production or other business or

Grow th in Sales 1999-2004

10

20

30

40

50

ales

in M

illion

s($)


1period) of beginningat (Value

period) of endat Value( nGM

other business or economic series

from one time period to another.

0

10

1999 2000 2001 2002 2003 2004

Year

Sa

Example 9

The total number of females enrolled in American colleges increased from 755,000 in

1992 to 835,000 in 2000.


0127.1000,755

000,8358 GM

The value 0.0127 indicates that the average annual growth over the last 8-year period was 1.27%.

Dispersionrefers to the

spread or variability in

the data.

Measures of Dispersion

0

5

10

15

20

25

30

0 2 4 6 8 10 12


the data.

Measures of dispersion include the following: Measures of dispersion include the following:

range, mean deviation, variance, and range, mean deviation, variance, and standard deviationstandard deviation..

Range = Largest value – Smallest value

0 2 4 6 8 10 12

The following represents the current year’s Return on Equity of the 25 companies in an investor’s 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 14.01 4 4 8 7 9 9 7 15 0

Example 10


-1.4 4.8 7.9 9.7 15.01.2 5.7 8.0 10.3 22.1

Highest value: 22.1 Lowest value: -8.1

Range = Highest value – lowest value= 22.1-(-8.1)

= 30.2

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MAD:The arithmetic

mean of the absolute l f th

The main features :

All values are used in the calculation.

It is not unduly influenced by large or small values

Mean Absolute Deviation (MAD)


values of the deviations from the

arithmetic mean.

by large or small values.

The absolute values are difficult to manipulate.

n

XXMAD

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

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

X = 102

Example 11


The mean deviation is:

4.25

541515

102103...102103

n

XXMD

Variance:the arithmetic mean of the

squared

Variance and standard Deviation


squared deviations from

the mean.

Standard deviation: The square root of the variance.

Not influenced by extreme values.

The units are awkward, the square of the

The major characteristics:The major characteristics:

Population Variance


, qoriginal units.

All values are used in the calculation.

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Population VariancePopulation Variance formula:

X is the value of an observation in the population

Variance and standard deviation

N

X

2

2


X is the value of an observation in the population

µ is the arithmetic mean of the population

N is the number of observations in the population

Population Standard Deviation formula:

2

In Example 10, the variance and standard deviation are:

(X - )2

N =

Example 10 (revisited)

62.6


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

25

= 42.227

= 6.498

Sample variance (s2):

s2 =(X - X)2

1

Sample variance and standard deviation


s2 = n-1

Sample standard deviation (s):

2ss

37X

Example 12

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

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

Find the sample variance and standard deviation.


40.75

37

n

XX

30.5

15

2.21

15

4.76...4.77

1

2222

n

XXs

30.230.52 ss

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Chebyshev’s theorem: For any set of observations (sample or population), the proportion of the values that lie within k standard deviations of

the mean is at least:

1

Chebyshev’s theorem


where k is any constant greater than 1.

2

11

k

Chebyshev’s theorem (cont’d)

The arithmetic mean biweekly amount by theDupree Paint employees to the company’s profit-sharing plan was $51.54, and the standarddeviation is $7.51. At least what percent of thecontributions lie within plus 3.5 standard deviationsand minus 3 5 standard deviations of the mean?


and minus 3.5 standard deviations of the mean?

92.0

25.12

11

5.3

11

11 22

k

About 92%.

Empirical RuleEmpirical Rule: For any symmetrical, bell-shaped distribution:

About 68% of the observations will lie within ±1s f th

Interpretation and Uses of theStandard Deviation


of the mean

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

Virtually all (99.7%) the observations will be within ± 3s of the mean

68%

Interpretation and Uses of the Standard Deviation

Bell-shaped Curve showing the relationship between µand σ


68%

95%99.7%

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The Mean of Grouped Data

The Mean of a sample of data organized in a frequency

distribution is computed by the following formula:


n

MfX

Example 13A sample of ten movie theaters

in a large metropolitan

area tallied the total number of movies showing

Movies showing

frequency f

class midpoint M

(f)(M)

1 up to 3 1 2 2

3 up to 5 2 4 8

5 up to 7 3 6 18


movies showing last week.

Compute the mean number of

movies showing.

5 up to 7 3 6 18

7 up to 9 1 8 8

9 up to 11 3 10 30

Total 10 66

6.610

66

n

MfX

The Median of Grouped Data

2CF

n

The Median of a sample of data organized in a frequency distribution is computed by:


)(2 if

LMedian

where L is the lower limit of the median class, CF is the cumulative frequency preceding the median class, f is the frequency of the median class, and i is the median class

interval.

Finding the Median Class

• Construct a cumulative frequency distribution.

• Decide the class that contains the median. MedianClass is the first class with the value of cumulativefrequency at least n/2


frequency at least n/2.

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Movies showing

Frequency Cumulative Frequency

1 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

Example 13 (cont’d)

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


33.6)2(3

32

10

5)(2

if

CFn

LMedian

The Mode of Grouped Data

The Mode for grouped data is approximated by the midpoint of the class

with the largest class frequency.

Movies h i

frequency f

class id i t


The modes in example 13 are 6 and 10 and so is

bimodal.

showing f midpoint M

1 up to 3 1 2

3 up to 5 2 4

5 up to 7 3 6

7 up to 9 1 8

9 up to 11 3 10

The Standard Deviation of Grouped Data

The Standard Deviation of a sample of data organized in

a frequency distribution is computed by the following


1

2

n

XMfs

formula:

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A sample of ten movie theaters in a large metropolitan

area tallied the total number of movies showing last week.

Compute the standard deviation of

Movies showing

frequency f class midpoint M

(M-X) f*(M-X)2

1 up to 3 1 2 -4.6 21.16

3 up to 5 2 4 -2.6 13.52

5 up to 7 3 6 -0.6 1.08

7 up to 9 1 8 1.4 1.96


standard deviation of movies showing.

p

9 up to 11 3 10 3.4 34.68

Total 10 72.40

8363.2

110

40.72

1

2

n

XMfs

Other Measures of Dispersion

• Quartiles divide a set of observations into four equalparts

• Deciles divide a set of observations into 10 equalparts


• Percentiles divide a set of observations into 100equal parts

Quartiles

Locate the median,

(50th percentile)

first quartile (25th percentile)

and the 3rd quartile


and the 3rd quartile

(75th percentile)

Location of a Percentile

P

100

where

Lp = (n+1)


P is the desired percentile

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80

90

100

Stock prices on twelveconsecutive days for a

majorpublicly traded company

Example 14


50

60

70

1 2 3 4 5 6 7 8 9 10 11 12

86, 79, 92, 84, 69, 88, 91

83, 96, 78, 82, 85.

Using the twelve stock prices, we can find the median, 25th, and 75th percentiles as follows:

L75 = (12 + 1) 75100

= 9.75th observationQuartile 3



L50 = (12 + 1) 50100 = 6.50th observation

L25 = (12+1) 25100

= 3.25th observationQuartile 1

Median

9692918886

12111098 50th percentile: Median

75th percentilePrice at 9.75 observation = 88 + .75(91-88)

= 90.25

Q3

Q4

Example 14 (cont’d)To locate the values, the first step is to organize the data in increasing order


8685848382797869

87654321

25th percentilePrice at 3.25 observation = 79 + .25(82-79)

= 79.75

50 percentile: MedianPrice at 6.50 observation = 84 + .5(85-84)

= 84.50

Q1

Q2

Q3

Interquartile Range

The Interquartilerange is the distance

between the third quartile Q3 and the

This distance will include the middle 50 percent of the


3

first quartile Q1.p

observations.

Interquartile range = Q3 - Q1

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Example 15For a set of

observations the third quartile is 24 and the

first quartile is 10. What is the quartile

deviation?


deviation?

The interquartile range is 24 - 10 = 14. Fifty

percent of the observations will occur

between 10 and 24.

Box Plots

Five pieces of data are needed

A box plot is a graphical display, based on quartiles, that helps to picture a set of

data.


to construct a box plot: the Minimum Value, the First Quartile, the Median, the Third Quartile, and the Maximum Value.

Example 16

Based on a sample of 20 deliveries, Buddy’s Pizza determined the following

information. The minimum delivery time was 13 minutes and the maximum 30 minutes. The first quartile was 15

minutes the median 18 minutes and the


minutes, the median 18 minutes, and the third quartile 22 minutes. Develop a box

plot for the delivery times.



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Q1 Q3MaxMin Median


12 14 16 18 20 22 24 26 28 30 32

Coefficient of Variation

The coefficient of variation is the ratio of the standard deviation to the arithmetic

mean expressed as a

Relative dispersion


%)100(X

sCV

mean, expressed as a percentage:

Mean

Skewness is the measurement of the lack of symmetry of the distribution.

The coefficient of skewness can range from -3 00 up to 3 00

Skewness


from 3.00 up to 3.00 when using the

following formula:A value of 0 indicates a symmetric distribution.

Some software packages use a different formula which results in a

wider range for the coefficient.

s

MedianXsk

3

Using the twelve stock prices, we find the mean to be 84.42, standard deviation, 7.18, median, 84.5.

Coefficient of variation:

Example 14 revisited

86 79 92 84 69 88 91 83 96 78 82 85


= 8.5%%)100(X

sCV

Coefficient of skewness:

= -.035

s

MedianXsk

3

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Relationship Between Two Variables

• Univariate Data (Single Variable)

• Bivariate Data (Two Variables)– Scatter diagram


– Contingency table

Scatter diagram :

A technique used to

show the

Variables must be at least interval scaled

Scatter diagram


show the relationship

between variables.

Relationship can be positive (direct) or negative (inverse)

96929188

PriceIndex(000s)

8.07.57.57.3

Relationship between Market Index and Stock Price

100

Example 14 revisitedThe twelve days of stock prices and the overall market index on each day

are given as follows:


8685848382797869

7.27.27.17.17.06.26.25.1

50

60

70

80

90

5 6 7 8 9 10

Index

Pri

ce

A contingency table is used to classify observations

according to two identifiable characteristics.

Contingency tables are used

Contingency table


A contingency table is a cross tabulation that

simultaneously summarizes two variables of interest.

g ywhen one or both variables are

nominally scaled.

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Weight Loss45 adults, all 60 pounds

overweight, are randomly assigned to three weight loss programs. Twenty weeks into

the program a researcher

Example 17


the program, a researcher gathers data on weight loss

and divides the loss into three categories: less than 20

pounds, 20 up to 40 pounds, 40 or more pounds. Here are

the results.

Weight

Loss

Plan

Less than 20 pounds

20 up to 40

pounds

40 pounds or more

Plan 1 4 8 3



4 8 3

Plan 2 2 12 1

Plan 3 12 2 1

Compare the weight loss under the three plans.

Practice Problems• Problem 11 (Page 62)

(Problem 13)

• Problem 21 (Page 68)

(Problem 25 (Page 69))











Assignment-2



• Problems 11, 13 (Page 108)

(Problems 11, 13 (Page 110))








bs-2

Documents

population data measures

number of sample values

ungrouped data measures

total number of values

set of data

population meanpopulation

mean mileage

dispersion measures