numerical descriptive techniques

1

Numerical Descriptive Techniques

Chapter 4

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4.2 Measures of Central LocationUsually, we focus our attention on two types of

measures when describing population characteristics: Central location (e.g. average) Variability or spread

The measure of central location reflects the locations of all the actual data points.

3

With one data pointclearly the central location is at the pointitself.

4.2 Measures of Central LocationThe measure of central location reflects the

locations of all the actual data points.How?

But if the third data point appears on the left hand-sideof the midrange, it should “pull”the central location to the left.

With two data points,the central location should fall in the middlebetween them (in order to reflect the location ofboth of them).

4

Sum of the observationsNumber of observations

Mean =

This is the most popular and useful measure of central location

The Arithmetic Mean

5

nx

x in

1i

Sample mean Population mean

Nx i

N1i

Sample size Population size

nx

x in

1i

The Arithmetic Mean

6

10

...

101021

101 xxxx

x ii

• Example 4.1The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet.

00 77 222211.011.0

• Example 4.2

Suppose the telephone bills of Example 2.1 representthe population of measurements. The population mean is

200x...xx

200x 20021i

2001i 42.1942.19 38.4538.45 45.7745.77

43.5943.59

The Arithmetic Mean

The arithmetic mean

7

Odd number of observations

0, 0, 5, 7, 8 9, 12, 14, 220, 0, 5, 7, 8, 9, 12, 14, 22, 330, 0, 5, 7, 8, 9, 12, 14, 22, 33

Even number of observations

Example 4.3

Find the median of the time on the internetfor the 10 adults of example 4.1

The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude.

The Median

Suppose only 9 adults were sampled (exclude, say, the longest time (33))

Comment

8.5, 8

8

The Mode of a set of observations is the value that occurs most frequently.

Set of data may have one mode (or modal class), or two or more modes.

The modal classFor large data setsthe modal class is much more relevant than a single-value mode.

The Mode

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Example 4.5Find the mode for the data in Example 4.1. Here are the data again: 0, 7, 12, 5, 33, 14, 8, 0, 9, 22

Solution

• All observation except “0” occur once. There are two “0”. Thus, the mode is zero.

• Is this a good measure of central location?• The value “0” does not reside at the center of this set

(compare with the mean = 11.0 and the mode = 8.5).

The Mode

The Mode The Mean, Median, Mode

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Relationship among Mean, Median, and Mode

If a distribution is symmetrical, the mean, median and mode coincide

If a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ.A positively skewed distribution

(“skewed to the right”)

MeanMedian

Mode

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If a distribution is symmetrical, the mean, median and mode coincide

If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ.

A positively skewed distribution(“skewed to the right”)

MeanMedian

Mode MeanMedian

Mode

A negatively skewed distribution(“skewed to the left”)

Relationship among Mean, Median, and Mode

12

This is a measure of the average growth rate.Let Ri denote the the rate of return in period i

(i=1,2…,n). The geometric mean of the returns R1, R2, …,Rn is the constant Rg that produces the same terminal wealth at the end of period n as do the actual returns for the n periods.

The Geometric Mean

13

If the rate of return was Rg in everyperiod, the nth period return wouldbe calculated by:

ng )R1( )R1)...(R1)(R1( n21

For the given series of rate of returns the nth period return iscalculated by:

Rg is selected such that…

1)R1)...(R1)(R1(R nn21g 1)R1)...(R1)(R1(R n

n21g

The Geometric MeanThe Geometric Mean

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4.3 Measures of variability

Measures of central location fail to tell the whole story about the distribution.

A question of interest still remains unanswered:

How much are the observations spread outaround the mean value?

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Observe two hypothetical data sets:

The average value provides a good representation of theobservations in the data set.

Small variability

This data set is now changing to...

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Observe two hypothetical data sets:

The average value provides a good representation of theobservations in the data set.

Small variability

Larger variability

The same average value does not provide as good representation of theobservations in the data set as before.

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The range of a set of observations is the difference between the largest and smallest observations.

Its major advantage is the ease with which it can be computed.

Its major shortcoming is its failure to provide information on the dispersion of the observations between the two end points.

? ? ?

But, how do all the observations spread out?

Smallestobservation

Largestobservation

The range cannot assist in answering this questionRange

The range

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This measure reflects the dispersion of all the observations

The variance of a population of size N x1, x2,…,xN

whose mean is is defined as

The variance of a sample of n observationsx1, x2, …,xn whose mean is is defined asx

N

)x( 2i

N1i2

N

)x( 2i

N1i2

1n

)xx(s

2i

n1i2

1n

)xx(s

2i

n1i2

The Variance

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Why not use the sum of deviations?

Consider two small populations:

1098

74 10

11 12

13 16

8-10= -2

9-10= -111-10= +1

12-10= +2

4-10 = - 6

7-10 = -3

13-10 = +3

16-10 = +6

Sum = 0

Sum = 0

The mean of both populations is 10...

…but measurements in Bare more dispersedthen those in A.

A measure of dispersion Should agrees with this observation.

Can the sum of deviationsBe a good measure of dispersion?

A

B

The sum of deviations is zero for both populations, therefore, is not a good measure of dispersion.

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Let us calculate the variance of the two populations

185

)1016()1013()1010()107()104( 222222B

25

)1012()1011()1010()109()108( 222222A

Why is the variance defined as the average squared deviation?Why not use the sum of squared deviations as a measure of variation instead?

After all, the sum of squared deviations increases in magnitude when the variationof a data set increases!!

The Variance

21

Which data set has a larger dispersion?Which data set has a larger dispersion?

1 3 1 32 5

A B

Data set Bis more dispersedaround the mean

Let us calculate the sum of squared deviations for both data sets

The Variance

22

1 3 1 32 5

A B

SumA = (1-2)2 +…+(1-2)2 +(3-2)2 +… +(3-2)2= 10SumB = (1-3)2 + (5-3)2 = 8

SumA > SumB. This is inconsistent with the observation that set B is more dispersed.

The Variance

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1 3 1 32 5

A B

However, when calculated on “per observation” basis (variance), the data set dispersions are properly ranked.

A2 = SumA/N = 10/5 = 2

B2 = SumB/N = 8/2 = 4

The Variance

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Example 4.7 The following sample consists of the number of jobs

six students applied for: 17, 15, 23, 7, 9, 13. Finds its mean and variance

Solution

2

2222

in

1i2

jobs2.33

)1413...()1415()1417(16

11n

)xx(s

jobs146

846

13972315176

xx i

61i

The Variance

25

2

2222

2i

n1i2

i

n

1i

2

jobs2.33

613...1517

13...151716

1

n)x(

x1n

1s

The Variance – Shortcut method

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The standard deviation of a set of observations is the square root of the variance .

2

2

:deviationandardstPopulation

ss:deviationstandardSample

2

2

:deviationandardstPopulation

ss:deviationstandardSample

Standard Deviation

27

Example 4.8 To examine the consistency of shots for a new

innovative golf club, a golfer was asked to hit 150 shots, 75 with a currently used (7-iron) club, and 75 with the new club.

The distances were recorded. Which 7-iron is more consistent?

Standard Deviation

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Example 4.8 – solution

Standard Deviation

Excel printout, from the “Descriptive Statistics” sub-menu.

Current Innovation

Mean 150.5467 Mean 150.1467Standard Error 0.668815 Standard Error 0.357011Median 151 Median 150Mode 150 Mode 149Standard Deviation 5.792104 Standard Deviation 3.091808Sample Variance 33.54847 Sample Variance 9.559279Kurtosis 0.12674 Kurtosis -0.88542Skewness -0.42989 Skewness 0.177338Range 28 Range 12Minimum 134 Minimum 144Maximum 162 Maximum 156Sum 11291 Sum 11261Count 75 Count 75

The innovation club is more consistent, and because the means are close, is considered a better club

The Standard Deviation

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Interpreting Standard Deviation

The standard deviation can be used to compare the variability of several distributions make a statement about the general shape of a

distribution. The empirical rule: If a sample of observations has a

mound-shaped distribution, the interval

tsmeasuremen the of 68%ely approximat contains )sx,sx(

tsmeasuremen the of 95%ely approximat contains )s2x,s2x( tsmeasuremen the of 99.7%ely approximat contains )s3x,s3x(

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Example 4.9A statistics practitioner wants to describe the way returns on investment are distributed. The mean return = 10% The standard deviation of the return = 8% The histogram is bell shaped.


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Example 4.9 – solutionThe empirical rule can be applied (bell shaped histogram)Describing the return distribution

Approximately 68% of the returns lie between 2% and 18% [10 – 1(8), 10 + 1(8)]

Approximately 95% of the returns lie between -6% and 26% [10 – 2(8), 10 + 2(8)]

Approximately 99.7% of the returns lie between -14% and 34% [10 – 3(8), 10 + 3(8)]


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The proportion of observations in any sample that lie within k standard deviations of the mean is at least 1-1/k2 for k > 1.

This theorem is valid for any set of measurements (sample, population) of any shape!!

K Interval Chebysheff Empirical Rule1 at least 0% approximately 68%2 at least 75% approximately 95%3 at least 89% approximately 99.7%

s2x,s2x sx,sx

s3x,s3x

The Chebysheff’s Theorem

(1-1/12)

(1-1/22)

(1-1/32)

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Example 4.10 The annual salaries of the employees of a chain of computer

stores produced a positively skewed histogram. The mean and standard deviation are $28,000 and $3,000,respectively. What can you say about the salaries at this chain?SolutionAt least 75% of the salaries lie between $22,000 and $34,000 28000 – 2(3000) 28000 + 2(3000)At least 88.9% of the salaries lie between $$19,000 and $37,000 28000 – 3(3000) 28000 + 3(3000)

The Chebysheff’s Theorem

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The coefficient of variation of a set of measurements is the standard deviation divided by the mean value.

This coefficient provides a proportionate measure of variation.

CV : variationoft coefficien Population

x

scv : variationoft coefficien Sample

A standard deviation of 10 may be perceivedlarge when the mean value is 100, but only moderately large when the mean value is 500

The Coefficient of Variation

35Your score

4.4 Measures of Relative Standing and Box Plots

Percentile The pth percentile of a set of measurements is the

value for which • p percent of the observations are less than that value• 100(1-p) percent of all the observations are greater than

that value. Example

• Suppose your score is the 60% percentile of a SAT test. Then 60% of all the scores lie here 40%

36

Commonly used percentiles First (lower)decile = 10th percentile First (lower) quartile, Q1, = 25th percentile Second (middle)quartile,Q2, = 50th percentile Third quartile, Q3, = 75th percentile Ninth (upper)decile = 90th percentile

Quartiles

37

Quartiles

Example

Find the quartiles of the following set of measurements 7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8

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SolutionSort the observations2, 4, 4, 5, 7, 8, 10, 12, 17, 18, 18, 21, 27, 29, 30

At most (.25)(15) = 3.75 observations should appear below the first quartile.Check the first 3 observations on the left hand side.

At most (.25)(15) = 3.75 observations should appear below the first quartile.Check the first 3 observations on the left hand side.

At most (.75)(15)=11.25 observations should appear above the first quartile.Check 11 observations on the right hand side.

At most (.75)(15)=11.25 observations should appear above the first quartile.Check 11 observations on the right hand side.

The first quartileThe first quartile

Comment:If the number of observations is even, two observations remain unchecked. In this case choose the midpoint between these two observations.

Comment:If the number of observations is even, two observations remain unchecked. In this case choose the midpoint between these two observations.

15 observations

Quartiles

39

Find the location of any percentile using the formula

Example 4.11Calculate the 25th, 50th, and 75th percentile of the data in Example 4.1

Location of Percentiles

percentilePtheoflocationtheisLwhere100P

)1n(L

thP

P

percentilePtheoflocationtheisLwhere100P

)1n(L

thP

P

40

2 3

0 5

1

0

LocationLocation

Values

Location 3

Example 4.11 – solution After sorting the data we have 0, 0, 5, 7, 8, 9, 22, 33.

75.210025

)110(L25 3.75

The 2.75th locationTranslates to the value(.75)(5 – 0) = 3.75

2.75


41

Example 4.11 – solution continued

The 50th percentile is halfway between the fifth and sixth observations (in the middle between 8 and 9), that is 8.5.


5.510050

)110(L50

42

Example 4.11 – solution continued

The 75th percentile is one quarter of the distance between the eighth and ninth observation that is14+.25(22 – 14) = 16.


25.810075

)110(L75

Eighth observation

Ninthobservation

43

Quartiles and Variability

Quartiles can provide an idea about the shape of a histogram

Q1 Q2 Q3

Positively skewedhistogram

Q1 Q2 Q3

Negatively skewedhistogram

44

This is a measure of the spread of the middle 50% of the observations

Large value indicates a large spread of the observations

Interquartile range = Q3 – Q1

Interquartile Range

45

1.5(Q3 – Q1) 1.5(Q3 – Q1)

This is a pictorial display that provides the main descriptive measures of the data set:

• L - the largest observation• Q3 - The upper quartile

• Q2 - The median

• Q1 - The lower quartile• S - The smallest observation

S Q1 Q2 Q3 LWhisker Whisker

Box Plot

46

Example 4.14 (Xm02-01)

Box Plot

Bills42.1938.4529.2389.35118.04110.46...

Smallest = 0Q1 = 9.275Median = 26.905Q3 = 84.9425Largest = 119.63IQR = 75.6675Outliers = ()

Left hand boundary = 9.275–1.5(IQR)= -104.226Right hand boundary=84.9425+ 1.5(IQR)=198.4438

9.2750 84.9425 198.4438119.63-104.22626.905

No outliers are found

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Additional Example - GMAT scoresCreate a box plot for the data regarding the GMAT scores of 200 applicants (see GMAT.XLS)

Box Plot

GMAT512531461515...

Smallest = 449Q1 = 512Median = 537Q3 = 575Largest = 788IQR = 63Outliers = (788, 788, 766, 763, 756, 719, 712, 707, 703, 694, 690, 675, )

537512449 575417.5512-1.5(IQR) 575+1.5(IQR)

669.5 788

48

Interpreting the box plot results• The scores range from 449 to 788.• About half the scores are smaller than 537, and about half are larger than

537.• About half the scores lie between 512 and 575.• About a quarter lies below 512 and a quarter above 575.

Q1

512Q2

537Q3

575

25% 50% 25%

449 669.5

Box PlotGMAT - continued

49

50%

25% 25%

The histogram is positively skewed

Q1

512Q2

537Q3

575

25% 50% 25%

449 669.5

Box PlotGMAT - continued

50

Example 4.15 (Xm04-15) A study was organized to compare the quality of

service in 5 drive through restaurants. Interpret the results

Example 4.15 – solution Minitab box plot

Box Plot

51

100 200 300

1

2

3

4

5

C6

C7

Wendy’s service time appears to be the shortest and most consistent.

Hardee’s service time variability is the largest

Jack in the box is the slowest in service

Box Plot

Jack in the Box

Hardee’s

McDonalds

Wendy’s

Popeyes

52

100 200 300

1

2

3

4

5

C6

C7

Popeyes

Wendy’s

Hardee’s

Jack in the Box

Wendy’s service time appears to be the shortest and most consistent.

McDonalds

Hardee’s service time variability is the largest

Jack in the box is the slowest in service

Box Plot

Times are positively skewed

Times are symmetric

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4.5 Measures of Linear Relationship

The covariance and the coefficient of correlation are used to measure the direction and strength of the linear relationship between two variables. Covariance - is there any pattern to the way two

variables move together? Coefficient of correlation - how strong is the linear

relationship between two variables

54

N

)y)((xY)COV(X,covariance Population yixi

N

)y)((xY)COV(X,covariance Population yixi

x (y) is the population mean of the variable X (Y).N is the population size.

1-n)yy)(x(x

y) cov(x,covariance Sample ii

1-n)yy)(x(x

y) cov(x,covariance Sample ii

Covariance

x (y) is the sample mean of the variable X (Y).n is the sample size.

55

Compare the following three sets

Covariance

xi yi (x – x) (y – y) (x – x)(y – y)267

132027

-312

-707

21014

x=5 y =20 Cov(x,y)=17.5

xi yi (x – x) (y – y) (x – x)(y – y)267

272013

-312

70-7

-210-14

x=5 y =20 Cov(x,y)=-17.5

xi yi

267

202713

Cov(x,y) = -3.5

x=5 y =20

56

If the two variables move in opposite directions, (one increases when the other one decreases), the covariance is a large negative number.

If the two variables are unrelated, the covariance will be close to zero.

If the two variables move in the same direction, (both increase or both decrease), the covariance is a large positive number.

Covariance

57

This coefficient answers the question: How strong is the association between X and Y.

yx

)Y,X(COV

ncorrelatio oft coefficien Population

yx

)Y,X(COV

ncorrelatio oft coefficien Population

yxss)Y,Xcov(

r

ncorrelatio oft coefficien Sample

yxss

)Y,Xcov(r

ncorrelatio oft coefficien Sample

The coefficient of correlation

58

COV(X,Y)=0 or r =

+1

0

-1

Strong positive linear relationship

No linear relationship

Strong negative linear relationship

or

COV(X,Y)>0

COV(X,Y)<0


59

If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship).

If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship).

No straight line relationship is indicated by a coefficient close to zero.


60

Compute the covariance and the coefficient of correlation to measure how GMAT scores and GPA in an MBA program are related to one another.

Solution We believe GMAT affects GPA. Thus

• GMAT is labeled X• GPA is labeled Y

The coefficient of correlation and the covariance – Example 4.16

61

1 599 9.6 358801 92.16 5750.4

2 689 8.8 474721 77.44 6063.2

3 584 7.4 341056 54.76 4321.6

4 631 10 398161 100 6310

11 593 8.8 351649 77.44 5218.4

12 683 8 466489 64 5464

Total 7,587 106.4 4,817,755 957.2 67,559.2

Student x y x2 y2 xy

………………………………………………….

n

xx

ns

n

yxyx

n

yx

i

iiii

222

1

1

1

1

),cov(

FormulasShortcut

The coefficient of correlation and the covariance – Example 4.16

cov(x,y)=(1/12-1)[67,559.2-(7587)(106.4)/12]=26.16

Sx = {(1/12-1)[4,817,755-(7587)2/12)]}.5=43.56Sy = similar to Sx = 1.12

r = cov(x,y)/SxSy = 26.16/(43.56)(1.12) = .5362

62

Use the Covariance option in Data Analysis If your version of Excel returns the population covariance and

variances, multiply each one by n/n-1 to obtain the corresponding sample values.

Use the Correlation option to produce the correlation matrix.

The coefficient of correlation and the covariance – Example 4.16 – Excel

GPA GMAT

GPA 1.15

GMAT 23.98 1739.52

GPA GMAT

GPA 1.25

GMAT 26.16 1897.66

1212-1

Variance-Covariance MatrixPopulation values

Sample values

Population values

Sample values

63

Interpretation The covariance (26.16) indicates that GMAT score

and performance in the MBA program are positively related.

The coefficient of correlation (.5365) indicates that there is a moderately strong positive linear relationship between GMAT and MBA GPA.

The coefficient of correlation and the covariance – Example 4.16 – Excel

64

We are seeking a line that best fits the data when two variables are (presumably) related to one another.

We define “best fit line” as a line for which the sum of squared differences between it and the data points is minimized. 2

ii

n

1i)yy(Minimize

The actual y value of point i The y value of point icalculated from the equation i10i xbby

The Least Squares Method

65

The least Squares Method

Errors

Different lines generate different errors, thus different sum of squares of errors.

X

Y

There is a line that minimizes the sum of squared errors

Errors

66

The least Squares Method

The coefficients b0 and b1 of the line that minimizes the sum of squares of errors are calculated from the data.

n

x

xandn

y

ywhere

xbyb

xx

yyxx

s

yxb

n

ii

n

ii

n

ii

n

iii

x

11

10

1

2

121 ,

)(

))((),cov(

n

x

xandn

y

ywhere

xbyb

xx

yyxx

s

yxb

n

ii

n

ii

n

ii

n

iii

x

11

10

1

2

121 ,

)(

))((),cov(

67145.)25.632)(0138(.87.8

87.812

4.106

25.63212

587,7

0138.2.1897

16.26),cov(

10

21

xbybn

yy

n

xx

s

yxb

i

i

x

145.)25.632)(0138(.87.8

87.812

4.106

25.63212

587,7

0138.2.1897

16.26),cov(

10

21

xbybn

yy

n

xx

s

yxb

i

i

x

Scatter Diagram

y = 0.1496 + 0.0138x

6

8

10

12

500 600 700 800

Scatter Diagram

y = 0.1496 + 0.0138x

6

8

10

12

500 600 700 800

Example 4.17 Find the least squares line for Example 4.16 (Xm04-16.xls)

The Least Squares Method

numerical descriptive techniques

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