Chapter 3 Descriptive Statistics: Numerical Measuresdscsss/teaching/mgs9920/slides/ch03 ver3.pdf · 1 Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Slide 2 Learning

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1Slide

Chapter 3Descriptive Statistics: Numerical Measures

2Slide

Learning objectives

1. Single variable – Part I (Basic)1.1. How to calculate and use the measures of location1.2. How to calculate and use the measures of variability

2. Single variable – Part II (Application)2.1. Understand what the measures of location (e.g., mean,

median, mode) tell us about distribution shape - Discuss its use in manipulating simulated experiments

2.2. How to detect outliers using z-score and empirical rule2.3. How to use Box plot to explore data 2.4. How to calculate weighted mean 2.5. How to calculate mean and variance for grouped data

3. Two variables3.1. How to calculate and use the measures of association

- Covariance, Correlation coefficient

2

3Slide

L.O. 1. Numerical measures – Part I

n Numerical measures

n Measures of Location•Mean, median, mode, percentiles, quartiles

n Measures of Variability•Range, interquartile range, variance, standard

deviation, coefficient of variation

4Slide

Numerical Measures

If the measures are computedfor data from a sample,

they are called sample statistics.

If the measures are computedfor data from a population,

they are called population parameters.

A sample statistic is referred toas the point estimator of the

corresponding population parameter.

3

5Slide

Mean

n The mean of a data set is the average of all the data values.

n The sample mean is the point estimator of the population mean µ.

xx

Number ofobservationsin the sample

Sum of the valuesof the n observations

ixx

n=

∑ ixx

n=

∑

Number ofobservations inthe population

Sum of the valuesof the N observations

ix

Nµ =

∑ ix

Nµ =

∑

•L.O. 1.1.•Mean

•Median•Mode

•Percentile•Quartile

6Slide

Median

n The median of a data set is the value in the middle when the data items are arranged in ascending order.• For odd number of observations:§ the median is the middle value

• For even number of observations:§ the median is the average of the middle two values.

n Whenever a data set has extreme values, the median is the preferred measure of central location.• Often used in annual income and property value data

•L.O. 1.1.•Mean

•Median•Mode


4

7Slide

Mode

n The mode of a data set is the value that occurs with the greatest frequency.

n The greatest frequency can occur at two ormore different values.• If the data have exactly two modes, the data are

bimodal.• If the data have more than two modes, the data are

multimodal.

•L.O. 1.1.•Mean

•Median•Mode


8Slide

Example

n Q4 (p. 84)

Compute the mean, median, and mode of the following sample:

53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53

ØMean = 59.727 ØMedian = 57ØMode = 53

nWhat is the median, if 59 is added to the data?

ØMedian = 57.5 = (57+58)/2

•L.O. 1.1.•Mean

•Median•Mode


5

9Slide

Percentiles

n A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.• Admission test scores for colleges and universities are

frequently reported in terms of percentiles.

n The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.

•L.O. 1.1.•Mean

•Median•Mode


10Slide

Percentiles

Arrange the data in ascending order.

Compute index i, the position of the pth percentile.

i = (p/100)n

If i is not an integer, round up. The pth percentileis the value in the ith position.

If i is an integer, the pth percentile is the averageof the values in positions i and i+1.

•L.O. 1.1.•Mean

•Median•Mode


6

11Slide

Quartiles

n Quartiles are specific percentiles.n First Quartile = 25th Percentilen Second Quartile = 50th Percentile = Mediann Third Quartile = 75th Percentile

•L.O. 1.1.•Mean

•Median•Mode


12Slide

Example: Percentiles and Quartiles

n Q4 (p. 84)

Find 25th and 75th percentiles from the sample below:

53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53

Ø 25th percentile = First quartile = 53

Ø 75th percentile = Third quartile = 68

•L.O. 1.1.•Mean

•Median•Mode


7

13Slide

Measures of Variability

n It is often desirable to consider measures of variability (dispersion), as well as measures of location.• For example, in choosing supplier A or supplier B we

might consider not only the average delivery time foreach, but also the variability in delivery time for each.

n Rangen Interquartile Rangen Variancen Standard Deviationn Coefficient of Variation

14Slide

Range

n The range of a data set is the difference between the largest and smallest data values.

n It is the simplest measure of variability.n It is very sensitive to the smallest and largest data

values.

n Range of the sample:53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53

= 70 – 53 = 17

•L.O. 1.2.•Range

•IQR•Variance

•St. Deviation•Coefficient of

variation

8

15Slide

Interquartile Range (IQR)

n The interquartile range of a data set is the difference between the third quartile and the first quartile.

n It is the range for the middle 50% of the data.

n It overcomes the sensitivity to extreme data values.

n IQR of the sample:53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53

= 68 – 53 = 15


•IQR•Variance


variation

16Slide

The variance is a measure of variability that utilizesall the data.

Variance

The variance is computed as follows:

The variance is the average of the squareddifferences between each data value and the mean.

for asample

for apopulation

σµ2

2=

−∑( )xNiσ

µ22

=−∑( )x

Nis

xi xn

22

1=

−∑−

( )s

xi xn

22

1=

−∑−

( )


•IQR•Variance


variation

9

17Slide

Standard Deviation

The standard deviation of a data set is the positivesquare root of the variance.

It is measured in the same units as the data, makingit more easily interpreted than the variance.

The standard deviation is computed as follows:

for asample

for apopulation

s s= 2s s= 2 σ σ= 2σ σ= 2


•IQR•Variance


variation

18Slide

The coefficient of variation is computed as follows:

Coefficient of Variation

100 %sx

×

100 %sx

×

The coefficient of variation indicates how large thestandard deviation is in relation to the mean.

for asample

for apopulation

100 %σµ

×

100 %σ

µ

×


•IQR•Variance


variation

10

19Slide

× = × = 6.74100 % 100 % 11.28%59.73

sx

× = × = 6.74100 % 100 % 11.28%59.73

sx

−= =

−∑ 2

2 ( ) 45.418

1ix x

sn

−= =

−∑ 2

2 ( ) 45.418

1ix x

sn

= = =2 33.82 6.74s s= = =2 33.82 6.74s s

the standarddeviation is

about 11% ofof the mean

n Variance

n Standard Deviation

n Coefficient of Variation

Example: Variance, Standard Deviation,And Coefficient of Variation

Consider the same data set:53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53


•IQR•Variance


variation

20Slide

L.O. 2. Numerical measure – Part II

n Measures of Distribution Shapen Detecting Outliers

• z-score, empirical rule

n Exploratory Data Analysis

n The Weighted Mean and Working with Grouped Data

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21Slide

Distribution Shape

n Symmetric (not skewed)• Skewness is zero.•Mean and median are equal.

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0

•L.O. 2.•Shape

•z-score•Empirical Rule

•Exploratory•Weighted mean

•Grouped data

22Slide

Distribution Shape

n Moderately Skewed Left• Skewness is negative.•Mean will usually be less than the median.

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0

•L.O. 2.•Shape



•Grouped data

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23Slide

Distribution Shape

n Moderately Skewed Right• Skewness is positive.•Mean will usually be more than the median.

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0

•L.O. 2.•Shape



•Grouped data

24Slide

Distribution Shape

n Highly Skewed Right• Skewness is positive (often above 1.0).•Mean will usually be more than the median.

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0

•L.O. 2.•Shape



•Grouped data

13

25Slide

The z-score is often called the standardized value.

It denotes the number of standard deviations a datavalue xi is from the mean.

z-Scores

z x xsi

i=−z x xsi

i=−

•L.O. 2.•Shape



•Grouped data

26Slide

z-Scores

n A data value less than the sample mean will have az-score less than zero.

n A data value greater than the sample mean will havea z-score greater than zero.

n A data value equal to the sample mean will have az-score of zero.

n An observation’s z-score is a measure of the relativelocation of the observation in a data set.

•L.O. 2.•Shape



•Grouped data

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27Slide

Empirical Rule

For data having a bell-shaped distribution:

of the values of a normal random variableare within of its mean.68.26%

+/- 1 standard deviation


+/- 2 standard deviations


+/- 3 standard deviations

•L.O. 2.•Shape



•Grouped data

28Slide

Empirical Rule

xµ – 3σ µ – 1σ

µ – 2σµ + 1σ

µ + 2σµ + 3σµ

68.26%95.44%99.72%

•L.O. 2.•Shape



•Grouped data

15

29Slide

Detecting Outliers

n An outlier is an unusually small or unusually largevalue in a data set.

n A data value with a z-score less than -3 or greaterthan +3 might be considered an outlier.

n It might be:• an incorrectly recorded data value• a data value that was incorrectly included in the

data set• a correctly recorded data value that belongs in

the data set

•L.O. 2.•Shape



•Grouped data

30Slide

Exploratory Data Analysis

• Five-Number Summary•Box Plot

n The techniques of exploratory data analysis consist ofsimple arithmetic and easy-to-draw pictures that canbe used to summarize data quickly.

•L.O. 2.•Shape



•Grouped data

16

31Slide

Five-Number Summary

1 Smallest Value

First Quartile

Median

Third Quartile

Largest Value

2

3

4

5

Sample: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53

53

53

57

68

70

•L.O. 2.•Shape



•Grouped data

32Slide

Box Plot

n A box plot is based on a five-number summary.

20 28 36 44 52 60 68 76 84 92 100

Q1 = 53 Q3 = 68Q2 = 57

Lower limit=30.5

1.5*IQR 1.5*IQR

Upper limit=90.5

Largest value (70)

Whisker

No whisker this side: smallest value = Q1

*

Outlier

•L.O. 2.•Shape



•Grouped data

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33Slide

The Weighted Mean andWorking with Grouped Data

n Weighted Meann Mean for Grouped Datan Variance for Grouped Datan Standard Deviation for Grouped Data

•L.O. 2.•Shape



•Grouped data

34Slide

Weighted Mean

n When the mean is computed by giving each datavalue a weight that reflects its importance, it isreferred to as a weighted mean.

n Class grade is usually computed by weighted mean.

n When data values vary in importance, the analystmust choose the weight that best reflects theimportance of each value.

weightIn class midterm exam Descriptive statistics and distributions 40% Final group project Statistical inference 30% Group project presentation 10% Homework 10% Participation 10%

•L.O. 2.•Shape



•Grouped data

18

35Slide

Weighted Mean

i i

i

w xx

w= ∑

∑i i

i

w xx

w= ∑

∑where:

xi = value of observation iwi = weight for observation i

•L.O. 2.•Shape



•Grouped data

36Slide

Grouped Data

n The weighted mean computation can be used toobtain approximations of the mean, variance, andstandard deviation for the grouped data.

n To compute the weighted mean, we treat themidpoint of each class as though it were the meanof all items in the class.

n We compute a weighted mean of the class midpointsusing the class frequencies as weights.

n Similarly, in computing the variance and standarddeviation, the class frequencies are used as weights.

•L.O. 2.•Shape



•Grouped data

19

37Slide

Mean for Grouped Data

i if Mx

n= ∑ i if M

xn

= ∑

NMf ii∑=µ

NMf ii∑=µ

where: fi = frequency of class i

Mi = midpoint of class i

n Sample Data

n Population Data

•L.O. 2.•Shape



•Grouped data

38Slide

Given below is the previous sample of monthly rentsfor 70 efficiency apartments, presented here as groupeddata in the form of a frequency distribution.

Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Sample Mean for Grouped Data•L.O. 2.•Shape



•Grouped data

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39Slide

Sample Mean for Grouped Data

This approximationdiffers by $2.41 fromthe actual samplemean of $490.80.

34,525 493.2170

x = =34,525 493.21

70x = =

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

f iM i

3436.07641.55634.03916.03566.52118.01099.02278.01179.03657.034525.0

•L.O. 2.•Shape



•Grouped data

40Slide

Variance for Grouped Data

s f M xn

i i22

1=

−∑−

( )s f M xn

i i22

1=

−∑−

( )

σ µ22

= −∑ f MN

i i( )σ µ22

= −∑ f MN

i i( )

n For sample data

n For population data

•L.O. 2.•Shape



•Grouped data

21

41Slide

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

Sample Variance for Grouped Data

M i - x-63.7-43.7-23.7-3.716.336.356.376.396.3116.3

f i(M i - x )2

32471.7132479.596745.97110.11

1857.555267.866337.13

23280.6618543.5381140.18

208234.29

(M i - x )2

4058.961910.56562.16

13.76265.36

1316.963168.565820.169271.76

13523.36

continued

•L.O. 2.•Shape



•Grouped data

42Slide

3,017.89 54.94s = =3,017.89 54.94s = =

s2 = 208,234.29/(70 – 1) = 3,017.89

This approximation differs by only $.20 from the actual standard deviation of $54.74.

Sample Variance for Grouped Data

n Sample Variance

n Sample Standard Deviation

•L.O. 2.•Shape



•Grouped data

22

43Slide

L.O. 3. Measures of Association Between Two Variables

n Covariancen Correlation Coefficient

44Slide

Covariance

Positive values indicate a positive relationship.

Negative values indicate a negative relationship.

The covariance is a measure of the linear associationbetween two variables.

•L.O. 3.•Covariance•Correlation

23

45Slide

Covariance

The correlation coefficient is computed as follows:

forsamples

forpopulations

s x x y ynxy

i i= − −∑−

( )( )1

s x x y ynxy

i i= − −∑−

( )( )1

σµ µ

xyi x i yx y

N=

− −∑( )( )σ

µ µxy

i x i yx yN

=− −∑( )( )


46Slide

Correlation Coefficient

Values near +1 indicate a strong positive linearrelationship.

Values near -1 indicate a strong negative linearrelationship.

The coefficient can take on values between -1 and +1.


24

47Slide

The correlation coefficient is computed as follows:

forsamples

forpopulations

rss sxy

xy

x y=r

ss sxy

xy

x y= ρ

σ

σ σxyxy

x y=ρ

σ

σ σxyxy

x y=

Correlation Coefficient•L.O. 3.

•Covariance•Correlation

48Slide

Correlation Coefficient

Just because two variables are highly correlated, it does not mean that one variable is the cause of theother.

Correlation is a measure of linear association and notnecessarily causation.


25

49Slide

In class Exercise

n Q45 (p. 112)n Q46 (p. 112)


50Slide

End of Chapter 3

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Chapter 3 Descriptive Statistics: Numerical Measuresdscsss/teaching/mgs9920/slides/ch03 ver3.pdf · 1 Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Slide 2 Learning