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Statistics. Descriptive Statistics – Numerical Measures. Contents. Measures of location Measures of variability Measures of distribution shape , relative location , and detecting outliers Exploratory data analysis Measures of association between two variables

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Statistics

Statistics

Descriptive Statistics – Numerical Measures


Contents

Contents

Measures of location

Measures of variability

Measures of distribution shape , relative location , and detecting outliers

Exploratory data analysis

Measures of association between two variables

The weighted mean and working with grouped data


Contents1

Contents

Measures of Distribution Shape, Relative Location, and Detecting Outliers

Exploratory Data Analysis

Measures of Association Between Two Variables

The Weighted Mean and Working with Grouped Data


Statistics in practice

STATISTICSin PRACTICE

  • Small Fry Design is a toy and accessory company that designs and imports products for infants.

  • Cash flow management is one of the

    most critical activities in the day-to-

    day operation of this company.


Statistics in practice1

STATISTICSin PRACTICE

  • A critical factor in cash flow management is the analysis and control of accounts receivable. By measuring the average age and dollar value of outstanding invoices.

  • The company set the following goals: the average age for outstanding invoices should not exceed 45 days, and the dollar value of invoices more than 60 days old should not exceed 5% of the dollar value of all accounts receivable.


Measures of location

Measures of Location

  • If the measures are computed for data from a sample , they are called sample statistics.

  • If the measures are computed for data from a

  • population , they are called population parameters.

  • A sample statistic is referred to as the point estimator of the corresponding population parameter.


Statistics

Mean

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

  • Population mean m.

  • Sample mean

  • The sample mean is the point estimator of the population mean m.


Statistics

Sample Mean x


Statistics

Population Mean m


Sample mean

Sample Mean

  • Example: Monthly Starting Salaries for a Sample of 12 Business School Graduates

  • Data


Sample mean1

Sample Mean

  • The mean monthly starting salary


Statistics

Sample Mean

  • Example: Apartment Rents

Seventy efficiency apartments

were randomly sampled in

a small college town. The

monthly rent prices for

these apartments are listed

in ascending order on the next slide.


Statistics

Sample Mean


Sample mean2

Sample Mean


Median

Median

  • The median of a data set is the value in the

  • middle when the data items are arranged in

  • ascending order.

  • Whenever a data set has extreme values, the

  • median is the preferred measure of central

  • location.


Median1

Median

  • The median is the measure of location most often

  • reported for annual income and property value

  • data.

  • A few extremely large incomes or property values

  • can inflate the mean.


Median2

  • 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325

  • Middle Two Values

Median

  • Example: Monthly Starting Salaries for a Sample of 12 Business School Graduates

  • We first arrange the data in ascending order.

  • Because n = 12 is even, we identify the middle two values: 2890 and 2920.


Statistics

Median

  • For an odd number of observations:

26

18

27

12

14

27

19

7 observations

in ascending order

27

12

14

18

19

26

27

the median is the middle value.

Median = 19


Statistics

Median

  • For an even number of observations:

8 observations

26

18

27

12

14

27

30

19

in ascending order

27

30

12

14

18

19

26

27

the median is the average of the middle two values.

Median = (19 + 26)/2 = 22.5


Statistics

Mode

  • Example: frequency distribution of 50 Soft Drink Purchases

  • The mode, or most frequently purchased soft drink, is Coke Classic.

Soft Drink Frequency

Coke Classic 19

Diet Coke 8

Dr. Pepper 5

Pepsi-Cola 13

Sprite 5

Total 50


Statistics

Mode

450 occurred most frequently (7 times)

Mode = 450


Percentiles

Percentiles

  • 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.


Percentiles1

Percentiles

  • 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.


Percentiles2

Percentiles

  • Example: Monthly Starting Salaries for a sample of 12 Business School Graduates

  • Let us determine the 85th percentile for the starting salary data


Percentiles3

Percentiles

  • Step 1. Arrange the data in ascending order.

    2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325

  • Step 2.

  • Step 3.

    Because iis not an integer, round up. The position of the 85th percentile is the next integer greater than 10.2, the 11th position.


Statistics

Percentiles

  • Arrange the data in ascending order.

  • Compute index i, the position of the pth

  • percentile.

  • If i is not an integer, round up. The p th

  • percentile is the value in the ith position.

  • If i is an integer, the p th percentile is the average

  • of the values in positionsiand i+1.

i = (p/100)n


90 th percentile

90th Percentile

i = (p/100)n = (90/100)70 = 63

Averaging the 63rd and 64th data values:

90th Percentile = (580 + 590)/2 = 585


Statistics

90th Percentile

“At least 90%

of the items

take on a value

of 585 or less.”

“At least 10%

of the items

take on a value

of 585 or more.”

63/70 = .9 or 90%

7/70 = .1 or 10%


Quartiles

Quartiles

  • Quartiles are specific percentiles.

  • First Quartile = 25th Percentile

  • Second Quartile = 50th Percentile = Median

  • Third Quartile = 75th Percentile


Quartiles1

Quartiles


Third quartile

Third Quartile

Third quartile = 75th percentile

i= (p/100)n = (75/100)70 = 52.5 = 53

Third quartile = 525


Measures of variability

Measures of Variability

  • 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 for each, but also the variability

  • in delivery time for each.


Measures of variability1

Measures of Variability

  • Range

  • InterquartileRange

  • Variance

  • Standard Deviation

  • Coefficient of Variation


Range

Range

  • The range of a data set is the difference between

  • the largest and smallest data values.

  • It is the simplest measure of variability.

  • It is very sensitive to the smallest and largest

  • data values.


Range1

Range

Range = largest value - smallest value

Range = 615 - 425 = 190


Interquartile range

Interquartile Range

  • The interquartile range of a data set is the

  • differencebetween the third quartile and the

  • first quartile.

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

  • It overcomes the sensitivity to extreme data

  • values.


Interquartile range1

Interquartile Range

3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

Interquartile Range = Q3 - Q1 = 525 - 445 = 80


Statistics

It is based on the difference between the value of

each observation (xi) and the mean ( for

a sample, μ for a population).

Variance

The variance is a measure of variability that

utilizes all the data.


Statistics

Variance

The variance is the average of the squared

differences between each data value and the mean.

The variance is computed as follows:

for a sample

for a population


Statistics

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.


Statistics

Standard Deviation

The standard deviation is computed as follows:

for a population

for a sample


Coefficient of variation

Coefficient of Variation

The coefficient of variation indicates how large

the standard deviation is in relation to the mean.

The coefficient of variation is computed as follows:

for a population

for a sample


Statistics

Variance, Standard Deviation,

And Coefficient of Variation

  • Variance

  • Standard Deviation


Statistics

Variance, Standard Deviation,

And Coefficient of Variation

  • Coefficient of Variation

the standard deviation is about 11% of

of the mean .


Measures of distribution shape relative location and detecting outliers

Measures of Distribution Shape,Relative Location, and Detecting Outliers

  • Distribution Shape

  • z-Scores

  • Chebyshev’s Theorem

  • Empirical Rule

  • Detecting Outliers


Distribution shape skewness

Distribution Shape: Skewness

  • An important measure of the shape of a distribution is called skewness.

  • The formula for computing skewness for a data set is somewhat complex.

  • Note: The formula for the skewnessof sample data


Distribution shape skewness1

Distribution Shape: Skewness

  • Skewness can be easily computed using statistical software.


Statistics

Distribution Shape: Skewness

  • Symmetric (not skewed)

  • Skewnessis zero.

  • Mean and median are equal.


Statistics

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = 0

Relative Frequency


Distribution shape skewness2

Distribution Shape: Skewness

  • Moderately Skewed Left

  • Skewnessis negative.

  • Mean will usually be less than the median.


Distribution shape skewness3

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = - .31

Relative Frequency


Distribution shape skewness4

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

  • Moderately Skewed Right

  • Skewnessis positive.

  • Mean will usually be more than the median.

Skewness = .31

Relative Frequency


Statistics

Distribution Shape: Skewness

  • Highly Skewed Right

  • Skewnessis positive (often above 1.0).

  • Mean will usually be more than the median.


Statistics

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = 1.25

Relative Frequency


Distribution shape skewness5

Distribution Shape: Skewness


Statistics

DistributionShape: Skewness

  • Example: Apartment Rents

Seventy efficiency apartments

were randomly sampled in

a small college town. The

monthly rent prices for

these apartments are listed

in ascending order on the next slide.


Statistics

Distribution Shape: Skewness


Statistics

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = .92

Relative Frequency


Statistics

z-Scores

The z-score is often called the standardized value.

It denotes the number of standard deviations a

data value xi is from the mean.


Statistics

z-Scores

  • An observation’s z-score is a measure of the

  • relative location of the observation in a data

  • set.

  • A data value less than the sample mean will

  • have a z-score less than zero.

  • A data value greater than the sample mean

  • will have a z-score greater than zero.

  • A data value equal to the sample mean will

  • have a z-score of zero.


Z scores

z-Scores

  • z-Score of Smallest Value (425)

Standardized Values for Apartment Rents


Chebyshev s theorem

Chebyshev’s Theorem

  • At least (1 - 1/z2) of the items in any data set will

  • be within z standard deviations of the mean,

  • where z is any value greater than 1.


Statistics

  • At least of the data values must be

  • within of the mean.

75%

z = 2 standard deviations

  • At least of the data values must be

  • within of the mean.

89%

z = 3 standard deviations

  • At least of the data values must be

  • within of the mean.

94%

z = 4 standard deviations

Chebyshev’s Theorem


Chebyshev s theorem1

Let z = 1.5 with = 490.80 and s = 54.74

- z(s) = 490.80 - 1.5(54.74) = 409

+ z(s) = 490.80 + 1.5(54.74) = 573

Chebyshev’s Theorem

For example:

At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%

of the rent values must be between

and

(Actually, 86% of the rent valuesarebetween

409 and 573.)


Empirical rule

of the values of a normal random variable

are within of its mean.

68.26%

+/- 1 standard deviation

of the values of a normal random variable

are within of its mean.

95.44%

+/- 2 standard deviations

of the values of a normal random variable

are within of its mean.

99.72%

+/- 3 standard deviations

Empirical Rule

For data having a bell-shaped distribution:


Empirical rule1

99.72%

95.44%

68.26%

Empirical Rule

x

m

m+ 3s

m –3s

m–1s

m+ 1s

m –2s

m+2s


Detecting outliers

Detecting Outliers

  • An outlier is an unusually small or unusually

  • large value in a data set.

  • A data value with a z-score less than -3 or

  • greater than +3 might be considered an outlier.


Detecting outliers1

Detecting Outliers

  • 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


Detecting outliers2

Detecting Outliers

  • The most extreme z-scores are -1.20 and 2.27

  • Using |z| > 3 as the criterion for an outlier,

  • there are no outliers in this data set.


Exploratory data analysis

Exploratory Data Analysis

  • Five-Number Summary

  • Box Plot


Five number summary

Five-Number Summary

1

Smallest Value

2

First Quartile

3

Median

4

Third Quartile

5

Largest Value


Five number summary1

  • 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325

  • Q1=2865 Q2=2905 Q3=3000

  • (Median)

Five-Number Summary

  • Example: Monthly Starting Salaries for a

    sample of 12 Business School Graduates

  • Five-Number Summary


Five number summary2

Five-Number Summary

Lowest Value = 425

First Quartile = 445

Median = 475

Largest Value = 615

Third Quartile = 525


Statistics

Box Plot

  • A box is drawn with its ends located at

  • the first and third quartiles.

  • A vertical line is drawn in the box at the

  • location of the median (second quartile).


Statistics

625

450

375

400

500

525

550

575

600

425

475

Box Plot

Q1 = 445

Q3 = 525

Q2 = 475


Statistics

Box Plot

  • Limits are located (not drawn) using the interquartile range (IQR).

  • Data outside these limits are considered outliers.

  • The locations of each outlier is shown with the symbol * .

  • … continued


Statistics

Box Plot

  • The lower limit is located 1.5(IQR) below Q1.

Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(75)

=332.5

  • The upper limit is located 1.5(IQR) above Q3.

Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75)

= 637.5

  • There are no outliers (values less than 332.5 or

  • greater than 637.5) in the apartment rent data.


Box plot

625

450

375

400

500

525

550

575

600

425

475

Box Plot

  • Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits.

Smallest value

inside limits = 425

Largest value

inside limits = 615


Box plot1

Box Plot

  • Example: Monthly Starting Salaries for a Sample of 12 Business School Graduates

  • Box Plot


Measures of association between two variables

Measures of Association Between Two Variables

  • Covariance

  • Correlation Coefficient


Covariance

Covariance

  • The covariance is a measure of the linear

  • association between two variables.

  • Positive values indicate a positive relationship.

  • Negative values indicate a negative relationship.


Statistics

Covariance

The correlation coefficient is computed as follows:

for

samples

for

populations


Covariance1

Covariance

  • Example: Sample Data for the Stereo and Sound Equipment Store

  • Data


Covariance2

Covariance

  • Scatter Diagram for the Stereo and Sound Equipment Store

  • Sample Covariance


Covariance3

Covariance

  • Partitioned Scatter Diagram for the Stereo and Sound Equipment Store


Correlation coefficient

Correlation Coefficient

  • The coefficient can take on values between

  • -1 and +1.

  • Values near -1 indicate a strong negative linear

  • relationship.

  • Values near +1 indicate a strong positive linear

  • relationship.


Statistics

Correlation Coefficient

The correlation coefficient is computed as follows:

for

samples

for

populations


Statistics

Correlation Coefficient

  • Correlation is a measure of linear association

  • and not necessarily causation.

  • Just because two variables are highly correlated

  • , it does not mean that one variable is the cause of

  • theother.


Statistics

Covariance and Correlation Coefficient

A golfer is interested in investigating

the relationship, if any, between driving

distance and 18-hole score.

Average Driving

Distance (yds.)

Average

18-Hole Score

69

71

70

70

71

69

277.6

259.5

269.1

267.0

255.6

272.9


Statistics

Covariance and Correlation Coefficient

x

y

69

71

70

70

71

69

-1.0

1.0

0

0

1.0

-1.0

277.6

259.5

269.1

267.0

255.6

272.9

10.65

-7.45

2.15

0.05

-11.35

5.95

-10.65

-7.45

0

0

-11.35

-5.95

Average

Total

267.0

70.0

-35.40

Std. Dev.

8.2192

.8944


Statistics

Covariance and Correlation Coefficient

  • Sample Covariance

  • Sample Correlation Coefficient


The weighted mean and working with grouped data

The Weighted Mean andWorking with Grouped Data

  • Weighted Mean

  • Mean for Grouped Data

  • Variance for Grouped Data

  • Standard Deviation for Grouped Data


Weighted mean

Weighted Mean

  • When the mean is computed by giving each

  • datavalue a weight that reflects its importance,

  • it is referred to as a weighted mean.

  • In the computation of a grade point average

  • (GPA), the weights are the number of credit

  • hours earned for each grade.

  • When data values vary in importance, the

  • analyst must choose the weight that best

  • reflects the importance of each value.


Weighted mean1

Weighted Mean

where:

xi= value of observation i

wi= weight for observation i


Grouped data

Grouped Data

  • The weighted mean computation can be

  • used to obtain approximations of the mean,

  • variance, and standard deviation for the

  • grouped data.

  • To compute the weighted mean, we treat the

  • midpoint of each class as though it were the

  • mean of all items in the class.


Grouped data1

Grouped Data

  • We compute a weighted mean of the class

  • midpoints using the class frequencies as

  • weights.

  • Similarly, in computing the variance and

  • standard deviation, the class frequencies are

  • used as weights.


Mean for grouped data

Mean for Grouped Data

  • Sample Data

  • Population Data

where:

fi= frequency of class i

Mi = midpoint of class i


Statistics

Sample Mean for Grouped

Data

Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped

data in the form of a

frequency distribution.


Statistics

Sample Mean for Grouped

Data

This approximation

differs by $2.41

fromthe actual

samplemean of

$490.80.


Variance for grouped data

Variance for Grouped Data

  • For sample data

  • For population data


Statistics

continued

Sample Variance for

Grouped Data


Statistics

Sample Variance for

Grouped Data

  • Sample Variance

s2 = 208,234.29/(70 – 1) = 3,017.89

  • Sample Standard Deviation

This approximation differs by only $.20

from the actual standard deviation of $54.74.


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