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Statistics

Descriptive Statistics – Numerical Measures

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

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

- 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.

- 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.

- 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.

- 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.

Sample Mean x

Population Mean m

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

- The mean monthly starting salary

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.

Sample Mean

- 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.

- 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.

- 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
- Middle Two Values

- 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.

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

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

- 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

450 occurred most frequently (7 times)

Mode = 450

- 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.

- 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.

- Example: Monthly Starting Salaries for a sample of 12 Business School Graduates
- Let us determine the 85th percentile for the starting salary data

- 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.

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

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

Averaging the 63rd and 64th data values:

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

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 are specific percentiles.

- First Quartile = 25th Percentile

- Second Quartile = 50th Percentile = Median

- Third Quartile = 75th Percentile

Third quartile = 75th percentile

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

Third quartile = 525

- 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.

- Range

- InterquartileRange

- Variance

- Standard Deviation

- Coefficient of Variation

- 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.

Range = largest value - smallest value

Range = 615 - 425 = 190

- 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.

3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

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

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.

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

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.

Standard Deviation

The standard deviation is computed as follows:

for a population

for a sample

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

Variance, Standard Deviation,

And Coefficient of Variation

- Variance

- Standard Deviation

Variance, Standard Deviation,

And Coefficient of Variation

- Coefficient of Variation

the standard deviation is about 11% of

of the mean .

- Distribution Shape
- z-Scores
- Chebyshev’s Theorem
- Empirical Rule
- Detecting Outliers

- 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

- Skewness can be easily computed using statistical software.

Distribution Shape: Skewness

- Symmetric (not skewed)
- Skewnessis zero.
- Mean and median are equal.

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = 0

Relative Frequency

- Moderately Skewed Left
- Skewnessis negative.
- Mean will usually be less than the median.

.35

.30

.25

.20

.15

.10

.05

0

Skewness = - .31

Relative Frequency

.35

.30

.25

.20

.15

.10

.05

0

- Moderately Skewed Right
- Skewnessis positive.
- Mean will usually be more than the median.

Skewness = .31

Relative Frequency

Distribution Shape: Skewness

- Highly Skewed Right
- Skewnessis positive (often above 1.0).
- Mean will usually be more than the median.

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = 1.25

Relative Frequency

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.

Distribution Shape: Skewness

.35

.30

.25

.20

.15

.10

.05

0

Distribution Shape: Skewness

Skewness = .92

Relative Frequency

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.

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-Score of Smallest Value (425)

Standardized Values for Apartment Rents

- 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.

- 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

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

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.)

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

For data having a bell-shaped distribution:

99.72%

95.44%

68.26%

x

m

m+ 3s

m –3s

m–1s

m+ 1s

m –2s

m+2s

- 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.

- 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

- 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.

- Five-Number Summary
- Box Plot

1

Smallest Value

2

First Quartile

3

Median

4

Third Quartile

5

Largest Value

- 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
- Q1=2865 Q2=2905 Q3=3000
- (Median)

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

- Five-Number Summary

Lowest Value = 425

First Quartile = 445

Median = 475

Largest Value = 615

Third Quartile = 525

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).

625

450

375

400

500

525

550

575

600

425

475

Box Plot

Q1 = 445

Q3 = 525

Q2 = 475

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

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.

625

450

375

400

500

525

550

575

600

425

475

- 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

- Example: Monthly Starting Salaries for a Sample of 12 Business School Graduates
- Box Plot

- Covariance
- Correlation Coefficient

- The covariance is a measure of the linear
- association between two variables.

- Positive values indicate a positive relationship.

- Negative values indicate a negative relationship.

Covariance

The correlation coefficient is computed as follows:

for

samples

for

populations

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

- Scatter Diagram for the Stereo and Sound Equipment Store
- Sample Covariance

- Partitioned Scatter Diagram for the Stereo and Sound Equipment Store

- 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.

Correlation Coefficient

The correlation coefficient is computed as follows:

for

samples

for

populations

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.

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

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

Covariance and Correlation Coefficient

- Sample Covariance

- Sample Correlation Coefficient

- Weighted Mean
- Mean for Grouped Data
- Variance for Grouped Data
- Standard Deviation for Grouped Data

- 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.

where:

xi= value of observation i

wi= weight for observation i

- 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.

- 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.

- Sample Data

- Population Data

where:

fi= frequency of class i

Mi = midpoint of class i

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.

Sample Mean for Grouped

Data

This approximation

differs by $2.41

fromthe actual

samplemean of

$490.80.

- For sample data

- For population data

continued

Sample Variance for

Grouped Data

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.