Business Statistics: A Decision-Making Approach, 7th edition · Chapter Topics Measures of Center and Location Mean, median, mode Other measures of Location Weighted mean, percentiles,

Chapter 3

Describing Data Using

Numerical Measures

Business Statistics

Department of Quantitative Methods & Information Systems

Dr. Mohammad Zainal QMIS 120

After completing this chapter, you should be able to:

Compute and interpret the mean, median, and mode for a

set of data

Compute the range, variance, and standard deviation and

know what these values mean

Construct and interpret a box and whisker graph

Compute and explain the coefficient of variation and z scores

Use numerical measures along with graphs, charts, and

tables to describe data

Chapter Goals

QMIS 120, by Dr. M. Zainal Chap 3-2

Chapter Topics

Measures of Center and Location

Mean, median, mode

Other measures of Location

Weighted mean, percentiles, quartiles

Measures of Variation

Range, interquartile range, variance and standard

deviation, coefficient of variation

Using the mean and standard deviation together

Coefficient of variation, z-scores


Summary Measures

Center and Location

Mean

Median

Mode

Other Measures

of Location

Weighted Mean

Describing Data Numerically

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range

Percentiles

Interquartile Range

Quartiles


Measures of Center and Location

Center and Location

Mean Median Mode Weighted Mean

N

x

n

x

x

N

i

i

n

i

i

1

1

i

ii

W

i

iiW

w

xw

w

xwX

Overview


Mean (Arithmetic Average)

The Mean is the arithmetic average of data

values

Population mean

Sample mean n = Sample Size

N = Population Size

n

xxx

n

x

x n

n

i

i

211

N

xxx

N

xN

N

i

i

211


Mean (Arithmetic Average)

The most common measure of central tendency

Mean = sum of values divided by the number of values

Affected by extreme values (outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

35

15

5

54321

4

5

20

5

104321


Median

In an ordered array, the median is the “middle”

number, i.e., the number that splits the

distribution in half

The median is not affected by extreme values

0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

Median = 3


Median

To find the median, sort the n data values

from low to high (sorted data is called a

data array)

Find the value in the i = (1/2)n position

The ith position is called the Median Index

Point

If i is not an integer, round up to next highest

integer

(continued)

Chap 3-9 QMIS 120, by Dr. M. Zainal

Median Example

Note that n = 13

Find the i = (1/2)n position:

i = (1/2)(13) = 6.5

Since 6.5 is not an integer, round up to 7

The median is the value in the 7th position:

Md = 12

(continued)

Data array:

4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24


Shape of a Distribution

Describes how data is distributed

Symmetric: identical on both sides of its central point

Skewed: the tail on one side is longer than the tail

on the other side


Shape of a Distribution

Mean = Median

Mean < Median Median < Mean

Right-Skewed Left-Skewed Symmetric

(Longer tail extends to left) (Longer tail extends to right)


Mode

A measure of location

The value that occurs most often

Not affected by extreme values

Used for either numerical or categorical data

There may be no mode

There may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 5

0 1 2 3 4 5 6

No Mode


Weighted Mean

Used when values are grouped by frequency or

relative importance

Days to

Complete Frequency

5 4

6 12

7 8

8 2

Example: Sample of

26 Repair Projects


Five houses on a hill by the beach

Review Example

$2,000 K

$500 K

$300 K

$100 K

$100 K

House Prices:

$2,000,000

500,000

300,000

100,000

100,000


Summary Statistics


Mean is generally used, unless

extreme values (outliers) exist

Then Median is often used, since

the median is not sensitive to

extreme values.

Example: Median home prices may be

reported for a region – less sensitive to

outliers

Which measure of location is the “best”?


Which measure of location is the “best”?

Example: Find the mean and median for the

following two sets of measurements

2, 9, 11, 5, 6 2, 9, 110, 5, 6


Measures of Variation

Variation

Variance Standard Deviation Coefficient of

Variation

Population

Variance

Sample

Variance

Population

Standard Deviation

Sample

Standard Deviation

Range

Interquartile

Range


Measures of variation give information on

the spread or variability of the data

values.

Variation

Same center,

different variation


Range

Simplest measure of variation

Difference between the largest and the smallest

observations:

Range = xmaximum – xminimum

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13

Example:


Ignores the way in which data are distributed

Sensitive to outliers

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

Disadvantages of the Range

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119


Average of squared deviations of values from

the mean

Population variance:

Sample variance:

Variance

N

μ)(x

σ

N

1i

2

i2

1- n

)x(x

s

n

1i

2

i2


Standard Deviation

Most commonly used measure of variation

Shows variation about the mean

Has the same units as the original data

Population standard deviation:

Sample standard deviation:

N

μ)(x

σ

N

1i

2

i

1-n

)x(x

s

n

1i

2

i


Calculation Example: Sample Standard Deviation

Sample

Data (Xi) : 10 12 14 15 17 18 18 24


Comparing Standard Deviations

Mean = 15.5

s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C

Same mean, but different

standard deviations:


Using Microsoft Excel

Descriptive Statistics are easy to obtain

from Microsoft Excel

Use menu choice:

Data / data analysis / descriptive statistics

Enter details in dialog box


Using Excel

Select:

Data / data analysis / descriptive statistics


Enter dialog box details

Check box for summary statistics

Click OK

Using Excel (continued)


Excel output

Microsoft Excel

descriptive statistics output,

using the house price data:

House Prices:

$2,000,000

500,000

300,000

100,000

100,000


Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Is used to compare two or more sets of data

measured in different units

100%x

sCV

100%

μ

σCV

Population Sample


Comparing Coefficients of Variation

Stock A:

Average price last year = $50

Standard deviation = $5

Stock B:

Average price last year = $100

Standard deviation = $5

Both stocks

have the same

standard

deviation, but

stock B is less

variable relative

to its price

10%100%$50

$5100%

x

sCVA

5%100%$100

$5100%

x

sCVB


Mean for grouped data

Once we group the data, we no longer know the values

of individual observations.

Thus, we find an approximation for the sum of these

values.


class. a offrequency theis andmidpoint theis Where

: samplefor Mean

:populationfor Mean

fm

n

mfx

N

mf

Variance and Standard

Deviation for grouped data

Also, we find an approximation for the variance and

standard deviation of grouped data.


class. a of frequency the is andmidpoint the is Where

1 : Sample

:Population

2

2

2

2

2

2

fm

n

n

mffm

s

N

N

mffm

22 and ss

Example: Grouped Data

The table below gives the frequency distribution of

the daily commuting times (in minutes) from home to

CBA for all 25 students in QMIS 120. Calculate the

mean and the standard deviation of the daily

commuting times.


f Daily commuting time (min)

4 0 to less than 10

9 10 to less than 20




25 Total

Example: Grouped Data


(continued)

Other Location Measures

Other Measures

of Location

Percentiles Quartiles

1st quartile = 25th percentile

2nd quartile = 50th percentile

= median

3rd quartile = 75th percentile

The pth percentile in a data array:

p% are less than or equal to this

value

(100 – p)% are greater than or

equal to this value

(where 0 ≤ p ≤ 100)


Percentiles

The pth percentile in an ordered array of n values is the

value in ith position, where

(n)100

pi

If i is not an integer,

round up to the next

higher integer value


Percentiles

Example: Find the 60th percentile in an ordered array of

the following 19 values.

6 7 2 3 5 1 8 6 7 3 9 4 2 1 5 4 3 9 6

Solution:


Quartiles

Quartiles split the ranked data into 4 equal

groups:

Note that the second quartile (the 50th percentile)

is the median

25% 25% 25% 25%

Q1 Q2 Q3


Quartiles

Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22

Example: Find the first quartile and third quartile

(n = 9)

Q1 = 25th percentile, so find i : i = (9) = 2.25

so round up and use the value in the 3rd position: Q1 = 13

Q3 = 75th percentile, so find i : i = (9) = 6.75

so round up and use the value in the 7th position: Q3 = 18

25 100

75 100


Box and Whisker Plot

A graphical display of data using a central “box” and extended “whiskers”:

Example:


25% 25% 25% 25%

* *

largest values* Smallest values*

Q1 Q2 Q3 Lower fence Upper fence

Outliers


Box and whisker plot gives a graphic presentation of

data using five measures:

Q1, Q2, Q3, smallest, and largest values*.

Can help to visualize the center, the spread, and the

skewness of a data set.

Very good tool of comparing more than a distribution.

* *



Can help in detecting outliers.

Detecting an outlier: Lower fence: Q1 – 1.5(Q3 - Q1)

Upper fence: Q3 + 1.5(Q3 - Q1)

If a data point is larger than the upper fence or smaller than the lower fence, it is considered to be an outlier.

25% 25% 25% 25%

* *


Q1 Q2 Q3 Lower fence Upper fence

Outliers

Constructing the Box and Whisker Plot

Lower 1st Median 3rd Upper

Limit Quartile Quartile Limit

* *

The lower limit is

Q1 – 1.5 (Q3 – Q1)

The upper limit is

Q3 + 1.5 (Q3 – Q1)

The center box extends from Q1 to Q3

The line within the box is the median

The whiskers extend to the smallest and largest values within

the calculated limits

Outliers are plotted outside the calculated limits


Shape of Box and Whisker Plots

The Box and central line are centered between the endpoints if data is symmetric around the median

(A Box and Whisker plot can be shown in either vertical or horizontal format)


Distribution Shape and Box and Whisker Plot

Right-Skewed Left-Skewed Symmetric

Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3


Box-and-Whisker Plot Example

Below is a Box-and-Whisker plot for the following data:

0 2 2 2 3 3 4 5 6 11 27

This data is right skewed, as the plot depicts

0 2 3 6 11 27

Min Q1 Q2 Q3 Max Outlier

*

Upper limit = Q3 + 1.5 (Q3 – Q1)

= 6 + 1.5 (6 – 2) = 12

27 is above the

upper limit so is

shown as an outlier


Interquartile Range

Can eliminate some outlier problems by using

the interquartile range

Eliminate some high-and low-valued

observations and calculate the range from the

remaining values.

Interquartile range = 3rd quartile – 1st quartile


Interquartile Range Example

Median

(Q2) X

maximum X minimum Q1 Q3

Example:

25% 25% 25% 25%

12 30 45 57 70

Interquartile range

= 57 – 30 = 27


A standardized data value refers to

the number of standard deviations a

value is from the mean

Standardized data values are

sometimes referred to as z-scores

Standardized Data Values


where:

x = original data value

μ = population mean

σ = population standard deviation

z = standard score

(number of standard deviations x is from μ)

Standardized Population Values

σ

μx z


where:

x = original data value

x = sample mean

s = sample standard deviation

z = standard score

(number of standard deviations x is from μ)

Standardized Sample Values

s

xx z


IQ scores in a large population have a bell-shaped distribution with mean μ = 100 and standard deviation σ = 15

Find the standardized score (z-score) for a person with an IQ of 121.

Standardized Value Example

Answer:


If the data distribution is bell-shaped, then

the interval:

contains about 68% of the values in

the population or the sample

The Empirical Rule

1σμ

μ

68%

1σμ


contains about 95% of the values in

the population or the sample

contains about 99.7% of the values

in the population or the sample

The Empirical Rule

2σμ

3σμ

3σμ

99.7% 95%

2σμ


Example: The age

distribution of a sample

of 5000 persons is bell-

shaped with a mean of

40 years and a standard

deviation of 12 years.

Determine the

approximate percentage

of people who are 16 to

64 years old.

The Empirical Rule


Sometimes a data set may have one or more

observation that is unusually small or large

value.

An experienced statistician may face the

following situations and need to take an action

Detecting Outliers


Outlier Action

A data value that was incorrectly recorded Correct it before any further

analysis

A data value that was incorrectly included Remove it before any further

analysis

A data value that belongs to the data set and

correctly recorded

Keep it !

This extreme value is called an outlier and can

be detected using the z-score and the empirical

rule for data with bell-shape distribution.

Detecting Outliers


3σμ

99.7% Outliers Outliers

Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean

Examples:

(1 - 1/12) = 0% ……..... k=1 (μ ± 1σ)

(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)

(1 - 1/32) = 89% ………. k=3 (μ ± 3σ)

Tchebysheff’s Theorem

within At least




k = 2 k = 3

Example: The average systolic blood pressure for 4000

women who were screened for high blood pressure was

found to be 187 with standard deviation of 22. Using

Chebyshev’s theorem, find at least what percentage of

women in this group have a systolic blood pressure

between 143 and 231.





(continued)

So far we have studied numerical methods to describe data with one variable.

Often decision makers are interested in the relationship between two variables.

To do so, we will use descriptive measure that is called covariance.

Covariance assigns a numerical value to the linear relationship between two variables (see scatter diagram)

Measures of association

between two variables


It is given by:

A big disadvantage of the covariance is that it depends on the units of measurement for x and y.




N

yx

n

yyxx

yixi

ii

xy

xy

: covariance Population

1S :covariance Sample

For the same data set, we will have two different covariance values depending on the units (i.e. height in meters or centimeters will make a big difference).

Pearson’s correlation coefficient is a good remedy to that problem as it can go only from -1 to 1.




It is given by




N

yx

n

yyxx

yixi

ii

xy

xy

: covariance Population

1S :covariance Sample

Example: A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score.

Example


Average Driving Distance (m)

Average 18-Hole Score

277.6 69

259.5 71

269.1 70

267.0 70

255.6 71

272.9 69

Example


Problems

Consider the following sample of four purchases of one stock in

the KSE. Find the average cost of the stock.

Purchase Price Quantity

1 .300 5,000

2 .325 15,000

3 .350 10,000

4 .295 20,000


Problems

Construct a box plot for the following data:

340 300 400 360 320 290 260 330


Problems

Example: You are given 8 measurements: 3, 5, 4, 6, 12, 5, 6, 7.

Find

a) The mean b) The median c) The mode


Problems

Find the covariance and Pearson’s correlation coefficient for

the following data


Week Number of commercials (x) Sales in $ (y)

1 2 50

2 5 57

3 1 41

4 3 54

5 4 54

6 1 38

7 5 63

8 3 48

9 4 59

10 2 46

Mean 3 51

St. Dev. 1.49 7.93

Measures of association between two variables


Chapter Summary

Described measures of center and location

Mean, median, mode, weighted mean

Discussed percentiles and quartiles

Created Box and Whisker Plots

Illustrated distribution shapes

Symmetric, skewed


Chapter Summary

Described measure of variation

Range, interquartile range, variance,

standard deviation, coefficient of variation

Discussed Tchebysheff’s Theorem

Calculated standardized data values

(continued)


Copyright

The materials of this presentation were mostly

taken from the PowerPoint files accompanied

Business Statistics: A Decision-Making Approach,

7e © 2008 Prentice-Hall, Inc.


Documents

Business Statistics: A Decision-Making Approach, 7th edition · Chapter Topics Measures of Center and Location Mean, median, mode Other measures of Location Weighted mean, percentiles,