Chap 3- 1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice

Chapter 3

Numerical Descriptive Measures
Business Statistics: A First Course 6th Edition

In this chapter, you learn:
To describe the properties of

Learning Objectives

Summary Definitions
The central tendency is the extent to which all

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Measures of Central Tendency: The Mean
The arithmetic mean (often just called

Sample size

Observed values

The ith value

Pronounced x-bar

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Measures of Central Tendency: The Mean
The most common measure of central

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11 12 13 14 15 16 17 18 19 20

Mean = 13

11 12 13 14 15 16 17 18 19 20

Mean = 14

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Measures of Central Tendency: The Median

In an ordered array, the median

Median = 13

Median = 13

11 12 13 14 15 16 17 18 19 20

11 12 13 14 15 16 17 18 19 20

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Measures of Central Tendency: Locating the Median
The location of the median

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Measures of Central Tendency: The Mode
Value that occurs most often
Not affected

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

Mode = 9

0 1 2 3 4 5 6

No Mode

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Measures of Central Tendency: Review Example
House Prices:

Mean: ($3,000,000/5)
= $600,000
Median: middle value of ranked data = $300,000
Mode: most frequent value = $100,000

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Measures of Central Tendency: Which Measure to Choose?
The mean is generally

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Measures of Central Tendency: Summary
Central Tendency
Arithmetic Mean
Median
Mode
Middle value in the ordered

Most frequently observed value

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Same center,
different variation
Measures of Variation
Measures of variation give information

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Measures of Variation: The Range
Simplest measure of variation
Difference between the largest

Range = Xlargest – Xsmallest

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

Range = 13 - 1 = 12

Example:

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Measures of Variation: Why The Range Can Be Misleading
Ignores the way

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

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

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

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Average (approximately) of squared deviations of values from the mean

Sample

Measures of Variation: The Sample Variance

Where

= arithmetic mean
n = sample size
Xi = ith value of the variable X

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Measures of Variation: The Sample Standard Deviation
Most commonly used measure of

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Measures of Variation: The Standard Deviation
Steps for Computing Standard Deviation

1. Compute the

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Measures of Variation: Sample Standard Deviation Calculation Example
Sample Data (Xi) :

n = 8 Mean = X = 16

A measure of the “average” scatter around the mean

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Measures of Variation: 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 = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5
S = 4.570

Data C

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Measures of Variation: Comparing Standard Deviations
Smaller standard deviation

Larger standard deviation
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Measures of Variation: Summary Characteristics
The more the data are spread out,

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Measures of Variation: The Coefficient of Variation
Measures relative variation
Always in percentage

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Measures of Variation: Comparing Coefficients of Variation
Stock A:
Average price last year

Both stocks have the same standard deviation, but stock B is less variable relative to its price

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Measures of Variation: Comparing Coefficients of Variation
Stock A:
Average price last year

Stock C has a much smaller standard deviation but a much higher coefficient of variation

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Locating Extreme Outliers: Z-Score
To compute the Z-score of a data value,

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Locating Extreme Outliers: Z-Score
where X represents the data value
X is

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Locating Extreme Outliers: Z-Score
Suppose the mean math SAT score is 490,

A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.

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Shape of a Distribution
Describes how data are distributed
Two useful shape

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Shape of a Distribution (Skewness)
Describes the amount of asymmetry in

Mean = Median

Mean < Median

Mean > Median

Right-Skewed

Left-Skewed

Symmetric

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Skewness
Statistic

< 0 0 >0

Shape of a Distribution (Kurtosis)
Describes relative concentration of values in

Sharper Peak
Than Bell-Shaped

Flatter Than
Bell-Shaped

Bell-Shaped

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Kurtosis
Statistic

< 0 0 >0

curve rises approaching the center of the distribution

Sharper Peak
Than

Flatter Than
Bell-Shaped
(Kurtosis < 0)

Bell-Shaped
(Kurtosis = 0)

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General Descriptive Stats Using Microsoft Excel Functions
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General Descriptive Stats Using Microsoft Excel Data Analysis Tool
Select Data.
Select

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General Descriptive Stats Using Microsoft Excel Data Analysis Tool
4. Enter

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Excel output
Microsoft Excel
descriptive statistics output,
using the house price

House Prices: $2,000,000
500,000 300,000 100,000 100,000

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Minitab Output
Descriptive Statistics: House Price

Quartile Measures
Quartiles split the ranked data into 4 segments with

The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% of the observations are smaller and 50% are larger)
Only 25% of the observations are greater than the third quartile

Q1

Q2

Q3

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Quartile Measures: Locating Quartiles
Find a quartile by determining the value in

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Quartile Measures: Calculation Rules
When calculating the ranked position use the following

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(n = 9)
Q1 is in the

Quartile Measures: Locating Quartiles

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

Q1 and Q3 are measures of non-central location
Q2 = median, is a measure of central tendency

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(n = 9)
Q1 is in the (9+1)/4 =

Quartile Measures Calculating The Quartiles: Example

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

Q1 and Q3 are measures of non-central location
Q2 = median, is a measure of central tendency

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Quartile Measures: The Interquartile Range (IQR)
The IQR is Q3 – Q1

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Calculating The Interquartile Range
Median
(Q2)
X
maximum
X
minimum
Q1
Q3
Example box plot for:
25%

12 30 45 57 70

Interquartile range
= 57 – 30 = 27

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The Five-Number Summary
The five numbers that help describe the center,

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Relationships among the five-number summary and distribution shape
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Five-Number Summary and The Boxplot
The Boxplot: A Graphical display of the

Example:

Xsmallest -- Q1 -- Median -- Q3 -- Xlargest

25% of data 25% 25% 25% of data
of data of data

Xsmallest Q1 Median Q3 Xlargest

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Five-Number Summary: Shape of Boxplots
If data are symmetric around the median

Xsmallest Q1 Median Q3 Xlargest

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Distribution Shape and The Boxplot
Right-Skewed
Left-Skewed
Symmetric
Q1
Q2
Q3
Q1
Q2
Q3
Q1
Q2
Q3
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Boxplot Example
Below is a Boxplot for the following data:
0

0 2 3 5 27

Xsmallest Q1 Q2 Q3 Xlargest

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Numerical Descriptive Measures for a Population
Descriptive statistics discussed previously described

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Numerical Descriptive Measures for a Population: The mean µ
The population

μ = population mean
N = population size
Xi = ith value of the variable X

Where

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Average of squared deviations of values from the mean

Population variance:
Numerical

Where

μ = population mean
N = population size
Xi = ith value of the variable X

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Numerical Descriptive Measures For A Population: The Standard Deviation σ
Most

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Sample statistics versus population parameters
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The empirical rule approximates the variation of data in a

The Empirical Rule

68%

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Approximately 95% of the data in a bell-shaped distribution lies

The Empirical Rule

99.7%

95%

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Using the Empirical Rule
Suppose that the variable Math SAT scores

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Regardless of how the data are distributed, at least (1

Chebyshev Rule

within

At least

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The Covariance
The covariance measures the strength of the linear relationship

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Covariance between two variables:


cov(X,Y) > 0 X

Interpreting Covariance

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Coefficient of Correlation
Measures the relative strength of the linear relationship

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Features of the Coefficient of Correlation
The population coefficient of correlation is

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Scatter Plots of Sample Data with Various Coefficients of Correlation
Y
X
Y
X
Y
X
Y
X
r

r = -.6

r = +.3

r = +1

Y

X

r = 0

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The Coefficient of Correlation Using Microsoft Excel Function
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The Coefficient of Correlation Using Microsoft Excel Data Analysis Tool
Select

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The Coefficient of Correlation Using Microsoft Excel Data Analysis Tool
Input data

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Chap 3-
Interpreting the Coefficient of Correlation Using Microsoft Excel
r = .733

There

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The Coefficient of Correlation Using Minitab

Pitfalls in Numerical Descriptive Measures
Data analysis is objective
Should report the

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Ethical Considerations
Numerical descriptive measures:

Should document both good and bad results
Should

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Chapter Summary
Described measures of central tendency
Mean, median, mode
Described measures of

Chapter Summary
Discussed covariance and correlation coefficient
Addressed pitfalls in numerical descriptive

(continued)

Lecture Slides
For use with Classroom Response Systems
Displaying and Describing

Business Statistics: A First Course

the shape of a distribution we typically do so in

shape of the distribution?
Slide 5-
Copyright © 2011 Pearson Education,

A. It is skewed left.
B. It is skewed right.
C. It is symmetric.
D. It is bimodal.

to use a stem-and-leaf display rather than a boxplot because

kept data records for the past 20 years. If they

a class took the same quiz. Section A had 15

more than two groups, a boxplot does a better job

following is not included in a five-number summary?
median

following are measures of the center of a distribution (circle

following are measures of the spread of a distribution (circle

find a reason for an outlier or remove it, you