business and economics statics principles

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-1
Statistics for
Business and Economics
7th Edition
Chapter 1-2
Basics

Dealing with Uncertainty
Everyday decisions are based on incomplete
information
Consider:
 Will the job market be strong when I graduate?
 Will the price of Yahoo stock be higher in six months
than it is now?
 Will interest rates remain low for the rest of the year if
the budget deficit is as high as predicted?
1.1

Dealing with Uncertainty
Numbers and data are used to assist decision
making
 Statistics is a tool to help process, summarize, analyze,
and interpret data
(continued)

Key Definitions
 A population is the collection of all items of interest or
under investigation
 N represents the population size
 A sample is an observed subset of the population
 n represents the sample size
 A parameter is a specific characteristic of a population
 A statistic is a specific characteristic of a sample
1.2

Population vs. Sample
a b c d
ef gh i jk l m n
o p q rs t u v w
x y z
Population Sample
Values calculated using
population data are called
parameters
Values computed from
sample data are called
statistics
b c
g i n
o r u
y

Examples of Populations
 Names of all registered voters in Turkey
 Incomes of all families living in Hawai
 Annual returns of all stocks traded on the New
York Stock Exchange
 Grade point averages of all the students in your
university

Random Sampling
Simple random sampling is a procedure in which
 each member of the population is chosen strictly by
chance,
 each member of the population is equally likely to be
chosen,
 every possible sample of n objects is equally likely to
be chosen
The resulting sample is called a random sample

Descriptive and Inferential Statistics
Two branches of statistics:
 Descriptive statistics
 Graphical and numerical procedures to summarize
and process data
 Inferential statistics
 Using data to make predictions, forecasts, and
estimates to assist decision making

Descriptive Statistics
 Collect data
 e.g., Survey
 Present data
 e.g., Tables and graphs
 Summarize data
 e.g., Sample mean =
i
X
n


Inferential Statistics
 Estimation
 e.g., Estimate the population
mean weight using the sample
mean weight
 Hypothesis testing
 e.g., Test the claim that the
population mean weight is 140
pounds
Inference is the process of drawing conclusions or
making decisions about a population based on
sample results

Types of Data
Data
Categorical Numerical
Discrete Continuous
Examples:
 Marital Status
 Are you registered to
vote?
 Eye Color
(Defined categories or
groups)
Examples:
 Number of Children
 Defects per hour
(Counted items)
Examples:
 Weight
 Voltage
(Measured characteristics)

Describing Data Numerically
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Central Tendency Variation
Ch. 2-12

Measures of Central Tendency
Central Tendency
Mean Median Mode
n
x
x
n
1
i
i



Overview
Midpoint of
ranked values
Most frequently
observed value
Arithmetic
average
Ch. 2-13
2.1

Arithmetic Mean
 The arithmetic mean (mean) is the most
common measure of central tendency
 For a population of N values:
 For a sample of size n:
Sample size
n
x
x
x
n
x
x n
2
1
n
1
i
i






  Observed
values
N
x
x
x
N
x
μ N
2
1
N
1
i
i






 
Population size
Population
values
Ch. 2-14

Arithmetic Mean
 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
3
5
15
5
5
4
3
2
1






4
5
20
5
10
4
3
2
1






Ch. 2-15

Median
 In an ordered list, the median is the “middle”
number (50% above, 50% below)
 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
Ch. 2-16

Finding the Median
 The location of the median:
 If the number of values is odd, the median is the middle number
 If the number of values is even, the median is the average of
the two middle numbers
 Note that is not the value of the median, only the
position of the median in the ranked data
data
ordered
the
in
position
2
1
n
position
Median


2
1
n 
Ch. 2-17

Mode
 A measure of central tendency
 Value that occurs most often
 Not affected by extreme values
 Used for either numerical or categorical data
 There may 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 = 9
0 1 2 3 4 5 6
No Mode
Ch. 2-18

Review Example
 Five houses on a hill by the beach
$2,000 K
$500 K
$300 K
$100 K
$100 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Ch. 2-19

Review Example:
Summary Statistics
 Mean: ($3,000,000/5)
= $600,000
 Median: middle value of ranked data
= $300,000
 Mode: most frequent value
= $100,000
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Sum 3,000,000
Ch. 2-20

Which measure of location
is the “best”?
 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
Ch. 2-21

Measures of Variability
Same center,
different variation
Variation
Variance Standard
Deviation
Coefficient of
Variation
Range Interquartile
Range
 Measures of variation give
information on the spread
or variability of the data
values.
Ch. 2-22
2.2

Range
 Simplest measure of variation
 Difference between the largest and the smallest
observations:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
Ch. 2-23

Disadvantages of the Range
 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
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
Ch. 2-24

Interquartile Range
 Can eliminate some outlier problems by using
the interquartile range
 Eliminate high- and low-valued observations
and calculate the range of the middle 50% of
the data
 Interquartile range = 3rd quartile – 1st quartile
IQR = Q3 – Q1

Interquartile Range
Median
(Q2)
X
maximum
X
minimum Q1 Q3
Example:
25% 25% 25% 25%
12 30 45 57 70
Interquartile range
= 57 – 30 = 27
Ch. 2-26

Population Variance
 Average of squared deviations of values from
the mean
 Population variance:
N
μ)
(x
σ
N
1
i
2
i
2




Where = population mean
N = population size
xi = ith value of the variable x
μ
Ch. 2-27

Sample Variance
 Average (approximately) of squared deviations
of values from the mean
 Sample variance:
1
-
n
)
x
(x
s
n
1
i
2
i
2




Where = arithmetic mean
n = sample size
Xi = ith value of the variable X
X
Ch. 2-28

Population Standard Deviation
 Most commonly used measure of variation
 Shows variation about the mean
 Has the same units as the original data
 Population standard deviation:
N
μ)
(x
σ
N
1
i
2
i




Ch. 2-29

Sample Standard Deviation
 Most commonly used measure of variation
 Shows variation about the mean
 Has the same units as the original data
 Sample standard deviation:
1
-
n
)
x
(x
S
n
1
i
2
i




Ch. 2-30

Calculation Example:
Sample Standard Deviation
Sample
Data (xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = x = 16
4.2426
7
126
1
8
16)
(24
16)
(14
16)
(12
16)
(10
1
n
)
x
(24
)
x
(14
)
x
(12
)
X
(10
s
2
2
2
2
2
2
2
2
























A measure of the “average”
scatter around the mean
Ch. 2-31

Measuring variation
Small standard deviation
Large standard deviation
Ch. 2-32

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
Ch. 2-33

Advantages of Variance and
Standard Deviation
 Each value in the data set is used in the
calculation
 Values far from the mean are given extra
weight
(because deviations from the mean are squared)

Coefficient of Variation
 Measures relative variation
 Always in percentage (%)
 Shows variation relative to mean
 Can be used to compare two or more sets of
data measured in different units
100%
x
s
CV 









Ch. 2-35

Comparing Coefficient
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
$5
100%
x
s
CVA 












5%
100%
$100
$5
100%
x
s
CVB 












Ch. 2-36

Using Microsoft Excel
 Descriptive Statistics can be obtained
from Microsoft® Excel
 Select:
data / data analysis / descriptive statistics
 Enter details in dialog box

Using Excel
 Select data / data analysis / descriptive statistics
Ch. 2-38

Using Excel
 Enter input
range details
 Check box for
summary
statistics
 Click OK

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
Ch. 2-40

The Sample Covariance
 The covariance measures the strength of the linear relationship
between two variables
 The population covariance:
 The sample covariance:
 Only concerned with the strength of the relationship
 No causal effect is implied
N
)
)(y
(x
y)
,
(x
Cov
N
1
i
y
i
x
i
xy









1
n
)
y
)(y
x
(x
s
y)
,
(x
Cov
n
1
i
i
i
xy







Ch. 2-41
2.4

Interpreting Covariance
 Covariance between two variables:
Cov(x,y) > 0 x and y tend to move in the same direction
Cov(x,y) < 0 x and y tend to move in opposite directions
Cov(x,y) = 0 x and y are independent

Coefficient of Correlation
 Measures the relative strength of the linear relationship
between two variables
 Population correlation coefficient:
 Sample correlation coefficient:
Y
X s
s
y)
,
(x
Cov
r 
Y
X σ
σ
y)
,
(x
Cov
ρ 
Ch. 2-43

Features of
Correlation Coefficient, r
 Unit free
 Ranges between –1 and 1
 The closer to –1, the stronger the negative linear
relationship
 The closer to 1, the stronger the positive linear
relationship
 The closer to 0, the weaker any positive linear
relationship

Scatter Plots of Data with Various
Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = +.3
r = +1
Y
X
r = 0
Ch. 2-45

Using Excel to Find
the Correlation Coefficient
 Select Data / Data Analysis
 Choose Correlation from the selection menu
 Click OK . . .

Using Excel to Find
the Correlation Coefficient
 Input data range and select
appropriate options
 Click OK to get output
(continued)
Ch. 2-47

Interpreting the Result
 r = .733
 There is a relatively
strong positive linear
relationship between
test score #1
and test score #2
 Students who scored high on the first test tended
to score high on second test
Scatter Plot of Test Scores
70
75
80
85
90
95
100
70 75 80 85 90 95 100
Test #1 Score
Test
#2
Score
Ch. 2-48

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