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business and economics statics principles
- 1. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-1
Statistics for
Business and Economics
7th Edition
Chapter 1-2
Basics
- 2. 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?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-2
1.1
- 3. Dealing with Uncertainty
Numbers and data are used to assist decision
making
Statistics is a tool to help process, summarize, analyze,
and interpret data
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-3
(continued)
- 4. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-4
1.2
- 5. Population vs. Sample
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-5
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
- 6. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-6
- 7. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-7
- 8. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-8
- 9. Descriptive Statistics
Collect data
e.g., Survey
Present data
e.g., Tables and graphs
Summarize data
e.g., Sample mean =
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-9
i
X
n
- 10. Inferential Statistics
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-10
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
- 11. 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)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 1-11
- 12. 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
- 13. Measures of Central Tendency
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 14. 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:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 15. Arithmetic Mean
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(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
- 16. Median
In an ordered list, the median is the “middle”
number (50% above, 50% below)
Not affected by extreme values
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 17. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
data
ordered
the
in
position
2
1
n
position
Median
2
1
n
Ch. 2-17
- 18. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 19. Review Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 20. Review Example:
Summary Statistics
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 21. Which measure of location
is the “best”?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 22. Measures of Variability
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 23. Range
Simplest measure of variation
Difference between the largest and the smallest
observations:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 24. Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 25. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-25
- 26. Interquartile Range
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 27. Population Variance
Average of squared deviations of values from
the mean
Population variance:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 28. Sample Variance
Average (approximately) of squared deviations
of values from the mean
Sample variance:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 29. 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:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
N
μ)
(x
σ
N
1
i
2
i
Ch. 2-29
- 30. 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:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
1
-
n
)
x
(x
S
n
1
i
2
i
Ch. 2-30
- 31. Calculation Example:
Sample Standard Deviation
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 33. Comparing Standard Deviations
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 34. 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)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-34
- 35. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
100%
x
s
CV
Ch. 2-35
- 36. Comparing Coefficient
of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 37. Using Microsoft Excel
Descriptive Statistics can be obtained
from Microsoft® Excel
Select:
data / data analysis / descriptive statistics
Enter details in dialog box
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-37
- 38. Using Excel
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Select data / data analysis / descriptive statistics
Ch. 2-38
- 39. Using Excel
Enter input
range details
Check box for
summary
statistics
Click OK
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-39
- 40. Excel output
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 41. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 42. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-42
- 43. Coefficient of Correlation
Measures the relative strength of the linear relationship
between two variables
Population correlation coefficient:
Sample correlation coefficient:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Y
X s
s
y)
,
(x
Cov
r
Y
X σ
σ
y)
,
(x
Cov
ρ
Ch. 2-43
- 44. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-44
- 45. Scatter Plots of Data with Various
Correlation Coefficients
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
- 46. Using Excel to Find
the Correlation Coefficient
Select Data / Data Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 2-46
Choose Correlation from the selection menu
Click OK . . .
- 47. Using Excel to Find
the Correlation Coefficient
Input data range and select
appropriate options
Click OK to get output
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
Ch. 2-47
- 48. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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