Newbold_chap03.ppt

Chap 3-1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 3
Describing Data: Numerical
Statistics for
Business and Economics
6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-2
After completing this chapter, you should be able to:
 Compute and interpret the mean, median, and mode for a
set of data
 Find the range, variance, standard deviation, and
coefficient of variation and know what these values mean
 Apply the empirical rule to describe the variation of
population values around the mean
 Explain the weighted mean and when to use it
 Explain how a least squares regression line estimates a
linear relationship between two variables
Chapter Goals

Chapter Topics
 Measures of central tendency, variation, and
shape
 Mean, median, mode, geometric mean
 Quartiles
 Range, interquartile range, variance and standard
deviation, coefficient of variation
 Symmetric and skewed distributions
 Population summary measures
 Mean, variance, and standard deviation
 The empirical rule and Bienaymé-Chebyshev rule

Chapter Topics
 Five number summary and box-and-whisker
plots
 Covariance and coefficient of correlation
 Pitfalls in numerical descriptive measures and
ethical considerations
(continued)

Describing Data Numerically
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Central Tendency Variation

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

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

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







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

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 

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

 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

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

 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”?

Shape of a Distribution
 Describes how data are distributed
 Measures of shape
 Symmetric or skewed
Mean = Median
Mean < Median Median < Mean
Right-Skewed
Left-Skewed Symmetric

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

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:

 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

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

Quartiles
 Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25% 25% 25% 25%
 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% are smaller, 50% are
larger)
 Only 25% of the observations are greater than the third
quartile
Q1 Q2 Q3

Quartile Formulas
Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position: Q1 = 0.25(n+1)
Second quartile position: Q2 = 0.50(n+1)
(the median position)
Third quartile position: Q3 = 0.75(n+1)
where n is the number of observed values

(n = 9)
Q1 = is in the 0.25(9+1) = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so Q1 = 12.5
Quartiles
Sample Ranked Data: 11 12 13 16 16 17 18 21 22
 Example: Find the first quartile

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

 Average (approximately) of squared deviations
of values from the mean
 Sample variance:
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

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:
1
-
N
μ)
(x
σ
N
1
i
2
i





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





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

Measuring variation
Small standard deviation
Large standard deviation

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

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)

 For any population with mean μ and
standard deviation σ , and k > 1 , the
percentage of observations that fall within
the interval
[μ + kσ]
Is at least
Chebyshev’s Theorem
)]%
(1/k
100[1 2


 Regardless of how the data are distributed,
at least (1 - 1/k2) of the values will fall
within k standard deviations of the mean
(for k > 1)
 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)
Chebyshev’s Theorem
within
At least
(continued)

 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
1σ
μ 
μ
68%
1σ
μ

 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
2σ
μ 
3σ
μ 
3σ
μ
99.7%
95%
2σ
μ

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 










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 













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

Using Excel
Use menu choice:
tools / 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

Weighted Mean
 The weighted mean of a set of data is
 Where wi is the weight of the ith observation
 Use when data is already grouped into n classes, with
wi values in the ith class
i
n
n
2
2
1
1
n
1
i
i
i
w
x
w
x
w
x
w
w
x
w
x


 



  

Approximations for Grouped Data
Suppose a data set contains values m1, m2, . . ., mk,
occurring with frequencies f1, f2, . . . fK
 For a population of N observations the mean is
 For a sample of n observations, the mean is
N
m
f
μ
K
1
i
i
i



n
m
f
x
K
1
i
i
i






K
1
i
i
f
N
where



K
1
i
i
f
n
where

Approximations for Grouped Data
Suppose a data set contains values m1, m2, . . ., mk,
occurring with frequencies f1, f2, . . . fK
 For a population of N observations the variance is
 For a sample of n observations, the variance is
N
μ)
(m
f
σ
K
1
i
2
i
i
2




1
n
)
x
(m
f
s
K
1
i
2
i
i
2






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








 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
Interpreting Covariance

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 

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

Using Excel to Find
the Correlation Coefficient
 Select
Tools/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)

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

Obtaining Linear Relationships
 An equation can be fit to show the best linear
relationship between two variables:
Y = β0 + β1X
Where Y is the dependent variable and X is the
independent variable

Least Squares Regression
 Estimates for coefficients β0 and β1 are found to
minimize the sum of the squared residuals
 The least-squares regression line, based on sample
data, is
 Where b1 is the slope of the line and b0 is the y-
intercept:
x
b
b
y 1
0
ˆ 

x
y
2
x
1
s
s
r
s
y)
Cov(x,
b 
 x
b
y
b 1
0 


Chapter Summary
 Described measures of central tendency
 Mean, median, mode
 Illustrated the shape of the distribution
 Symmetric, skewed
 Described measures of variation
 Range, interquartile range, variance and standard deviation,
coefficient of variation
 Discussed measures of grouped data
 Calculated measures of relationships between
variables
 covariance and correlation coefficient

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