MBA 7020 Business Analysis Foundations Descriptive Statistics June 20, 2005

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MBA 7020Business Analysis Foundations
Descriptive StatisticsJune 20, 2005
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Agenda
Confidence Interval
1. Measures of Central Location Mean, Median,
Mode 2. Measures of Variation The Range,
Variance and Standard Deviation 3. Measures of
Association Covariance and Correlation
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Describing Data Summary Measures
1. Measures of Central Location Mean,
Median, Mode 2. Measures of Variation
The Range, Variance and Standard
Deviation 3. Measures of Association
Covariance and Correlation
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Mean
1. It is the Arithmetic Average of data
values 2. The Most Common Measure of
Central Tendency 3. Affected by Extreme Values
(Outliers)
Sample Mean
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
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Mean 5
Mean 6
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Median

• Important Measure of Central Tendency
• In an ordered array, the median is the middle
  number.
• If n is odd, the median is the middle number.
• If n is even, the median is the average of the 2
• middle numbers.
• Not Affected by Extreme Values

0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
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Median 5
Median 5
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Mode

• A Measure of Central Tendency
• Value that Occurs Most Often
• Not Affected by Extreme Values
• There May Not be a Mode
• There May be Several Modes
• Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
0 1 2 3 4 5 6
No Mode
Mode 9
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Agenda
Confidence Interval
1. Measures of Central Location Mean, Median,
Mode 2. Measures of Variation The Range,
Variance and Standard Deviation 3. Measures of
Association Covariance and Correlation
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Measures of Variability
Variation
Variance / Standard Deviation
Coefficient of Variation
Range / Percentiles
Population
Sample
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The Range

• Measure of Variation
• Difference Between Largest Smallest
• Observations
• Range
• Ignores How Data Are Distributed

Range 12 - 7 5
Range 12 - 7 5
7 8 9 10 11 12
7 8 9 10 11 12
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Percentile Scores

• Arrange data in ascending order.
• The middle number is the median.
• The number halfway to the median is the first
  quartile.
• The number halfway past the median is the 3rd
  quartile.
• A number with (no more than) 66 of the values
  less than it is the 66th percentile, and so forth.

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Box Plot
Smallest
Largest
Q1
Q3
Median
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Variance

• Important Measure of Variation
• Shows Variation About the Mean
• For the Population
• For the Sample

For the Population use N in the denominator.
For the Sample use n - 1 in the denominator.
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Standard Deviation

• Most Important Measure of Variation
• Shows Variation About the Mean
• For the Population
• For the Sample

For the Population use N in the denominator.
For the Sample use n - 1 in the denominator.
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Sample Standard Deviation
For the Sample use n - 1 in the denominator.
s
Data 10 12 14
15 17 18 18 24
n 8 Mean 16
s
4.2426
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Comparing Standard Deviations
Data A
Mean 15.5 s 3.338
11 12 13 14 15 16 17 18
19 20 21
Data B
Mean 15.5 s .9258
11 12 13 14 15 16 17 18
19 20 21
Data C
Mean 15.5 s 4.57
11 12 13 14 15 16 17 18
19 20 21
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Agenda
Confidence Interval
1. Measures of Central Location Mean, Median,
Mode 2. Measures of Variation The Range,
Variance and Standard Deviation 3. Measures of
Association Covariance and Correlation
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Coefficient of Variation

• Measure of Relative Variation
• Always a
• Shows Variation Relative to Mean
• Used to Compare 2 or More Groups
• Formula (for Sample)

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Comparing Coefficient of Variation

• Stock A Average Price last year 50
• Standard Deviation 5
• Stock B Average Price last year 100
• Standard Deviation 5

Coefficient of Variation Stock A CV
10 Stock B CV 5
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Shape

• Describes How Data Are Distributed
• Measures of Shape
• Symmetric or skewed

Right-Skewed
Left-Skewed
Symmetric
Mean


Median


Mode
Mean

Median

Mode

Median

Mean
Mode
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Agenda
Confidence Interval
Descriptive Summary Measures
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Confidence Interval

• Sample Mean Margin of Error (MOE)
• Called a Confidence Interval
• To Compute Margin of Error, One of Two Conditions
  Must Be True
• The Distribution of the Population of Incomes
  Must Be Normal, or
• The Distribution of Sample Means Must Be Normal.

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A Side-Trip Before Constructing Confidence
Intervals

• What is a Population Distribution?
• What is a Distribution of the Sample Mean?
• How Does Distribution of Sample Mean Differ From
  a Population Distribution?
• What is the Central Limit Theorem?

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Estimating the Population Mean Income of Lexus
Owners
Assume Small Population of Lexus Owners Incomes
(N 200)
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Distribution of N 200 Incomes (Population Mean )
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75 125 175 225 275 325
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Constructing a Distribution of Samples of Size 5
from N 200 Owners
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Distribution of Sample Mean Incomes (Column 7)
Distribution of Sample Means Near Normal!
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Central Limit Theorem

• Even if Distribution of Population is Not Normal,
  Distribution of Sample Mean Will Be Near Normal
  Provided You Select Sample of Five or Ten or
  Greater From the Population.
• For a Sample Sizes of 30 or More, Distribution of
  the Sample Mean Will Be Normal, with
• mean of sample means population mean, and
• standard error population deviation /
  sqrt(n)
• Thus Can Use Expression

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Why Does Central Limit Theorem Work?

• As Sample Size Increases
• Most Sample Means will be Close to Population
  Mean,
• Some Sample Means will be Either Relatively Far
  Above or Below Population Mean.
• A Few Sample Means will be Either Very Far Above
  or Below Population Mean.

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Impact of Side-Trip on MOE

• Determine Confidence, or Reliability, Factor.
• Distribution of Sample Mean Normal from Central
  Limit Theorem.
• Use a Normal-Like Table to Obtain Confidence
  Factor.
• Determine Spread in Sample Means (Without Taking
  Repeated Samples)

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Drawing Conclusions about a Population Mean
Using a Sample Mean
Select Simple Random Sample
Compute Sample Mean and Std. Dev. For n lt 10,
Sample Bell-Shaped? For n gt10 CLT Ensures Dist of
Normal
Draw Conclusion about Population Mean
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Federal Aid Problem

• Suppose a census tract with 5000 families is
  eligible for aid under program HR-247 if average
  income of families of 4 is between 7500 and
  8500 (those lower than 7500 are eligible in a
  different program). A random sample of 12
  families yields data on the next page.

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Federal Aid Study Calculations
Representative Sample
7,300 7,700 8,100 8,400 7,800 8,300 8,500
7,600 7,400 7,800 8,300 8,600
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Estimated Standard Error

• Measures Variation Among the Sample Means If We
  Took Repeated Samples.
• But We Only Have One Sample! How Can We Compute
  Estimated Standard Error?
• Based on Constructing Distribution of Sample Mean
  Slide, Will Estimated Standard Error Be Smaller
  or Larger Than Sample Standard Deviation (s)?
• Estimated Std. Error ______ than s.
• Estimated Standard Error Expression

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Confidence Factor for MOE
Can Use t-Table Provided Distribution of Sample
Mean is Normal
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95 Confidence Interval

• Interpretation of Confidence Interval
• 95 Confident that Interval 7,983 280
  Contains Unknown Population (Not Sample) Mean
  Income.
• If We Selected 1,000 Samples of Size 12 and
  Constructed 1,000 Confidence Intervals, about 950
  Would Contain Unknown Population Mean and 50
  Would Not.
• So Is Tract Eligible for Aid???

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Sample Means versus Sample Proportion
Mean
Proportion of

• Americans Who Believe that Japan is 1 Economic
  Power
• Circuit Boards with One or More Failed Solder
  Connections
• African-Americans Who Pass CPA

• Income/Loss
• Time to Complete Loan Papers
• Number of Fat Calories in Burger
• Breaking Strength of Cellular Phone Housing

Means and Proportions Not the Same!!!!
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Similarities and Differences Between Sample
Means and Proportions

• Sample Means
• Computed from Data that Are Measured.
• Estimate Population Means.
• Sample Proportions
• Computed from Data that Are Counted.
• Estimate Population Proportions.

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Drawing Conclusions about a Population Proportion
From a Sample Proportion
Select Simple Random Sample
Compute Sample Proportion Check for Normality -
Table 7.8
Draw Conclusion About Population Proportion, p
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Japan Business Survey

• N 200 Californians
• Yes 116
• No 84

Is Japan the Foremost Economic Power Today?
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90 Confidence Interval on P