Statistics for Managers Using Microsoft Excel, 2/e

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Online Course 1
Descriptive Statistics
Roger L. Brown, Ph.D. Medical Research
Consulting Middleton, WI
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This online course is a FREE service to all MRC
clients
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Purpose of this series

• To assist researchers in the interpretation and
  application of statistical analyses

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Statistics ?
The Science of collecting, organizing, analyzing,
interpreting and presenting data
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Topics we will review

• Descriptive Statistics
• Frequency Distributions and Histograms
• Relative / Cumulative Frequency
• Measures of Central Tendency
• Mean, Median, Mode, Midrange

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Topics (continued)

• Measures of Dispersion (Variation)
• Range, Standard Deviation,
• Variance and Coefficient of variation
• Shape
• Symmetric, Skewed, using Box-and-
• Whisker Plots
• Quartile
• Statistical Relationships
• Correlation , Covariance

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Descriptive Statistics
A collection of quantitative measures and ways of
describing data. This includes Frequency
distributions histograms, measures of central
tendency and measures of dispersion
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Descriptive Statistics

• Collect Data e.g. Survey
• Present Data e.g. Tables and Graphs
• Characterize Data e.g. Mean

A Characteristic of a Population is
a Parameter Sample is a Statistic.
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Collection of Data

• Survey/questionnaires/interviews
• Direct observation
• Secondary data source (e.g., Medical charts)

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Presenting DataGraphics
The visual representation of data may be used not
only to present results/findings in the data, but
may also be used to learn about the data.
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Summary Measures in Descriptive Statistics
Summary Measures
Variation
Central Tendency
Quartile
Mean
Mode
Coefficient of Variation
Range
Median
Variance
Midrange
Standard Deviation
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Measures of Central Tendency
Central Tendency
Mean
Median
Mode
Midrange
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The Mean (Arithmetic Average)

• It is the Arithmetic Average of data values
• The Most Common Measure of Central Tendency
• 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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The 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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The 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
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
No Mode
Mode 9
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Midrange

• A Measure of Central Tendency
• Average of Smallest and Largest
• Observation
• Affected by Extreme Value

Midrange
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Midrange 5
Midrange 5
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Summary Measures in Descriptive Statistics
Summary Measures
Variation
Central Tendency
Quartile
Mean
Mode
Coefficient of Variation
Range
Median
Variance
Midrange
Standard Deviation
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Quartiles

• Not a Measure of Central Tendency
• Split Ordered Data into 4 Quarters
• Position of i-th Quartile position of point

25
25
25
25
Q1
Q2
Q3
i(n1)
Q

i
4
Data in Ordered Array 11 12 13 16 16
17 18 21 22

1(9 1)

Position of Q1
2.50
Q1
12.5
4
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Quartiles

• Not a Measure of Central Tendency
• Split Ordered Data into 4 Quarters
• Position of i-th Quartile position of point

25
25
25
25
Q1
Q2
Q3
i(n1)
Q

i
4
Data in Ordered Array 11 12 13 16 16
17 18 21 22

3(9 1)

Position of Q3
7.50
Q3
19.5
4
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Summary Measures
Summary Measures
Central Tendency
Quartile
Variation
Mean
Mode
Coefficient of Variation
Range
Median
Variance
Midrange
Standard Deviation
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Measures of Dispersion (Variation)
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
Population Variance
Population Standard Deviation
Sample Variance
Sample Standard Deviation
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Understanding Variation

• The more Spread out or dispersed data
• the larger the measures of variation
• The more concentrated or homogenous the data
  the smaller the measures of variation
• If all observations are equal
• measures of variation Zero
• All measures of variation are Nonnegative

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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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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 10 12 14
15 17 18 18 24
N 8 Mean 16
s
4.2426
3.9686
Value for the Standard Deviation is larger for
data considered as a Sample.
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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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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

• Group A Average Health Measure 50
• Standard Deviation 5
• Group B Average Health Measure 100
• Standard Deviation 5

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

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

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Shape

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

-0.5 lt0 lt 0.5
Symmetric
Mean


Median


Mode
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Shape

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

lt -1
-0.5 lt0 lt 0.5
Left-Skewed
Symmetric
Mean

Median

Mode
Mean


Median


Mode
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Shape

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

gt 1
lt -1
-0.5 lt0 lt 0.5
Right-Skewed
Left-Skewed
Symmetric
Mean

Median

Mode
Mean


Median


Mode
Mode

Median

Mean
Negatively Skewed
Positively Skewed
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Box-and-Whisker Plot

• Graphical Display of Data Using 5-Number
  Summary

Median
Q
Q
X
X
smallest
3
1
largest
12
4
6
8
10
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Distribution Shape Box-and-Whisker Plots
Right-Skewed
Left-Skewed
Symmetric
Q


Median


Q
Q

Median

Q
Q

Median

Q
1
3
1
3
3
1
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Summary

• Discussed Measures of Central Tendency
• Mean, Median, Mode, Midrange
• Quartiles
• Addressed Measures of Variation
• The Range, Interquartile Range, Variance,
  
• Standard Deviation, Coefficient of
  Variation
• Determined Shape of Distributions
• Symmetric, Skewed, Box-and-Whisker Plot

Mean

Median

Mode
Mean
Median
Mode
Mode
Median
Mean