descriptive statistics roger l. brown, ph.d. medical research consulting middleton, wi online course...

Descriptive Statistics

Roger L. Brown, Ph.D.Roger L. Brown, Ph.D.

Medical Research ConsultingMedical Research Consulting

Middleton, WIMiddleton, WI

Online Course #1

This online course is a FREE This online course is a FREE service to all MRC clientsservice to all MRC clients

Purpose of this seriesPurpose of this series

To assist researchers in the To assist researchers in the interpretation and application of interpretation and application of

statistical analysesstatistical analyses

Statistics ?Statistics ?

The Science of collecting, The Science of collecting, organizing, analyzing, organizing, analyzing,

interpreting and presenting datainterpreting and presenting data

Topics we will reviewTopics we will review

• Descriptive Statistics

• Frequency Distributions and Histograms

Relative / Cumulative Frequency

• Measures of Central Tendency

Mean, Median, Mode, Midrange

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

A collection of quantitative measures and

ways of describing data. This includes:

Frequency distributions & histograms, measures of central tendency

and

measures of dispersion



•Collect Data e.g. Survey

•Present Data e.g. Tables and Graphs

•Characterize Data e.g. Mean

nx i

A Characteristic of a: Population is a Parameter

Sample is a Statistic.

Collection of DataCollection of Data

Survey/questionnaires/interviewsSurvey/questionnaires/interviewsDirect observationDirect observationSecondary data source (e.g., Medical charts)Secondary data source (e.g., Medical charts)

Presenting DataPresenting DataGraphicsGraphics

The visual representation of data may be used not The visual representation of data may be used not only to present results/findings in the data, but only to present results/findings in the data, but may also be used to learn about the data.may also be used to learn about the data.

Summary Measures in Descriptive Statistics

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Measures of Central Tendency

Central Tendency

Mean Median Mode

Midrange

The Mean (Arithmetic Average)

•It is the Arithmetic Average of data values:

•The Most Common Measure of Central Tendency

•Affected by Extreme Values (Outliers)

n

xn

1ii

n

xxx n2i

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

xSample Mean

The Median

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

•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

The Mode

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

•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

No Mode

Midrange

•A Measure of Central Tendency

•Average of Smallest and Largest

Observation:

•Affected by Extreme Value

2

xx smallestestl arg

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

Summary Measures in Descriptive Statistics

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation


Range

QuartilesQuartiles

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

Q i(n+1)i 4

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

Position of Q1 = 2.50 Q1 =12.5= 1•(9 + 1)4

QuartilesQuartiles

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

Q i(n+1)i 4

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

Position of Q3 = 7.50 Q3 =19.5= 3•(9 + 1)4

Summary Measures

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation


Range

Measures of Dispersion (Variation)

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

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

• Measure of Variation

• Difference Between Largest & Smallest Observations:

Range =

• Ignores How Data Are Distributed:

The Range

SmallestrgestLa xx

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

•Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Variance

N

X i

22

1

22

n

XXs i

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

•Most Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Standard Deviation

N

X i

2

1

2

n

XXs i

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

Sample Standard Deviation

1

2

n

XX i For the Sample : use n - 1 in the denominator.

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

1624161816171615161416121610 2222222

)()()()()()()(

= 4.2426

s

:X i

Comparing Standard Deviations

1

2

n

XX is =

= 4.2426

N

X i

2 = 3.9686

Value for the Standard Deviation is larger for data considered as a Sample.

Data : 10 12 14 15 17 18 18 24:X i

N= 8 Mean =16

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 = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Data C

Coefficient of VariationCoefficient of Variation

Measure of Relative Variation

Always a %

Shows Variation Relative to Mean

Used to Compare 2 or More Groups

Formula ( for Sample):

100%

X

SCV

Comparing Coefficient of VariationComparing Coefficient of Variation

Group A: Average Health Measure = 50 Standard Deviation = 5 Group B: Average Health Measure = 100 Standard Deviation = 5

100%

X

SCV

Coefficient of Variation:

Group A: CV = 10%

Group B: CV = 5%

ShapeShape

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

ShapeShape


SymmetricMean = Median = Mode

-0.5 <0 < 0.5

ShapeShape


Left-Skewed SymmetricMean = Median = ModeMean Median Mode

< -1 -0.5 <0 < 0.5

ShapeShape


Right-SkewedLeft-Skewed SymmetricMean = Median = ModeMean Median Mode Median MeanMode

< -1 > 1 -0.5 <0 < 0.5

Negatively Skewed Positively Skewed

Box-and-Whisker PlotBox-and-Whisker Plot

Graphical Display of Data Using5-Number Summary

Median

4 6 8 10 12

Q3Q1 XlargestXsmallest

Distribution Shape & Distribution Shape & Box-and-Whisker PlotsBox-and-Whisker Plots

Right-SkewedLeft-Skewed Symmetric

Q1 Median Q3Q1 Median Q3 Q1

Median Q3

SummarySummary 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 = ModeMean Median Mode Mode Median Mean

descriptive statistics roger l. brown, ph.d. medical research consulting middleton, wi online course...

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