1. presenting and summarizing data

7/29/2019 1. Presenting and Summarizing Data

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Presenting and

summarizing data

Spori Goran, PhD.

http://kif.hr/predmet/mki

http://www.science4performance.com/
http://kif.hr/predmet/mkihttp://kif.hr/predmet/mki


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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 ofCentral Tendency

Mean, Median, Mode, Midrange


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Topics (continued)

Measures ofDispersion (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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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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Collect Data e.g. Survey

Present Data e.g. Tables and Graphs

Characterize Data e.g. Meann

xi

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 Data

Graphics

The visual representation of data may be used not

only to present results/findings in the data, butmay also be used to learn about the data.


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Summary Measures in Descriptive

Statistics

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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Measures of Central Tendency

Central Tendency

Mean Median Mode

Midrange


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The Mean (Arithmetic Average)

It is theArithmetic Averageof data values:

The Most Common Measure of Central Tendency

Affected by Extreme Values(Outliers)

n

xn

1i

in

xxxn2i

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


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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 isodd, the median is the middle number.

If n iseven, the median is the average of the 2

middle numbers.

Not Affected by Extreme Values


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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


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Midrange

A Measure of Central Tendency

Average of Smallest and Largest

Observation:

Affected by Extreme Value

2

xxsmallestestl

argMidrange

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

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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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

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


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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

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


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Summary Measures

Central Tendency

Mean

Median

Mode

Midrange

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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Measures of Dispersion (Variation)

Variation

Variance Standard Deviation Coefficient ofVariation

Population

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range


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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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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


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Important Measure of Variation

Shows Variation About the Mean:

For the Population:

For the Sample:

Variance

N

Xi 22

1

2

2

n

XXs i

For the Population: use N in the

denominator.

For the Sample : use n - 1

in the denominator.


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Most Important Measure of Variation

Shows Variation About the Mean:

For the Population:

For the Sample:

Standard Deviation

N

Xi 2

1

2

n

XXs i

For thePopulation:use N in the

denominator.

For theSample : use n - 1

in the denominator.


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Sample Standard Deviation

1

2

n

XX iFor the Sample : use n - 1

in the denominator.

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

16241618161716151614161216102222222

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

= 4.2426

s

:Xi


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Comparing Standard Deviations

1

2

n

XXis = = 4.2426

N

X i

2 = 3.9686

Value for the Standard Deviation islargerfor data considered as aSample.

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

N= 8 Mean =16


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Comparing Standard Deviations

Mean = 15.5

s = 3.33811 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

DataB

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


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Coefficient of Variation

Measure ofRelative Variation

Always a%

Shows Variation Relative to Mean

Used toCompare 2 or More Groups

Formula ( for Sample):

100%

X

SCV


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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

100%

X

SCV

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


Measures of Shape:

Symmetric or skewed

Symmetric

Mean = Median = Mode

-0.5


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Shape


Measures of Shape:

Symmetric or skewed

Left-Skewed Symmetric

Mean = Median = ModeMean Median Mod

e

< -1 -0.5


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Shape


Measures of Shape:

Symmetric or skewed

Right-SkewedLeft-Skewed Symmetric

Mean = Median = ModeMean Median Mode Median MeanMode

< -1 > 1-0.5


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Box-and-Whisker Plot

Graphical Display of Data Using

5-Number Summary

Median

4 6 8 10 12

Q3

Q1

Xlargest

Xsmallest

S &


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Distribution Shape &Box-and-Whisker Plots

Right-SkewedLeft-Skewed Symmetric

Q1

Median Q3

Q1 Median Q3 Q1 Median Q3


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Summary

Discussed Measures ofCentral Tendency

Mean, Median, Mode, Midrange

Quartiles

Addressed Measures ofVariation The Range, Interquartile Range, Variance, Standard Deviation, Coefficient ofVariation

Determined Shape of Distributions

Symmetric, Skewed, Box-and-Whisker PlotMean= Median =ModeMean Median Mode Mode Median Mean

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