Measures of Dispersion Measures of Dispersion

oror

Measures of VariabilityMeasures of Variability

Measures of VariabilityMeasures of Variability

A single summary figure that describes A single summary figure that describes the spread of observations within a the spread of observations within a distribution.distribution.

MEASURES OF DESPERSIONMEASURES OF DESPERSION

RANGERANGE INTERQUARTILE RANGEINTERQUARTILE RANGE VARIANCEVARIANCE STANDARD DEVIATIONSTANDARD DEVIATION

Measures of VariabilityMeasures of Variability

RangeRange Difference between the smallest and largest Difference between the smallest and largest

observations.observations. Interquartile RangeInterquartile Range

Range of the middle half of scores.Range of the middle half of scores. VarianceVariance

Mean of all squared deviations from the mean.Mean of all squared deviations from the mean. Standard DeviationStandard Deviation

Measure of the average amount by which Measure of the average amount by which observations deviate from the mean. The square root observations deviate from the mean. The square root of the variance.of the variance.

Variability Example: Variability Example: RangeRange

Las Vegas Hotel RatesLas Vegas Hotel Rates

52, 76, 100, 136, 186, 196, 205, 150, 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891750, 791, 891

Range:Range: 891-52 = 839 891-52 = 839

Pros and Cons of the Pros and Cons of the RangeRange

ProsPros Very easy to Very easy to

compute.compute. Scores exist in the Scores exist in the

data set.data set.

ConsCons Value depends only Value depends only

on two scores.on two scores. Very sensitive to Very sensitive to

outliers.outliers. Influenced by sample Influenced by sample

size (the larger the size (the larger the sample, the larger the sample, the larger the range).range).

Inter quartile RangeInter quartile Range

The inter quartile range is QThe inter quartile range is Q33-Q-Q11

50% of the observations in the 50% of the observations in the distribution are in the inter quartile range.distribution are in the inter quartile range.

The following figure shows the interaction The following figure shows the interaction between the quartiles, the median and between the quartiles, the median and the inter quartile range.the inter quartile range.

Inter quartile RangeInter quartile Range

Quartiles: 1 = Q n+1

th4

2 = Q 2(n+1) n+1 = th

4 2

3 = Q 3(n+1) th

4Inter quartile : IQR = Q3 – Q1

Pros and Cons of the Pros and Cons of the Interquartile RangeInterquartile Range

ProsPros Fairly easy to Fairly easy to

compute.compute. Scores exist in the Scores exist in the

data set.data set. Eliminates influence Eliminates influence

of extreme scores.of extreme scores.

ConsCons Discards much of the Discards much of the

data.data.

Percentiles and QuartilesPercentiles and Quartiles

Maximum is 100th percentile: 100% of Maximum is 100th percentile: 100% of values lie at or below the maximumvalues lie at or below the maximum

Median is 50th percentile: 50% of values Median is 50th percentile: 50% of values lie at or below the medianlie at or below the median

Any percentile can be calculated. But the Any percentile can be calculated. But the most common are 25most common are 25thth (1 (1stst Quartile) and Quartile) and 7575thth (3 (3rdrd Quartile) Quartile)

Locating Percentiles in a Locating Percentiles in a Frequency DistributionFrequency Distribution

A percentile is a score below which a specific A percentile is a score below which a specific percentage of the distribution falls.percentage of the distribution falls.

The 75th percentile is a score below which 75% of The 75th percentile is a score below which 75% of the cases fall.the cases fall.

The median is the 50th percentile: 50% of the cases The median is the 50th percentile: 50% of the cases fall below itfall below it

Another type of percentile :The lower quartile is Another type of percentile :The lower quartile is 25th percentile and the upper quartile is the 75th 25th percentile and the upper quartile is the 75th percentilepercentile

NUMBER OF CHILDREN

260 26.6 26.6 26.6

161 16.4 16.5 43.1

260 26.6 26.6 69.7

155 15.8 15.9 85.6

70 7.2 7.2 92.7

31 3.2 3.2 95.9

21 2.1 2.1 98.1

11 1.1 1.1 99.2

8 .8 .8 100.0

977 99.8 100.0

2 .2

979 100.0

0

1

2

3

4

5

6

7

EIGHT OR MORE

Total

Valid

NAMissing

Total

Frequency PercentValid

PercentCumulative

Percent

50th percentile

80th percentile

50% included here

80% included

here

25th percentile

25% included here

Locating Percentiles in a Frequency Distribution

Five Number SummaryFive Number Summary

Minimum ValueMinimum Value 1st Quartile1st Quartile MedianMedian 3rd Quartile3rd Quartile Maximum ValueMaximum Value

VARIANCE:VARIANCE:

Deviations of each observation from Deviations of each observation from the mean, then averaging the sum of the mean, then averaging the sum of squares of these deviations.squares of these deviations.

STANDARD DEVIATION:STANDARD DEVIATION:

“ “ ROOT- MEANS-SQUARE-DEVIATIONS”ROOT- MEANS-SQUARE-DEVIATIONS”

VarianceVariance

The average amount that a score deviates from The average amount that a score deviates from the typical score.the typical score. Score – Mean = Difference ScoreScore – Mean = Difference Score Average of Difference Scores = 0Average of Difference Scores = 0 In order to make this number not 0, square the In order to make this number not 0, square the

difference scores ( negatives to become positives).difference scores ( negatives to become positives).

Variance: Variance: Computational FormulaComputational Formula

PopulationPopulation SampleSample

2

222

)(

N

XXN

2

222

)(

n

XXnS

VarianceVariance

Use the computational formula to calculate the variance.Use the computational formula to calculate the variance.

X X2

3 94 164 164 166 367 497 498 648 649 81

Sum: 60 Sum: 4002

222

)(

n

XXnS

0.4

100

3600400010

)60()400(10

2

2

2

22

S

S

S

Variability Example: Variability Example: VarianceVariance

Las Vegas Hotel RatesLas Vegas Hotel Rates

88.44760

1225

17918499623401707035

)13386()6686202(35

2

2

2

22

S

S

S

2

222

)(

n

XXnS

X X2

472 222784303 91809280 78400282 79524417 173889400 160000254 64516205 42025384 147456264 69696317 10048976 5776

643 413449480 230400136 18496250 62500100 10000732 535824317 100489264 69696384 147456750 562500402 161604422 178084373 139129325 105625313 97969749 561001791 625681196 38416891 793881283 8008952 2704

186 34596693 480249

Sum: 13386 Sum: 6686202

Pros and Cons of Pros and Cons of VarianceVariance

ProsPros Takes all data into Takes all data into

account.account. Lends itself to Lends itself to

computation of other computation of other stable measures (and stable measures (and is a prerequisite for is a prerequisite for many of them). many of them).

ConsCons Hard to interpret.Hard to interpret. Can be influenced by Can be influenced by

extreme scores.extreme scores.

Standard DeviationStandard Deviation

To “undo” the squaring of difference To “undo” the squaring of difference scores, take the square root of the scores, take the square root of the variance.variance.

Return to original units rather than Return to original units rather than squared units.squared units.

Quantifying UncertaintyQuantifying Uncertainty

Standard deviationStandard deviation: measures the : measures the variation of a variable in the sample.variation of a variable in the sample. Technically, Technically,

s x xN ii

N

1

12

1

( )

Standard DeviationStandard Deviation

PopulationPopulation SampleSample

Measure of the average amount by which observations deviate on either side of the mean. The square root of the variance.

2 s s2

(X )N

2 2)(

n

XXS

N X2 ( X )

N2

22

2

2 )(

n

XXnS

Variability Example: Variability Example: Standard DeviationStandard Deviation

Mean: 6

Standard Deviation: 20.2

0.4

100

36004000

10

)60()400(102

2

S

S

S

S

0.210

40

10

)69()68()68()67()67()66()64()64()64()63(

)(

2222222222

2

S

S

n

XXS

2

2

2 )(

n

XXnS

Variability Example: Variability Example: Standard DeviationStandard Deviation

Las Vegas Hotel Rates

0

1

2

3

4

5

6

7

8

9

0-99

100-199

200-299

300-399

400-499

500-599

600-699

700-799

800-899

Rates

Fre

quen

cy

hotel rates

Mean: $371.60

Standard Deviation: 57.211$88.44760

1225

179184996234017070

35

)13386()6686202(352

2

S

S

S

Pros and Cons of Pros and Cons of Standard DeviationStandard Deviation

ProsPros Lends itself to computation Lends itself to computation

of other stable measures of other stable measures (and is a prerequisite for (and is a prerequisite for many of them).many of them).

Average of deviations Average of deviations around the mean.around the mean.

Majority of data within one Majority of data within one standard deviation above or standard deviation above or below the mean.below the mean.

ConsCons Influenced by extreme Influenced by extreme

scores.scores.

Mean and Standard Mean and Standard DeviationDeviation

Using the mean and standard deviation Using the mean and standard deviation together:together: Is an efficient way to describe a distribution with just Is an efficient way to describe a distribution with just

two numbers.two numbers. Allows a direct comparison between distributions Allows a direct comparison between distributions

that are on different scales.that are on different scales.

A 100 samples were selected. Each of the sample contained 100 normal individuals. The mean Systolic BP of each sample is presented

110, 120, 130, 90, 100, 140, 160, 100, 120, 120, 110, 130, etc

Systolic BP level No. of samples90 - 5

100 - 10 110 - 20 120 - 34 130 - 20 140 - 10 150 - 5 160 - 2

Mean = 120Sd., = 10

•The curve describes probability of getting any range of values ie., P(x>120), P(x<100), P(110 <X<130) •Area under the curve = probability•Area under the whole curve =1•Probability of getting specific number =0, eg P(X=120) =0

Normal Distribution

120110100 130 140 15090

Mean = 120SD = 10

15 30 45 60 75

95% Confidence Interval for Mu

28 33 38 43

95% Confidence Interval for Median

Variable: Age

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

32.3851

13.3380

28.0000

0.9620.014

36.450015.7356247.608

0.6796268.51E-02

60

11.000025.000031.500046.750079.0000

40.5149

19.1921

42.0000

Anderson-Darling Normality Test

95% Confidence Interval for Mu

95% Confidence Interval for Sigma

95% Confidence Interval for Median

Descriptive Statistics

WHICH MEASURE TO USE ?WHICH MEASURE TO USE ?

DISTRIBUTION OF DATA IS SYMMETRICDISTRIBUTION OF DATA IS SYMMETRIC

---- USE MEAN & S.D.,---- USE MEAN & S.D.,

DISTRIBUTION OF DATA IS SKEWEDDISTRIBUTION OF DATA IS SKEWED

---- USE MEDIAN & QUARTILES---- USE MEDIAN & QUARTILES

ANY QUESTIONSANY QUESTIONS

THANK YOUTHANK YOU