variability

Range and Interquartile Range

Standard Deviation and Variance

Variability is a measure of how different scores are from one another within a set of data.

Synonyms: spread,dispersion.

Does the amount of variability (spread, dispersion) make a difference? Do we care?

How could we measure the amount of variability?

Bookhaven by waffler at http://www.flickr.com/photos/adrian_s/23441729/

Julian and Delia ask for help

Their mean quiz score is the same:M = 15 (out of 25, on 20 quizzes)

Do we know enough to help them?

Let’s look at the actual scores for each student

Best score = 18

Worst score = 13

Range=18.5-12.5=6

Range of middle 50% is IQR=16.5-13.5 =3

Scores are pretty similaracross all 20 quizzes

Julian’s Quiz Scores (Mean = 15.0)

16 14 15 14 16 14 14 17 14 13 15 15 16 15 18 15 14 16 14 15

Delia’s Quiz Scores (Mean = 15.0)

15 22 10 11 16 13 20 13 17 8 18 16 12 14 9 19 18 13 21 15

Best score = 22

Worst score = 8

Range=22.5 – 7.5 = 15

Range of middle 50% isIQR=18.5-12.5 =6

Scores seem to differ quite a bit from quiz to quiz

Range: Distance from highest number to lowest number (may require real limits)

Interquartile Range: Distance from 25% point to 75% point (range of middle 50% of scores)

• Quartile: 25th, 50th (median), and 75th percentiles

• The 25th and 75th are midpoint of each half

• May be an exact value, or may be between two values, using the same rules as the median.

Essentials of Statistics for Behavioral Science, 6th Edition by Frederick Gravetter and Larry Wallnau Copyright 2008 Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Developed by John Tukey to display central tendency & variability efficiently

Box = middle 50% of cases Top = 75th percentileBottom = 25th percentileHeight = Interquartile Range IQR

Line inside the box = MEDIANIf line is not centered, data arenot perfectly symmetric.

Line (“whiskers”) extend to minimum or maximum values within 1.5 IQR

Outliers: Beyond 1.5 IQR from edge of box

Extremes: More than 3 IQRs from edge of box

http://web.anglia.ac.uk/numbers/common_folder/graphics/fig6_single_box.jpg

Range: Distance from highest number to lowest number (may require real limits)

Interquartile Range: Distance from 25% point to 75% point (range of middle 50% of scores)

Average deviation: Sum of deviations of scores from M, divided by N = (X-M) / N

Range: Distance from highest number to lowest number (may require real limits)

Interquartile Range: Distance from 25% point to 75% point (range of middle 50% of scores)

Average deviation: Sum of deviations of scores from M, divided by N = (X-M) / NDOESN’T WORK: ALWAYS EQUALS 0.

Square the deviations so all scores positive

Sum of Squares (SS) used in most inferential statistics

SS is in the numerator of a fraction for both Variance and Standard Deviation

X Mean (X–Mean) (X-Mean)2

Jim 48 138.75 -90.75 8235.5625

Orlend 27 138.75 -111.75 12488.0625

Ellen 189 138.75 50.25 2525.0625

Steve 136 138.75 -2.75 7.5625

Jose 250 138.75 111.25 12376.5625

Tabia 218 138.75 79.25 6280.5625

Kaleb 151 138.75 12.25 150.0625

Lisa 201 138.75 62.25 3875.0625

Pavlik 78 138.75 -60.75 3690.5625

Kris 163 138.75 24.25 588.0625

Emma 106 138.75 -32.75 1072.5625

Michael 98 138.75 -40.75 1660.5625

Mean= 138.75 Sum= 0 52950.25

Called SS or “Sum of Squares” which

means Sum of Squared Deviations from the Mean

Julian Mean X-M (X-M)2

16 15 1 114 15 -1 115 15 0 014 15 -1 116 15 1 114 15 -1 114 15 -1 117 15 2 414 15 -1 113 15 -2 415 15 0 015 15 0 016 15 1 115 15 0 018 15 3 915 15 0 014 15 -1 116 15 1 114 15 -1 115 15 0 0

Sum 300 Sum 0 28 SSMean 15 1.474 Variance

1.214 Std Dev

2)( MXSS

Delia Scores X X2

15 225

22 484

10 100

11 121

16 256

13 169

20 400

13 169

17 289

8 64

18 324

16 256

12 144

14 196

9 81

19 361

18 324

13 169

21 441

15 225

Sum 300 4798

N 20 60.1 SS

3.163Variance

1.779Std Dev

N

XXSS

2

2)(

Range: Distance from highest number to lowest number (may require real limits)

Interquartile Range: Distance from 25% point to 75% point (range of middle 50% of scores)

Average deviation: Sum deviations from M, divide by N. Doesn’t work. Always = 0.

Variance: Average of squared deviation scores

2

n

SS

n

XX

2

2)(

Variance: Average of squared deviation scores.

Standard Deviation: Square root of Variance

n

SS

n

MX

2

2)(

n

SS

n

MX

2)(

STANDARD DEVIATIONFOR A SAMPLE

The data we use are from a randomly selected sample

Numerator of fraction is Sum of Squares (SS)

Denominator of fraction is n – 1

Symbols: s or SD

Excel function: STDEV

STANDARD DEVIATIONFOR A POPULATION

The data we use are from all members of population

Numerator of fraction is Sum of Squares (SS)

Denominator of fraction is n

Symbols: σ (Greek sigma = s)

Excel function: STDEVP

1

)(

1

2

n

XX

n

SSs

n

XX

n

SS

2)(

Range and Interquartile Range use only a few scores.

Standard deviation and variance use all scores

Measure of Variability

Can be used with … Best or most commonly used for …

Percentages Any type of data Categorical / Nominal data

Range or Interquartile Range

Data with order (low to high)and equal intervals(Interval or Ratio data)

•Open-ended categories•Indeterminate values•Extreme or skewed values

Boxplot Interval or Ratio dataGood for graph

•Comparing variability in two or more groups (Ch 8-14)

Standard Deviation or Variance

Interval or Ratio data with no open-ended or indeterminate values

•Any situation in which it can be appropriately computed•Inferential Statistics

Range and Interquartile Range

Standard Deviation and Variance