variability
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Range and Interquartile Range
Standard Deviation and Variance
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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/
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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
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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
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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
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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.
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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.
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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
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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
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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
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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.
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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
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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
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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)(
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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
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2)(
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Variance: Average of squared deviation scores.
Standard Deviation: Square root of Variance
n
SS
n
MX
2
2)(
n
SS
n
MX
2)(
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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)(
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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
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Range and Interquartile Range
Standard Deviation and Variance