Range and Interquartile Range Standard Deviation and Variance
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