Measures of Variability Measures of Variability n Why

Measures of Variability

Measures of Variability n Why are measures of variability important? Why not just stick with the mean? q Ratings of attractiveness (out of 10) – Mean = 5 q Everyone rated you a 5 (low variability) n q What could we conclude about attractiveness from this? People’s ratings fell into a range from 1 – 10, that averaged a 5 (high variability) n What could we conclude about attractiveness from this?

Measures of Variability n n n Range Interquartile Range Average Deviation Variance Standard Deviation

Measures of Variability n Range q q The difference between the highest and lowest values in a dataset Heavily biased by outliers n Dataset #1: 5 7 11 Range = 6 Dataset #2: 5 7 11 million Range = 10, 999, 995

Measures of Variability n Interquartile Range q q The difference between the highest and lowest values in the middle 50% of a dataset Less biased by outliers than the Range Based on sample with upper and lower 25% of the data “trimmed” However this kind of trimming essentially ignores half of your data – better to trim top and bottom 1 or 5%

Measures of Variability n Average Deviation q q For each score, calculate deviation from the mean, then sum all of these scores However, this score will always equal zero n Dataset: 19, 16, 20, 17, 20, 19, 7, 11, 10, 19, 14, 11, 6, 11, 14, 19, 20, 17, 4, 11 X = 285/20 = 14. 25

Data n n n Diff. from Mean = 0 0/N = 0 www. randomizer. org Mean Diff. from Mean 19 14. 25 4. 75 16 14. 25 1. 75 20 14. 25 5. 75 17 14. 25 2. 75 20 14. 25 5. 75 19 14. 25 4. 75 7 14. 25 -7. 25 11 14. 25 -3. 25 10 14. 25 -4. 25 19 14. 25 4. 75 14 14. 25 -. 25 11 14. 25 -3. 25 6 14. 25 -8. 25 11 14. 25 -3. 25 14 14. 25 -. 25 19 14. 25 4. 75 20 14. 25 5. 75 17 14. 25 2. 75 4 14. 25 -10. 25 11 14. 25 -3. 25

Measures of Variability n Variance q q Sample Variance (s 2) = (X - )2/(n -1) Population Variance (σ2) = (X - )2/N n n n Note the use of squared units! Gets rid of the positive and negative values in our “Diff. from Mean” column before that added up to 0 However, because we’re squaring our values they will not be in the metric of our original scale q If we calculate the variance for a test out of 100, a variance of 100 is actually average variability of 10 pts. ( 100 = 10) about the mean of the test

Data n n (Diff. from Mean)2 = 493. 75 Variance = 493. 75/(20 -1) = 25. 99 Mean Diff. from Mean (Diff. from Mean)2 19 14. 25 4. 75 22. 56 16 14. 25 1. 75 3. 06 20 14. 25 5. 75 33. 06 17 14. 25 2. 75 7. 56 20 14. 25 5. 75 33. 06 19 14. 25 4. 75 22. 56 7 14. 25 -7. 25 52. 56 11 14. 25 -3. 25 10. 56 10 14. 25 -4. 25 18. 06 19 14. 25 4. 75 22. 56 14 14. 25 -. 25 . 06 11 14. 25 -3. 25 10. 56 6 14. 25 -8. 25 68. 06 11 14. 25 -3. 25 10. 56 14 14. 25 -. 25 . 06 19 14. 25 4. 75 22. 56 20 14. 25 5. 75 33. 06 17 14. 25 2. 75 7. 56 4 14. 25 -10. 25 105. 06 11 14. 25 -3. 25 10. 56

Measures of Variability n Standard Deviation q q Sample Standard Deviation (s) = √ [ (X - )2/(n -1)] Population Standard Deviation (σ) = √ [ (X - )2/N] n n q q Note that the formula is identical to the Variance except that after everything else you take the square-root! You can interpret the standard deviation without doing any mental math, like you did with the variance Variance = 25. 99 Standard Deviation = √(25. 99) = 5. 10

Measures of Variability n Standard Deviation q q Example: Bush/Cheaney – 55% Kerry/Edwards – 40% Margin of Error = 30% Bush/Cheaney – 25% – 85% Kerry/Edwards – 10% - 70%

Computational Formula for Variability n Definitional Formula q n designed more to illustrate how the formula relates to the concept it underlies Computational Formula q q q identical to the definitional formula, but different in form allows you to compute your variable with less effort particularly useful with large datasets

Computational Formula for Variability n Definitional Formula for Variance: q s 2 = (X – )2 N– 1 n Computational Formula for Variance: q s 2 = n n All you need to plug in here is X 2 and X Standard deviation still = √ s 2, no matter how it is calculated

Computational Formula for Variability n Definitional Formula for Standard Deviation: q s = √ [(X – )2] [ N– 1 ] n Computational Formula for Standard Deviation q s=√ ( )

Computational Formula for Variability n Example: q For the following dataset, compute the variance and standard deviation. 1 q q q 2 2 3 3 3 4 X = 23 X 2 = 77 s 2 = 77 – (23)2 _____8__ 8– 1 s 2 = 77 – 66. 125 = 1. 55 7 5 Data (X) X 2 1 1 2 4 3 9 3 9 4 16 5 25

Measures of Variability n What do you think will happen to the standard deviation if we add a constant (say 4) to all of our scores? n What if we multiply all the scores by a constant?

Measures of Variability n Characteristics of the Standard Deviation q Adding a constant to each score will not alter the standard deviation n n i. e. add 3 to all scores in a sample and your s will remain unchanged Let’s say our scores originally ranged from 1 – 10 q q Add 5 to all scores, the new data ranges from 6 – 15 In both cases the range is 9

Measures of Variability q However, multiplying or dividing each score by a constant causes the s to be similarly multiplied or divided by that constant (and s 2 by the square of the constant) n n i. e. divide each score by 2 and your s will decrease from 10 to 5 in multiplication, higher numbers increase more than lower ones do, increasing the distance between the highest and lowest score, which increases the variability q i. e. 2 x 5 = 10 – difference of 8 pts. 5 x 5 = 25 – difference of 20 pts.

Measures of Variability n Characteristics of the Standard Deviation q Generally, the larger the dataset, the smaller the range/standard deviation n More scores = more clustering in the middle – REMEMBER: more central scores are more likely to occur

Smaller Dataset Larger Dataset s = 3. 96482 s = 2. 75609

Graphically Depicting Variability n Boxplot/Box-and. Whisker Plot q Median q Hinges/1 st & 3 rd Quartiles q H-Spread q Whisker q Outlier

Graphically Depicting Variability n Boxplot/Box-and. Whisker Plot q Median q Hinges/1 st & 3 rd Quartiles q H-Spread q Whisker q Outlier {

Graphically Depicting Variability n Boxplot/Box-and. Whisker Plot q Median q Hinges/1 st & 3 rd Quartiles q H-Spread q Whisker q Outlier

Graphically Depicting Variability n Percentile – the point below which a certain percent of scores fall q i. e. If you are at the 75 th%ile (percentile), then 75% of the scores are at or below your score

Graphically Depicting Variability n Quartile – similar to %ile, but splits distribution into fourths q i. e. 1 st quartile = 0 -25% of distribution, 2 nd = 26 -50%, 3 rd = 51 -75%, 4 th = 76 -100%

Graphically Depicting Variability n Interpreting a Boxplot/Box-and. Whisker Plot q q q Off-center median = Nonsymmetry Longer top whisker = Positively-skewed distribution Longer bottom whisker = Negatively-skewed distribution

Graphically Depicting Variability

Graphically Depicting Variability n Boxplot/Box-and-Whisker Plot q q Hinge/Quartile Location = (Median Location+1)/2 Data: 1 3 3 5 8 8 9 12 13 16 17 17 18 20 21 40 n n n Median Location = (16+1)/2 = 8. 5 Hinge Location = (8. 5+1)/2 = 4. 75 (4 since we drop the fraction) Hinges = 5 and 18

Graphically Depicting Variability q H-Spread = Upper Hinge – Lower Hinge n q H-Spread = 18 -5 = 13 Whisker = H-Spread x 1. 5 n n Since the whisker always ends at an actual data point, if we, say calculated the whisker to end at a value of 12, but the data only has a 10 and a 15, we would end the whisker at the 10. Whiskers = 12 x 1. 5 = 19. 5 Lower whisker from 5 to 1 Higher whisker from 18 to 21 q Outliers n Value of 40 extends beyond upper whisker

Graphically Depicting Variability