Understanding Variability: Range, IQR, Standard Deviation, Variance

1. Part IISigma Freud & Descriptive Statistics Chapter 3     Viva La Difference: Understanding Variability

2. What you will learn in Chapter 3 • Variability is valuable as a descriptive tool • Difference between variance & standard deviation • How to compute: • Range • Inter-quartile Range • Standard Deviation • Variance

3. Why Variability is Important • Variability • how different scores are from one particular score • Spread • Dispersion • What is the “score” of interest here? • Ah ha!! It’s the MEAN!! • So…variability is really a measure of how each score in a group of scores differs from the mean of that set of scores.

4. Measures of Variability • Four types of variability that examine the amount of spread or dispersion in a group of scores… • Range • Inter-quartile Range • Standard Deviation • Variance • Typically report the average and the variability together to describe a distribution.

5. Computing the Range • Range is the most “general” estimate of variability… • Two types… • Exclusive Range • R = h - l • Inclusive Range • R = h – l + 1 (Note: R is the range, h is the highest score, l is the lowest score)

6. Measures of variation Range • Range • The difference between the highest and lowest numbers in a set of numbers. 2, 35, 77, 93, 120, 540 540 – 2 = 538

7. Measures of variationRange • What is the range of: 2, 3, 3, 3, 4, 5, 6, 6, 7, 9, 11, 13, 15, 15, 15, 16 24, 57, 81, 96, 107, 152, 179, 211 1001, 1467, 1479, 1680, 1134

8. Interquartile range • Difference between upper (third) and lower (first) quartiles • Quartiles divide data into four equal groups • Lower (first) quartile is 25th percentile • Middle (second) quartile is 50th percentile and is the median • Upper (third) quartile is 75th percentile

9. Calculating the interquartile range for high temperatures interquartile range = 52 – 35 = 17

10. Stem and Leaf 0730 Q1 Fall 2010 (N=22) • 2|349 • 3|03344555666677779 • 4|01 • Q1= .25 (22)=5.5 data point round up to 6th data point=value of 33 • Q2= n+1/2=23/2=11.5 = avg of 11th and 12th data pt = 35.5 • Q3= .75(22)=16.5 =round up to17th data point= • Value of 37

11. Interquartile range and outliers • Value can be considered to be an outlier if it falls more than 1.5 times the interquartile range above the upper quartile or more than 1.5 times the range below the lower quartile • Example for high temperatures • Interquartile range is 17 • 1.5 times interquartile range is 25.5 • Outliers would be values • Above 52 + 25.5 = 77.5 (none) • Below 35 – 25.5 = 9.5 (none)

12. Review: Steps to Quartiles, Interquartile Range,and Checking for Outliers • 1) Put values in ascending OR descending order • 2) Multiply .25 (n) for Q1 • 3) Multiply .75 (n) for Q3 • 4) Q3 - Q1 = IQR • 5) Q1 – 1.5 (IQR)= value below smallest value in data set; • 6) Q3 + 1.5 (IQR)= value above largest value in data set;

13. Let’s practice Finding Outliers • What is the median, Q1, Q3, range, and IQR for the following? Then check for outliers. 10, 25, 35, 65, 100, 255, 350, 395 (n=8) • 10, 65, 75, 99, 299 (n=5) • 5, 39, 45, 59, 64, 74 (n=6)

14. Computing Standard Deviation • Standard Deviation (SD) is the most frequently reported measure of variability • SD = average amount of variability in a set of scores • What do these symbols represent?

15. Why n – 1? • The standard deviation is intended to be an estimate of the POPULATION standard deviation… • We want it to be an “unbiased estimate” • Subtracting 1 from n artificially inflates the SD…making it larger • In other words…we want to be “conservative” in our estimate of the population

16. Things to Remember… • Standard deviation is computed as the average distance from the mean • The larger the standard deviation the greater the variability • Like the mean…standard deviation is sensitive to extreme scores • If s = 0, then there is no variability among scores…they must all be the same value.

17. Computing Variance • Variance = standard deviation squared • So…what do these symbols represent? Does the formula look familiar?

18. Standard Deviation or Variance • While the formulas are quite similar…the two are also quite different. • Standard deviation is stated in original units • Variance is stated in units that are squared • Which do you think is easier to interpret???

19. Same mean, different standard deviation; Sample variance and Sample standard deviation: {20,31,50,69,80}

20. Then square each distance from mean and add together… • (-30)2 + (-19)2 + (0)2+ (19)2 + (30)2 • 900+ 361+ 0+ 361 +900= • 2522 • Divide by N-1 (N=5) • 2522/4=630.5= Sample Variance • To find sample standard deviation, take square root of variance= 25.11

21. Same mean, different standard deviation: {39,44,50,56,61}

22. Which data set has more variability? • (-11)2 + (-6)2 + (0)2 + (11)2 + (6)2 • 121+ 36+ 0+ 121+ 36= • 314 • Divide by N-1 gives us sample variance • 314/4=78.5 • Square root of 78.5 gives us sample standard deviation=8.86

23. Measures of variationStandard deviation • How about a more user-friendly equation?

24. Using Excel’s VAR Function

25. Using the Computer to Compute Measures of Variability

26. Glossary Terms to Know • Variability • Range • Standard deviation • Mean deviation • Unbiased estimate • Variance