Lab 2: Central Tendency and Variability

1. Lab 2:Central Tendency and Variability

2. Learning Objectives • Describe Central Tendency • Properties of the Mean, Median, & Mode • Hand calculation of Mean, Median & Mode • SPSS calculation of Mean, Median & Mode

3. Learning Objectives (2) • Describe Dispersion • Properties of the range, average deviation, standard deviation, and variance • Calculate range, SD and variance by hand • Calculate them with SPSS

4. Central Tendency • Central tendency means the middle of the distribution. Three main indicators: • The Mode • The Median • The Mean

5. The Mode Defined • The most frequently occurring score or category • Most useful with categorical data. • Can be more than one mode (bimodal, etc.). • Unstable (based on limited scores). • To calculate: find the most frequently occurring score.

6. Calculating the Mode • 1122333455555556667789 • The mode is 5 because there are 7 fives in the distribution. • What is the mode for these data? • 12345677778889

7. The Median Defined • The middlemost score; separates the top 50 % of scores from the bottom 50% • More stable than the mode • Most useful for ordinal data

8. Calculating the Median – Even # of scores • If even number of scores, use the average of two middle scores • 1 2 3 4 5 6 7 8 (Order) • 2 2 3 3 4 5 5 6 (Number) • Median is 3.5 = (3+4)/2

9. Calculating Median (Even) • 1 2 3 4 5 6 7 8 (Order) • 1 2 2 2 2 3 3 3 (Number) • Median is 2 = (2+2)/2 • 1 2 3 4 5 6 7 8 9 10 (Order) • 1 2 2 5 6 8 9 9 9 10 (Number) • Median is 7 = (6+8)/2

10. Calculating Median (Odd) • If there are an odd number of scores in the distribution, the median is the middle score [(N+1)/2]. • 1 2 3 4 5 6 7 8 9 (Order) • 1 2 4 5 8 9 9 9 10 (Number) • Median is 8 (fifth score)[(9+1)/2]

11. Calculating Median (Test) • What is the median? • 1 2 3 4 5 6 7 8 9 (Order) • 2 3 4 5 6 8 9 9 9 (Number) • 1 2 3 4 5 6 (Order) • 0 5 7 9 9 9 (Number)

12. The Mean Defined • The mean is the arithmetic average. It is the sum of scores divided by the number of scores. • That is, add ‘em all up and divide by how many you’ve got.

13. Mean Defined • In the sample, the mean is: • In the population, the mean is:

14. Properties of the Mean • Stable • Basis for many statistical procedures • Unbiased estimator – will not over or underestimate population mean on average. • Deviations from mean sum to zero.

15. Calculating the Mean • 1 2 3 4 5 • Mean is (1+2+3+4+5)/5 = 15/5 = 3 • 2 2 2 3 3 3 • Mean is (2+2+2+3+3+3) = 15/6 = 2+3/6 = 2.5

16. Comparing Measures of Central Tendency • Use mode when • Need speed – fastest • Need score that occurs most often • Use Median when • Have small but skewed distribution • Missing/arbitrarily determined scores • Use Mean when • Estimating the population mean

17. Dispersion Defined • Dispersion refers to the spread of scores around the mean. Hug or hate the mean. • Four main indices of dispersion: • Range • Average Deviation • Variance • Standard Deviation

18. The Range • The Range is the difference between the top and bottom scores (Max – min). • Gives a quick approximation of variability in the data • Depends on quality of extreme scores

19. Calculating the Range • What is the range? • 1 2 2 3 3 5 6 7 8 9 • The range is 9-1 = 8. • 5 7 9 12 • The range is 12-5 = 7. • 2 2 2 2 3 3 3 4 4 13 • The range is 13-2 = 11.

20. Average Deviation Defined • The average deviation is the average absolute difference between the scores and the mean. • The formula for the AD is:

21. Calculating the AD • To find the Average Deviation • Find the mean • Subtract the mean from each score • Take the absolute value of the difference • Sum the absolute values • Divide by N, the number of scores

22. Calculating the AD • Example of AD • 2 2 3 3 4 4 • Mean is 3 = (18/6) • Absolute deviations: 1 1 0 0 1 1 • Sum is 4 • AD is 4/6 = 2/3 = .67 • Nice conceptually, but not very useful in practice

23. Variance Defined • The average of the unsigned deviations is zero • The average of the absolute deviations is bigger than zero unless all observations are at the mean • Instead of taking absolute value, we can square the deviations

24. Variance Defined • The variance is the average or mean of the squared deviations. • The population formula: • The sample formula:

25. Calculating the Variance

26. Calculating the Variance • From the last slide, the sum of squared deviations is 8. • The average is 8/3 = 2.67 • This is the variance.

27. Calculating the Variance

28. Calculating the Variance • What is the variance of the following distribution? • 5 6 7 8 9 10 • Answer: 2.92

29. Calculating the Variance • For reasons we will describe later, SPSS calculates the sample variance like this: • The only difference is (N-1) instead of N. You need this to compare your answer to SPSS. Use N-1 to compare to SPSS so you agree.

30. Standard Deviation Defined • The standard deviation is the square root of the variance. • The variance is the average squared deviation. • The standard deviation is in the original, unsquared units of measurement. • It’s easier to interpret than the variance.

31. Standard Deviation Defined • The population formula: • The sample formula {SPSS uses (N-1)}:

32. Calculating the SD

33. A Look Ahead • The mean, variance and standard deviation form the basis of most of the statistics we will be using. • If we do an experiment, we will look at the means and variance to decide if the treatment had an effect.

34. Using SPSS • Next we will wake up SPSS and see how to get it to provide the estimates of central tendency and variability

35. Homework • Compute estimates of central tendency (mode, median, mean). • Compute estimates of variability (range, variance, SD). • Explain them. • Compare the two distributions. • Due at the beginning of next lab.