Formula

3. Formula

4. Compute a standard deviation with the Raw-Score Method • Previously learned the deviation formula • Good to see “what's going on” • Raw score formula • Easier to calculate than the deviation formula • Not as intuitive as the deviation formula • They are algebraically the same!!

5. Raw-Score Formula Note: This is the formula for both  and S

6. Step 1: Create a table

7. Step 2: Square each value

8. Step 3: Sum

9. Step 4: Plug in values N = 5 X = 44  X2 = 640

10. Step 4: Plug in values 5 5 N = 5 X = 44  X2 = 640

11. Step 4: Plug in values 44 5 5 N = 5 X = 44  X2 = 640

12. Step 4: Plug in values 44 640 5 5 N = 5 X = 44  X2 = 640

13. Step 5: Solve! 1936 44 640 5 5

14. Step 5: Solve! 1936 44 640 387.2 5 5

15. Step 5: Solve! 1936 44 50.56 640 387.2 5 5 Answer = 7.11

16. Practice • Use the raw score formula and find the standard deviation of: 6, 3, 4, 10, 8

18. 31 225 5 5 N = 5 X = 31  X2 = 225

19. 1936 44 225 192.2 2.56 = 5 5

21. Ŝ • What if we want to use a sample standard deviation to estimate the population ? • We need to make one small change to the formula to do this • You need to make the s an “unbiased estimator”

22. Ŝ • To do that you use Ŝ • This provides an estimate of the populations variability

23. Remember

24. Just “ - 1” Ŝ

25. Remember S =

26. Just “ - 1” Ŝ= -1

27. Why? • The first formula is biased -- its answer tends to be too small • Don’t worry about why -- unless you want too!!

28. Practice! • Below is data from 5 people in this class. What is the estimated standard deviation of all the students in this class? Use the Ŝ raw score formula. • Neuroticism scores 12, 15, 22, 10, 9

30. 68 1034 5 5 - 1 N = 5 X = 68  X2 = 1034

31. 1936 44 1034 924.8 5.22 = 5 4

33. Variance • The last step in calculating a standard deviation is to find the square root • The number you are fining the square root of is the variance!  2 = population variance Ŝ2 = sample variance used to estimate  2

34. Variance S 2,  2 = Ŝ2 =

35. Variance S 2,  2 = Ŝ2 = - 1

36. There are 12 different formulas! • Standard Deviation • Deviation Formula , S, Ŝ • Raw Formula , S, Ŝ • Variance • Deviation Formula  2, S 2, Ŝ2 • Raw Formula  2, S 2, Ŝ2

37. Review -- Important Formulas • Standard Deviation -- Deviation Formula  = Ŝ =

38. Review -- Important Formulas • Standard Deviation -- Deviation Formula

39. Review -- Important Formulas • Variance -- Deviation Formula 2 = Ŝ2 =

40. Review -- Important Formulas • Variance -- Deviation Formula 2

41. Review -- Important Formulas • Standard Deviation -- Raw Formula  and S = Ŝ = - 1

42. Review -- Important Formulas • Variance -- Raw Formula 2 and S2 = Ŝ2 = - 1

43. How to know which to use • 1) Does the question want a standard deviation or a variance (most of the time standard deviations are used) • 2) Is the group of scores a sample or population? • 3) If it’s a sample, do you want to generalize the findings to a population?

44. Practice • You are interested in how citizens of the US feel about the president. You asked 8 people to rate the president on a 10 point scale. Describe how the country feels about the president -- be sure to report a measure of central tendency and the standard deviation. 8, 4, 9, 10, 6, 5, 7, 9

45. Central Tendency 8, 4, 9, 10, 6, 5, 7, 9 4, 5, 6, 7, 8, 9, 9, 10 Mean = 7.25 Median = (4.5) = 7.5 Mode = 9

46. Standard Deviation • Want to use Ŝ

47. Standard Deviation • Want to use Ŝ -1

48. Standard Deviation • Want to use Ŝ 452 58 8 8 - 1 -1

49. Standard Deviation • Want to use Ŝ 58 452 8 8 - 1 -1

50. Standard Deviation • Want to use Ŝ 452 420.5 7 -1