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
Ŝ 2 = sample variance used to estimate  2

34. Variance
S 2,  2 =
Ŝ 2 =

35. Variance
S 2,  2 =
Ŝ 2 =
- 1

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

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 =
Ŝ 2 =

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 =
Ŝ 2 =
- 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

51. Standard Deviation
Want to use Ŝ
2.12 -1

53. Boxplots
The boxplot graphically displays three different characteristics of the distribution
Extreme scores
Interquartile range
Median

54. Boxplot

55. Boxplot
Interquartile range
25th - 75th percentile

56. Boxplot
Extreme Scores

57. Boxplot
Median

58. Boxplot
Skew -- Look at the “whiskers” to determine if the distribution is skewed

60. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

61. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median =
25th =
75th =
Lowest =
Highest =

62. Create a boxplot Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median = 22.5
25th = 6
75th = 62
Lowest = 2
Highest = 99

64. Neuroticism

65. Extraversion

66. Conscientiousness

67. Which distribution has a positive skew?

68. Which distribution has a negative skew?

69. Which distribution is most compact?E A B D

70. Which distribution has a median close to 25?

71. Which distribution is most symmetrical?

72. Which distribution has has the largest range?

73. Practice
Make a box plot using the simple frequency distribution of Satisfaction with Life scores on page 27

74. Practice
Page 67
3.25
3.28

75. Practice
3.25
Use S “hat”
33.16

76. Practice
3.28
Use S “hat”
A = 1.90
B = 1.55
B is more consistent