1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
Welcome

2. A note on doodling

3. Before next exam (March 3rd)
Schedule of readings
Before next exam (March 3rd)
Please read chapters in OpenStax textbook
Please read Chapters 10, 11, 12 and 14 in Plous
Chapter 10: The Representativeness Heuristic
Chapter 11: The Availability Heuristic
Chapter 12: Probability and Risk
Chapter 14: The Perception of Randomness

4. By the end of lecture today 2/15/17
Counting ‘standard deviationses’ – z scores
Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve
Percentiles

5. Homework Assignment Assignment 10
Please complete the homework worksheet
Finding z scores and areas under the curve.
Due: Friday, February 17th
Assignment 11
Please complete the homework modules on the D2L website
HW10-Normal Curve, z scores and probabilities

6. Lab sessions Everyone will want to be enrolled
in one of the lab sessions
Labs continue
With Project 2

7. Hand out z tables

8. z score = raw score - mean standard deviation
If we go up one standard deviation
z score = +1.0 and raw score = 105
z = -1
z = +1
68%
If we go down one standard deviation
z score = -1.0 and raw score = 95
If we go up two standard deviations
z score = +2.0 and raw score = 110
z = -2
95%
z = +2
If we go down two standard deviations
z score = -2.0 and raw score = 90
If we go up three standard deviations
z score = +3.0 and raw score = 115
99.7%
z = -3
z = +3
If we go down three standard deviations
z score = -3.0 and raw score = 85
z score: A score that indicates how many standard deviations
an observation is above or below the mean of the distribution
z score = raw score - mean
standard deviation

9. If score is within 2 standard deviations (z < 2)
“not unusual score”
If score is beyond 2 standard deviations (z = 2 or up to 3)
“is unusual score”
If score is beyond 3 standard deviations (z = 3 or up to 4)
“is an outlier”
If score is beyond 4 standard deviations (z = 4 or beyond)
“is an extreme outlier”

10. Scores, standard deviations, and probabilities
What is total percent under curve?
What proportion of curve is above the mean?
.50
100%
The normal curve always has the same shape.
They differ only by having different means and standard deviation

11. Scores, standard deviations, and probabilities
What percent of curve is below a score of 100?
What score is associated with 50th percentile?
50%
median
Mean = 100
Standard deviation = 5

12. Raw scores, z scores & probabilities
Distance from the mean
(z scores)
convert
convert
Raw Scores
(actual data)
Proportion of curve
(area from mean)
68%
z = -1
z = 1
We care about this!
What is the actual number on this scale? “height” vs “weight”
“pounds” vs “test score”
We care about this!
“percentiles”
“percent of people”
“proportion of curve” “relative position”
68%
Raw Scores
(actual data)
Proportion of curve
(area from mean)
z = -1
z = 1
Distance from the mean
(z scores)
convert
convert

13. z table Formula Normal distribution Raw scores z-scores probabilities
Have z
Find raw score Z
Scores
Have z
Find area
z table
Formula
Have area
Find z
Area &
Probability
Have raw score
Find z
Raw Scores

14. Find z score for raw score of 60
z score: A score that indicates how many standard deviations
an observation is above or below the mean of the distribution
z score = raw score - mean
standard deviation 60 50 10
z = 1
Mean = 50
Standard deviation = 10

15. Find the area under the curve that falls between 50 and 60
34.13%
1) Draw the picture
2) Find z score
3) Go to z table - find area under correct column
4) Report the area 60 50 10
z = 1

16. Find the area under the curve that falls between 40 and 60
Mean = 50 Standard deviation = 10
68.26%
Find the area under the curve that falls between 40 and 60
34.13%
34.13%
z score = raw score - mean
standard deviation
Hint always draw a picture!
z score = 10
z score = 10
z score = = 1.0 10
z score = = -1.0 10
z table
z table
z score of 1 = area of .3413
z score of 1 = area of .3413
= .6826

17. Find the area under the curve that falls between 30 and 50
Mean = 50 Standard deviation = 10
1) Draw the picture
2) Find z score
3) Go to z table - find area under correct column
4) Report the area
Find the area under the curve that falls between 30 and 50
z score = raw score - mean
standard deviation
z score = 10
z score = = 10
Hint always draw a picture!

18. Find the area under the curve that falls between 30 and 50
Mean = 50 Standard deviation = 10
1) Draw the picture
2) Find z score
3) Go to z table - find area under correct column
4) Report the area
47.72%
Find the area under the curve that falls between 30 and 50
z score = raw score - mean
standard deviation
z score = 10
z score = = 10
z table
z score of - 2 = area of .4772
Hint always draw a picture!
Hint always draw a picture!

19. Find the area under the curve that falls between 70 and 50
Mean = 50 Standard deviation = 10
47.72%
Find the area under the curve that falls between 70 and 50
z score = raw score - mean
standard deviation
z score = 10
z score = = +2.0 10
z table
z score of 2 = area of .4772
Hint always draw a picture!

20. Find the area under the curve that falls between 30 and 70
Mean = 50 Standard deviation = 10
.4772
.4772
95.44%
z score of 2 = area of .4772
Find the area under the curve that falls between 30 and 70
= .9544
Hint always draw a picture!

21. Scores, standard deviations, and probabilities
Actually 68.26
Actually 95.44
To be exactly 95% we will use z = 1.96

22. Writing Assignment Let’s do some problems
Mean = 50 Standard deviation = 10
Writing Assignment Let’s do some problems

23. Find the percentile rank for score of 60
Mean = 50 Standard deviation = 10 ?
Let’s do some problems 60
Find the area under the curve that falls below 60
means the same thing as
Find the percentile rank for score of 60
Problem 1

24. Find the percentile rank for score of 60
Mean = 50 Standard deviation = 10 ?
Let’s do some problems
Find the percentile rank for score of 60 60
.5000
.3413
1) Find z score
z score = 10
= 1
2) Go to z table
- find area under correct column (.3413)
3) Look at your picture - add to = .8413
4) Percentile rank or score of 60 = 84.13%
Problem 1
Hint always draw a picture!

25. Find the percentile rank for score of 75
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 75 75
.4938
1) Find z score
z score = 10
z score = 10
= 2.5
2) Go to z table
Problem 2
Hint always draw a picture!

26. Find the percentile rank for score of 75
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 75 75
.4938
.5000
1) Find z score
z score = 10
z score = 10
= 2.5
2) Go to z table
3) Look at your picture - add to = .9938
4) Percentile rank or score of 75 = 99.38%
Problem 2
Hint always draw a picture!

27. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45 45
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
Problem 3

28. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45
.1915 45 ?
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
Problem 3

29. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45
.1915 45
.3085
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
3) Look at your picture - subtract = .3085
Problem 3
4) Percentile rank or score of 45 = 30.85%

30. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
Problem 4

31. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
.1915
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
Problem 4

32. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
.1915 .5
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
3) Look at your picture - add = .6915
4) Percentile rank or score of 55 = 69.15%
Problem 4

33. Find the score that is associated
Mean = 50 Standard deviation = 10 ?
Find the score for z = -2 30
Hint always draw a picture!
Find the score that is associated
with a z score of -2
raw score = mean + (z score)(standard deviation)
Raw score = 50 + (-2)(10)
Raw score = (-20) = 30
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

34. percentile rank of 77%ile
Mean = 50 Standard deviation = 10 ?
Find the score for
percentile rank of 77%ile
.7700 ?
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 5

35. percentile rank of 77%ile
Mean = 50 Standard deviation = 10
.27 ?
Find the score for
percentile rank of 77%ile .5
= .77 .5
.27
.7700 ?
1) Go to z table - find z score for
for area ( ) = .27
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .2704
(closest I could find to .2700)
z = 0.74
Problem 5

36. percentile rank of 77%ile
Mean = 50 Standard deviation = 10
.27 ?
Find the score for
percentile rank of 77%ile .5
x = 57.4 .5
.27
.7700 ?
2) x = mean + (z)(standard deviation)
x = 50 + (0.74)(10)
x = 57.4
x = 57.4
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 5

37. percentile rank of 55%ile
Mean = 50 Standard deviation = 10 ?
Find the score for
percentile rank of 55%ile
.5500 ?
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 6

38. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5
= .55 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .05
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .0517
(closest I could find to .0500)
z = 0.13
Problem 7

39. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .05
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .0517
(closest I could find to .0500)
z = 0.13
Problem 7

40. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5
x = 51.3 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .0500
area = .0517
(closest I could find to .0500)
z = 0.13
2) x = mean + (z)(standard deviation)
x = 50 + (0.13)(10)
x = 51.3
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
x = 51.3
Problem 7

41. Not included in class lecture
Normal Distribution has a mean of 50 and standard deviation of 4.
Determine value below which 95% of observations will occur Note: sounds like a percentile rank problem
Go to
table
.4500
nearest z = 1.64
x = mean + z σ = 50 + (1.64)(4) = 56.56
.9500
.4500
.5000
Additional practice
Problem 8 38 42 46 50 54
56.60 ? 58 62
Not included in class lecture

42. Not included in class lecture
Normal Distribution has a mean of $2,100 and s.d. of $250.
What is the operating cost for the lowest 3% of airplanes Note: sounds like a percentile rank problem = find score for 3rd percentile
Go to
table
.4700
nearest z =
x = mean + z σ = (-1.88)(250) = 1,630
.0300
.4700
Additional practice
Problem 9
1,630 ?
2100
Not included in class lecture

43. Not included in class lecture
Normal Distribution has a mean of 195 and standard deviation of 8.5.
Determine value for top 1% of hours listened.
Go to
table
.4900
nearest z = 2.33
x = mean + z σ = (2.33)(8.5) =
.4900
.5000
.0100
Additional practice
Problem 10
195
214.8 ?
Not included in class lecture

44. Not included in class lecture.
Find score associated with the 75th percentile
75th percentile
Go to
table
nearest z = .67
.2500
x = mean + z σ = 30 + (.67)(2) = 31.34
.7500
.25
.5000 24 26 28 30 ? 34 36
31.34
Additional practice
Problem 11
z = .67
Not included in class lecture

45. Not included in class lecture.
Find the score associated with the 25th percentile
25th percentile
Go to
table
nearest z = -.67
.2500
x = mean + z σ = 30 + (-.67)(2) = 28.66
.2500
.25
.25 24 26
28.66 ? 28 30 34 36
Additional practice
Problem 12
z = -.67
Not included in class lecture

46. .
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (1.96)(2) = 33.92
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-1.96)(2) = 26.08
.9500
.475
.475
Additional practice
Problem 13
26.08
33.92 24 ? 28 32 ? 30 36
Not included in class lecture

47. .
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 100 and standard deviation of 5
Go to
table
.4750
nearest z = 1.96
mean + z σ = (1.96)(5) =
Go to
table
.4750
nearest z = -1.96
mean + z σ = (-1.96)(5) = 90.20
.9500
.475
.475
Additional practice
Problem 14
90.2
109.8 85 ? 95
105 ?
100
115
Not included in class lecture

48. .
Try this one:
Please find the (2) raw scores that border exactly the middle 99% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (2.58)(2) = 35.16
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-2.58)(2) = 24.84
.9900
.495
.495
Additional practice
Problem 15
24.84 ?
35.16 28 32 ? 30
Not included in class lecture

49. Thank you!
See you next time!!