1. Reasoning in Psychology Using Statistics
2017

2. Annoucement Quiz 3 is posted, due Friday, Feb. 24 at 11:59 pm
Covers
Tables and graphs
Measures of center
Measures of variability
You may want to have a calculator handy
Exam 2 is two weeks from today (Wed. Mar. 8th)
Annoucement

3. Outline for next 2 classes
Transformations: z-scores
Normal Distribution
Using Unit Normal Table
Combines 2 topics
Today
Statistical Snowball:
For the rest of the course, new concepts build upon old concepts
So if you feel like you don’t understand something now, ask now, don’t wait. X X X X
Outline for next 2 classes

4. Characteristics of a standard deviation
Change/add/delete a given score, then the standard deviation will change.
May change the mean and (if adding or subtracting) the number of scores (n or N)
Characteristics of a standard deviation

5. Characteristics of a standard deviation
Change/add/delete a given score, then the standard deviation will change.
Add/subtract a constant to each score, then the standard deviation will NOT change.
Looking at a numerical example. (subtract 1 from every score)
Original sample
2, 4, 6, 8
New sample
1, 3, 5, 7
Original mean 5
New mean 4
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
1 - 4 = -3
3 - 4 = -1
5 - 4 = +1
7 - 4 = +3
Original SS 20
Characteristics of a standard deviation

6. Characteristics of a standard deviation
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the standard deviation will NOT change.
Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. X
= -1
(-1)2
= +1
(+1)2
s =
Characteristics of a standard deviation

7. Characteristics of a standard deviation
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the standard deviation will NOT change.
Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant.
= -2
(-2)2 X
= +2
(+2)2
s =
Sold=1.41
Characteristics of a standard deviation

8. Extreme scores: Range is most affected, IQR is least affected
Sample size: Range tends to increase as n increases, IQR & s do not
With open-ended distributions, one cannot even compute the Range or σ, so the IQR is the only option
Range is unstable when you repeatedly sample from the same population, but the IQR & σ are stable and tend not to fluctuate.
When to use which

9. Location Where is Bone student center? Reference point Direction
CVA Rotunda
Direction
North (and 10o West)
Distance
Approx ft.
1625 ft.
Location

10. Locating a score Where is a score within distribution? Reference point
Direction
Distance
Obvious choice is mean μ
Negative or positive sign on deviation score
Subtract mean from score (deviation score).
Value of deviation score
Locating a score

11. Locating a score μ Reference point X1 = 162 X1 - 100 = +62 Direction

12. Locating a score μ Below Above X1 = 162 X1 - 100 = +62 Direction

13. Locating a score μ Distance Distance X1 = 162 X1 - 100 = +62 X2 = 57

14. Transforming a score Direction and Distance
Deviation score is valuable,
BUT measured in units of measurement of score
AND lacks information about average deviation
SO, convert raw score (X) to standard score (z).
Raw score
Population mean
Population standard deviation
Transforming a score

15. Transforming scores μ X1 - 100 = +1.24 50 If X1 = 162, z =
z-score: standardized location of X value within distribution
X = +1.24 50
If X1 = 162,
z =
Direction. Sign of z-score (+ or -): whether score is above or below mean
Distance. Value of z-score: distance from mean in standard deviation units
X = -0.86 50
If X2 = 57,
z =
Transforming scores

16. Transforming scores μ μ = 20 σ = 5 X1 - 20 = +1.2 5 If X1 = 26, z =
z-score: standardized location of X value within distribution
X = +1.2 5
If X1 = 26,
z =
Direction. Sign of z-score (+ or -): whether score is above or below mean
Distance. Value of z-score: distance from mean in standard deviation units
X = -0.8 5
If X2 = 16,
z =
Transforming scores

17. Transforming distributions
Can transform all of scores in distribution
Called a standardized distribution
Has known properties (e.g., mean & stdev)
Used to make dissimilar distributions comparable
Comparing your height and weight
Combining GPA and GRE scores
z-distribution
One of most common standardized distributions
Can transform all observations to z-scores if know distribution mean & standard deviation
Transforming distributions

18. Properties of z-score distribution
Shape:
Mean:
Standard Deviation:
Properties of z-score distribution

19. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score.
transformation μ
150 50 μZ
original
z-score
Note: this is true for other shaped distributions too:
e.g., skewed, mulitmodal, etc.
Properties of z-score distribution

20. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score
Mean
If X = μ, z = ?
Meanz always = 0
transformation μ μZ
Xmean = 100 50
150
= 0
= 0
Properties of z-score distribution

21. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score
Mean: always = 0
Standard Deviation:
Properties of z-score distribution

22. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score
Mean: always = 0
Standard Deviation:
For z, μ μ
transformation +1
X+1std = 150 50
150
= +1
z is in standard deviation units
Properties of z-score distribution

23. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score
Mean: always = 0
Standard Deviation:
For z, μ μ
transformation -1
X-1std = 50 50
150 +1
X+1std = 150
= +1
= -1
Properties of z-score distribution

24. Properties of z-score distribution
Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score
Mean: always = 0
Standard Deviation: always = 1, so it defines units of z-score
Properties of z-score distribution

25. If know z-score and mean & standard deviation of original distribution, can find raw score (X)
have 3 values, solve for 1 unknown
 (z)( σ) = (X - μ)
 X = (z)( σ) + μ μ +1 -1 μ
150 50
transformation
 z = -0.60
X = 70
 X = (-0.60)( 50) + 100 =
From z to raw score:

26. SAT examples Population parameters of SAT: μ= 500, σ= 100 Example 1
Another student got 420. What is her z-score?
A student got 580 on the SAT. What is her z-score?
Example 1
SAT examples

27. SAT examples Population parameters of SAT:
μ= 500, σ= 100
Example 2
Student said she got 1.5 SD above mean on SAT. What is her raw score?
X = z σ + μ
= (1.5)(100) + 500
= = 650
Standardized tests often convert scores to:
μ = 500, σ = 100 (SAT, GRE)
μ = 50, σ = 10 (Big 5 personality traits)
SAT examples

28. SAT examples SAT: μ = 500, σ = 100 ACT: μ = 21, σ = 3 Example 3
Suppose you got 630 on SAT & 26 on ACT. Which score should you report on your application?
Example 3
z-score of 1.67 (ACT) is higher than z-score of 1.3 (SAT), so report your ACT score.
SAT examples

29. Example with other tests
On Aptitude test A, a student scores 58, which is .5 SD below the mean. What would his predicted score be on other aptitude tests (B & C) that are highly correlated with the first one?
Test B: μ = 20, σ = XB < or > 20? How much: 1? 2.5? 5? 10?
Test C: μ = 100, σ = 20 XC < or > 100? How much: 20? 10?
If XA = -.5 SD, then zA = -.5
XB = zB σ + μ
XC = zC σ + μ = (-.5)(20) = = 90
Find out later that this is true only if perfectly correlated; if less so, then XB and XC closer to mean.
= (-.5)(5) + 20
= = 17.5
Example with other tests

30. Population
Sample
Mean
Standard Deviation
Z-score X
Formula Summary

31. Wrap up In lab Questions?
Using SPSS to convert raw scores into z-scores; copy formulae with absolute reference
Questions?
Applications: Statistics and the Weather
Brian’s Climate Blog
NOAA – Jan 2014 Temperature Anomalies
Brandon Foltz: Understanding z-scores (~22 mins)
StatisticsFun (~4 mins)
Chris Thomas. How to use a z-table (~7 mins)
Dr. Grande. Z-scores in SPSS (~7 mins)
Wrap up