1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Describing Distributions &
Locating scores & Transforming distributions

2. Announcements Homework #1: due today Quiz problems
Quiz 1 is now posted, due date extended to Tu, Jan 26th (by 11:00)
Quiz 2 is now posted, due Th Jan 28th (1 week from today)
Don’t forget Homework 2 is due Tu (Jan 26)

3. Outline (for week) Characteristics of Distributions
Finishing up using graphs
Using numbers (center and variability)
Descriptive statistics decision tree
Locating scores: z-scores and other transformations

4. Standard deviation
The standard deviation is the most commonly used measure of variability.
The standard deviation measures how far off all of the scores in the distribution are from the mean of the distribution.
Essentially, the average of the deviations. m

5. Computing standard deviation (population)
To review:
Step 1: compute deviation scores
Step 2: compute the SS
SS = Σ (X - μ)2
Step 3: determine the variance
take the average of the squared deviations
divide the SS by the N
Step 4: determine the standard deviation
take the square root of the variance

6. Computing standard deviation (sample)
The basic procedure is the same.
Step 1: compute deviation scores
Step 2: compute the SS
Step 3: determine the variance
This step is different
Step 4: determine the standard deviation

7. Computing standard deviation (sample)
Step 1: Compute the deviation scores
subtract the sample mean from every individual in our distribution.
Our sample
2, 4, 6, 8 X
X - X = deviation scores
2 - 5 = -3
6 - 5 = +1
4 - 5 = -1
8 - 5 = +3

8. Computing standard deviation (sample)
Step 2: Determine the sum of the squared deviations (SS).
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
= (-3)2
+ (-1)2
+ (+1)2
+ (+3)2
= = 20
X - X = deviation scores
SS = Σ (X - X)2
Apart from notational differences the procedure is the same as before

9. Computing standard deviation (sample)
Step 3: Determine the variance
Recall:
Population variance = σ2 = SS/N
The variability of the samples is typically smaller than the population’s variability μ X 3 X 1 X 4 X 2

10. Computing standard deviation (sample)
Step 3: Determine the variance
Recall:
Population variance = σ2 = SS/N
The variability of the samples is typically smaller than the population’s variability
To correct for this we divide by (n-1) instead of just n
Sample variance = s2

11. Computing standard deviation (sample)
Step 4: Determine the standard deviation
standard deviation = s =

12. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Changes the total and the number of scores, this will change the mean and the standard deviation

13. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
All of the scores change by the same constant. X
old

14. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
All of the scores change by the same constant. X
old

15. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
All of the scores change by the same constant. X
old

16. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
All of the scores change by the same constant. X
old

17. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
All of the scores change by the same constant.
But so does the mean X
new

18. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

19. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

20. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

21. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

22. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

23. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

24. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old

25. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
No change
It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X
old X
new

26. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
No change
Multiply/divide a constant to each score
= -1
(-1)2 X
= +1
(+1)2
s =

27. Properties of means and standard deviations
Change/add/delete a given score
changes
changes
Add/subtract a constant to each score
changes
No change
Multiply scores by 2
Multiply/divide a constant to each score
changes
changes
= -2
(-2)2 X
= +2
(+2)2
Sold=1.41
s =

28. Locating a score Where is our raw score within the distribution?
The natural choice of reference is the mean (since it is usually easy to find).
So we’ll subtract the mean from the score (find the deviation score).
The direction will be given to us by the negative or positive sign on the deviation score
The distance is the value of the deviation score

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

30. Locating a score μ Reference point Below Above X1 = 162 X1 - 100 = +62

31. Transforming a score The distance is the value of the deviation score
However, this distance is measured with the units of measurement of the score.
Convert the score to a standard (neutral) score. In this case a z-score.
Raw score
Population mean
Population standard deviation

32. Transforming scores μ X1 = 162 X1 - 100 = +1.20 50 X2 = 57
A z-score specifies the precise location of each X value within a distribution.
Direction: The sign of the z-score (+ or -) signifies whether the score is above the mean or below the mean.
Distance: The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and σ.
X1 = 162
X = +1.20 50
X2 = 57
X = -0.86 50

33. Transforming a distribution
We can transform all of the scores in a distribution
We can transform any & all observations to z-scores if we know either the distribution mean and standard deviation.
We call this transformed distribution a standardized distribution.
Standardized distributions are used to make dissimilar distributions comparable.
e.g., your height and weight
One of the most common standardized distributions is the Z-distribution.

34. Properties of the z-score distributionμ μ
transformation
Xmean = 100 50
150
= 0

35. Properties of the z-score distributionμ μ
transformation +1
X+1std = 150 50
150
Xmean = 100
= 0
= +1

36. Properties of the z-score distributionμ μ
transformation -1
X-1std = 50 50
150 +1
Xmean = 100
= 0
X+1std = 150
= +1
= -1

37. Properties of the z-score distribution
Shape - the shape of the z-score distribution will be exactly the same as the original distribution of raw scores. Every score stays in the exact same position relative to every other score in the distribution.
Mean - when raw scores are transformed into z-scores, the mean will always = 0.
The standard deviation - when any distribution of raw scores is transformed into z-scores the standard deviation will always = 1.

38. From z to raw score m m Z = -0.60 X = 70 X = (-0.60)( 50) + 100
We can also transform a z-score back into a raw score if we know the mean and standard deviation information of the original distribution. m
150 50 m +1 -1
transformation
Z = -0.60
X = 70
X = (-0.60)( 50) + 100

39. Why transform distributions?
Known properties
Shape - the shape of the z-score distribution will be exactly the same as the original distribution of raw scores. Every score stays in the exact same position relative to every other score in the distribution.
Mean - when raw scores are transformed into z-scores, the mean will always = 0.
The standard deviation - when any distribution of raw scores is transformed into z-scores the standard deviation will always = 1.
Can use these known properties to locate scores relative to the entire distribution
Area under the curve corresponds to proportions (or probabilities)

40. SPSS
There are lots of ways to get SPSS to compute measures of center and variability
Descriptive statistics menu
Compare means menu
Also typically under various ‘options’ parts of the different analyses
Can also get z-score transformation of entire distribution using the descriptives option under the descriptive statistics menu