1. Normal Distributions and Standard Scores
PSYC 214
Normal Distributions and Standard Scores

2. Agenda Percentiles Distributions Standard (-ization of) Scores
Appendix B
Examples

3. Few Reminders Rounding Greater than or less than > or <
Usually round to the hundredth’s place (two numbers after decimal point e.g = 1.23, = 1.24)
Below 5 round down
5 and above round up
Greater than or less than > or <
Greater than > (e.g. 6 > 5)
Less than < (e.g. 1 < 4)
Alligator eats the biggest number

4. Percentile Rank
Percentage of scores in the distribution that are at or below a specific value; Pxx
Describes the position of a score
In relation to other scores
What type of measurement is this?
Example: Baby at 90th percentile for height & weight is tall & heavy: 90% of other babies are shorter/lighter and only 10% are longer/heavier p

5. 95th percentile Baby

6. Another View (7 months)

7. My Nieces 70th Height and 57th Weight (4 yrs)
55th Height and 32nd Weight (18mths)

8. 2 years 3 years 5 years

9. 9 years, 11 years, and 7 years

10. How to find Pxx
A score’s percentile: “If you were 45th from the bottom of 60 people, you’d be at the _____ percentile.”
Score position/total number of scores X 100 = percentile
45/60 X 100 = 75th percentile or P75
If you scored 50th from the bottom of a class of 160 people, at what percentile would your score be?

11. Practice 23th from the bottom of 375 scores?
What’s the percentile rank for a score that’s:
23th from the bottom of 375 scores?
7th from the end out of 34 scores?
95th from the bottom of 180 scores?
What’s the position (from the bottom) for a score with a percentile rank of:
P25 among 775 scores?
P90 among 425 scores?

12. Percentile Rank as a z-Score
But remember:
Pxx can reflect an ordinal measure of relative standing (position relative to the group of scores)
As a rank it does not take into consideration means or standard deviations (no math!)
Standardized scores are often used to present percentile ranks, also referred to as z-scores
They describe how far a given score is from the other scores, in terms of the z distribution

13. Purpose of z-Scores
Identify and describe location of every score in the distribution
Standardize an entire distribution
Takes different distributions and makes them equivalent and comparable

14. Two distributions of exam scores

15. Locations and Distributions
Exact location is described by z-score
Sign tells whether score is located above or below the mean
Number tells distance between score and mean in standard deviation units

16. Relationship of z-scores and locations

17. Traits of the normal distribution
Sections on the left side of the distribution have the same area as corresponding sections on the right
Because z-scores define the sections, the proportions of area apply to any normal distribution
Regardless of the mean
Regardless of the standard deviation

18. Normal Distribution with z-scores

19. z-Scores for Comparisons
All z-scores are comparable to each other
Scores from different distributions can be converted to z-scores
The z-scores (standardized scores) allow the comparison of scores from two different distributions along

20. Properties of z scores
Mean ALWAYS = 0 Standard deviation ALWAYS = 1 Positive z score is ABOVE the mean Negative z score is BELOW the mean

21. Standard Scores
Give a score’s distance above or below the mean in terms of standard deviations
Positive z-scores are always above the mean
Negative z-scores are always below the mean
Negative & positive z-scores only indicate directionality on the x-axis, not necessarily a change in the value of the score

22. Standard Scores
An index of a score’s relative standing Standard (converted) scores are referred to as z-scores
x = raw score
µ = population mean
σ = population SD
Backward formula: x = z (σ) + µ
Use this one to solve for x
p.105

23. Equation for z-score Numerator is a deviation score
Denominator expresses deviation in standard deviation units

24. Determining raw score from z-score
Numerator is a deviation score
Denominator expresses deviation in standard deviation units

25. Order of Operation Regular formula Backward formula Raw score (x)
Std. score (z)
% or Proportion
Raw (X) z % or Proportion
May not go directly from X to % or % to X – always go through z

26. Standard Normal Distribution
50%
of total area under the curve
+1 σ is where the line turns from concave to convex
Symmetrical
Asymptotic
μ = 0
σ = 1.0
p.110, Fig. 4.1

27. Distributions
Normal distribution: all normal distributions are symmetrical and bell shaped. Thus, the mean, median & mode are all equal!
“Ideal world” bell curve
μ = 0
σ = 1.0
Completely symmetrical
Asymptotic

28. Standard Normal Distribution Fig 4.1

29. IQ Standard Normal Distribution
68%
95%
95%
99%
99%
Genius
Gifted
Above average
Higher average
Lower average
Below average
Borderline low
Low
>144
85-99
70-84
55-69
<55
0.13%
2.14%
13.59%
34.13%
Adapted from

30. Shape of a Distribution
Researchers describe a distribution’s shape in words rather than drawing it
Symmetrical distribution: each side is a mirror image of the other
Skewed distribution: scores pile up on one side and taper off in a tail on the other
Tail on the right (high scores) = positive skew
Tail on the left (low scores) = negative skew

31. Skewed Distributions
Mean, influenced by extreme scores, is found far toward the long tail (positive or negative)
Median, in order to divide scores in half, is found toward the long tail, but not as far as the mean
Mode is found near the short tail.
If Mean – Median > 0, the distribution is positively skewed.
If Mean – Median < 0, the distribution is negatively skewed

32. Distributions
Positively skewed distribution: tendency for scores to cluster below the mean
p.90

33. Distributions
Negatively skewed distribution: tendency for scores to cluster above the mean
p.90

34. z z Appendix B Z or Z μ or μ p.617
Used to find proportions under the normal curve
We only use Columns 1, 3 and 5
Column 3 gives proportion from the z-score towards either tail
Column 5 gives proportion between z-score and the mean z
Z or z
μ or μ Z

35. Standard Scores Standard scores = z scores (in pop)
µ = mu = population mean
σ = sigma = population standard deviation
x = z (σ) + µ (Backward formula)

36. Finding a proportion from a raw score
1: Sketch the normal distribution 2: Shade the general region corresponding to the required proportion 3: Using the “forward” formula compute the corresponding z-score from the raw score (x). 4: Locate the proportion in the correct column of the table

37. What proportion of people have an IQ of 85 or less?
What about x < 70?
Step 1: draw curve
Step 2: use formula
z = x – μ σ
Step 3: Appendix B
Which column do you use?
For IQ scores µ = 100 & σ =15

38. Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B
What proportion of people have an IQ of or above?
Of x > 120?
Step 1: draw curve
Step 2: use formula
z = x – μ σ
Step 3: Appendix B
Which column do you use?
For IQ scores µ = 100 & σ =15

39. Finding a z-score from a proportion
1: Sketch the normal distribution 2: Shade the general region corresponding to the required proportion 3: Locate the proportion in the correct column of the table 4: Identify corresponding z-score in Col.1 5: Calculate raw score (x) with “backwards” formula

40. What is the lowest IQ score you can earn and still be in the top 5% of the population?
Step 1: draw curve
Step 2: Appendix B
Which column do you use?
Step 3: use formula
X = (z)(σ) + µ
For IQ scores µ = 100 & σ =15

41. What proportion of test takers score between 300 and 650 on the SAT?
Step 1: draw curve
Step 2: use formula
z = x – μ σ
Step 3: Appendix B
Which column do you use?
For SAT scores µ = 500 & σ =100

42. Applications of z-Scores
Any observation from a normal distribution can be converted to a z-score
If we have an actual, theoretical, or hypothesized normal distribution of many means we can determine the position of one particular mean relative to the other means in this distribution
This logic is the basis of many hypothesis tests

43. Z-score Practice Lecture Handout