1. Z - Scores and Probability
Lecture 5:
Z - Scores and Probability

2. What we Know We know all about descriptive statistics:
Frequency distributions
Histograms
Measures of Central Tendency
Measures of Variability
Important because they help us get an overall picture of our data
But they don’t tell us if our sample says anything important about the population (Our GOAL)

3. Foundation of Inferential Statistics
Reminder - Inferential Stats allows us to look at samples and make generalizations from them about the population.
Before we can proceed to inferential stats…we still need to know a bit more about samples
How they are related to each other
How they are related to the population they were taken from
System for representing samples:
Are they representative of the population?
Are they extreme?

4. Cars, Shoe Sizes and Much More
Standardizing scores to help make data more meaningful.
Example IQ testing:
Lots of different types of IQ tests that are all standardized to form a distribution with a mean = 100 and a standard deviation of 15
We know if an IQ is 98 points that they are slightly below the mean
We know if an IQ is 140 points the IQ is substantially above the mean
This allows us to look at a standardized score and know where it is in the distribution
Uses the mean and the standard distribution

5. Standardizing: z-scores
Each x score is transformed using the mean and the standard deviation
Each score will tell the exact location of the original X value within the distribution.
For example. You receive score of 45 on an exam, but don’t really know how your score measures up until you have the mean (40) and the sd (2).
The z-scores will form a distribution that can be directly compared to other distributions that have also been transformed
For example. You received the 45 in a stats class, but are also taking a course on gender and receive a 75 on that exam. Without knowing where the 75 lies in the distribution you don’t know in which class you are performing better.
Back to IQ

6. Z - scores
Will describe the exact location of a score w/in a distribution by transforming each x valued into a signed number
The sign tells us whether the score is located above (+) OR below (-) the mean.
The number tells us the distance between the score and the mean using number of standard deviations as a unit
For example: IQ
Mean = 100, SD = 15
X score of 115 is transformed into a z-score of +1.0
X score of 70 is transformed into a z-score of -2.0

7. Calculating Z-Scores from a Raw Score
 = 10
 = 50
If we go up one standard deviation
z score = +1.0 and raw score = 60
z - scores
If we go down one standard deviation
z score = and raw score = 40
What happens if you go up 2 SD or +2.0 z-scores?
How about if you go down 3 SD or +3.0 z-scores

8. Z - scores
The locations identified by z-scores are the same for ALL distributions regardless of the original mean and standard deviation.
For more complicated values:
z = (X - ) / 
What is the z-score given an x score of 98 in a distribution where  = 50 and  = 35?
z = ( ) / 35
z = 48 /35 = 1.37

9. Example
Calculate z-scores for the following distribution where  = 78.9 and  = 22.6
X-scores: 58 86
102 49

10. Calculating Z-scores from a raw score: Why should we bother?
When 2 scores come from 2 distributions they are hard to compare; z-scores let us do that.
Bob earned a 60 on his psychology 101 exam and a 56 on his biology exam. For which course should he expect a better grade?
 = 50  = 10 Psyc z = ( ) / 10 = +1.0
 = 48  = 4 Bio z = ( ) / 4 = + 2.0
Bob would expect a better grade in Bio.
It is always meaningful to compare 2 scores that have been standardized because a z-score of +2.0 always indicates a higher position than a z-score of +1.0

11. Calculating Z-scores from a raw score: Why should we bother?
When 2 scores come from 2 distributions they are hard to compare; z-scores let us do that.
What if we want to design a new measure for IQ…if we scale it with Z-scores we can see how people perform on our new measure compared to the old measure.
Traditional IQ:  =  = 15
New measure IQ:  = 60  = 6
Sally’s score: traditional =120; new measure = 72
traditional z = ( ) / 15 = 2
new measure z = ( ) / 6 = 2
New measure works well!

12. Calculating Raw Scores from a Z-score
z = (X - ) / 
Just algebra…
If Bobby has an a z-score of +3.83, then what is his IQ score, given that:
IQ :  =  = 15
3.83 = X / 15
3.83 * 15 = X - 100
= X
= X

13. Calculating Raw Scores from a Z-score
z = (X - ) / 
Just algebra…
Given a z-score of and that  = 42  = 8, what is X?
-1.26 = X - 42 / 8
-1.26 * 8 = X - 42
= X
31.92 = X

14. Example
Calculate raw scores for the following distribution where  = 78.9 and  = 22.6
z-scores:
+2.6
+1.4
-2.2
-1.8

15. Characteristics of Z-Transformations
Mean: z distribution has a mean = 0
 = 100 and  = 15
z = ( ) / 15 = 0
Standard deviation = 1
Shape: z transforming a distribution does not normalize it
Shape does not change x-axis is simply relabeled

16. x f  x f  
z - scores
Z-transformations do not change the shape of the distribution. Important to use normal distributions.
Convenient = Mean = 0 and SD = 1
Mean = = 0
MS = ( ) / 6 = 1
SD = 1(1/2) = 1

17. Remember our Goal How scores relate to each other
How scores are related to their population
Are the scores representative of their population? Are they extreme?

18. Probability
Relationship between a score (or sample) and its population based on probability
Probability - likelihood of a particular outcome from a varied set of outcomes
Example:
There are 40 strawberry jellybeans in a jar and 80 anise flavored jellybeans. What’s the likelihood of randomly grabbing a anise flavored jellybean?
p(anise) = number of outcomes classified as ()
number of possible outcomes
p(anise) = 80/120
Here probability is defined as a proportion (it can just as easily be represented by a percentage)

19. What’s the likelihood of drawing a queen from a deck of cards?
p() = outcome ()/total possible outcomes
p(queen) = 4/52 = .08 = 8%
What’s the likelihood of grabbing a pb cookie out of the jar?
p = f/N
p(pb) = 20/98 = .2 = 20%
What’s the likelihood of obtain a value of 9 from the table?
p(X = 9) = 12/49 = .24 = 24%
Cookie
Peanut butter
Choc. Chip
Oatmeal f 20 45 33 x 10 9 8 7 f 12 17

20. Rules of Probability Must use a random sample:
Each individual has an equal chance of being selected
e.g. There are aliens; there are no aliens. There is not an equal chance of either so p cannot equal 1/2 here
If more than one individual is selected there must a constant probability
p(queens) 4/52
After the first card is pulled it must be replaced before the next card is pulled

21. Inferential Statistics
Begin with a sample and answer questions about a population
Two cookie jars:
Jar 1
Jar 2
Cookie
Peanut butter
Choc. Chip
Oatmeal f 20 45 33
Cookie
Peanut butter
Choc. Chip
Oatmeal f 82 8
You close your eyes and have to pick four cookies out of one of the jars (you don’t know which), you pick 4 peanut butter cookies. Based on what you now know about probability which jar do you think you chose from?

22. Probability and the Normal Distribution
Synonyms: bell curve, Gaussian distribution.
Normal distribution has a wider or steeper curve depending on the amount of variability.
We will devote considerable attention to the normal distribution because:
Often naturally occurring and, even, guaranteed in some circumstances
Many statistical procedures assume that the population values follow a normal distribution.

23. The Normal Curve Characteristics Symmetrical Unimodal
Asymptotic to abscissa
Limits are 
M = Mdn = Mo
Most score are piled up around the mean (mean = mode) and extreme scores are relatively rare (low frequency)
50% of the scores are below the mean/median and 50% of the scores are above the mean/median

24. The Probability of Obtaining a Score
Mean = 50
 = 10
So, most scores in a normal distribution should be +/- 2 standard deviations OR +/- 2 z-scores from the mean.
Gives us an indication of how probable obtaining a particular score is:
Given this distribution is a score of 90 reasonable?
How about a score of 70?

25. Scores, Standard Deviations, & Probabilities
The Normal distribution is defined mostly by its mean and standard deviation
Given any of these values (score, probability of occurrence or distance from the mean) you can figure out the other 2.
Scores
Probability
Distance
(Also percent, proportion, or area)
from mean
(z scores)

26. The Standard Normal Curve or Z distribution
A normal curve whose:
Mean = 0
Standard deviation = 1
Total area under the curve = 1.0 or 100%
Can be described by the proportions of area contained in each section of the distribution.
(-) below the mean
(+) above the mean
Note: 2 SD away from the mean encompasses 95% of the distributions

27. Uses of the Standard Normal Curve
Scores by themselves don’t give us very much information
Exact location of the x value in a distribution
We can’t compare scores from various tests without the standard normal
Locations identified are the same for all distributions no matter their original mean or standard deviation
Find probabilities/proportions
What proportion of the distribution is above, below, or in between 2 scores in a distribution?
Find scores
What score is at a given probability?

28. What’s the probability of obtaining a score?
* Remember you could make these probabilities proportions simply by dividing by 100.
So what proportion of the curve is greater than 1? 0.16
Given the above graph what is the probability of obtaining a z-score that is greater than + 1?
~ = 15.74%
Given the above graph what is the probability of obtaining a z-score that is greater than -2.0?
~ = 97.59%

29. What’s the probability of obtaining a score?
Given the above graph what is the probability of obtaining a z-score that is between 1 and 2?
~13.59%
Given the above graph what is the probability of obtaining a z-score that is between - 3 and 0?
~ = 49.87

30. But, what if the z-score is a fraction of decimal value
But, what if the z-score is a fraction of decimal value? How do we determine the probability?
Unit normal table - This table provides the proportions of the normal distribution for a full range of possible z-scores (Appendix B - Table B.1- pg. 690).
Column A = z-score
Column B = Proportion in the Body
Column C = Proportion in Tail
Column D = Proportion between Mean and z
Rules for the Unit Normal Table:
Normal distribution is symmetrical, so the proportion on the right side = corresponding portion on the left side. (E.g. proportion in the tail beyond z = +1.0 is the same as proportion in the tail beyond z = -1.0
Although the z-score values will change sign (+/-) the proportions always stay the same and will be positive
If a section you are working with is > 50% of the distribution then it is called the body. If < 50% then it is called the tail
The 2 proportions (tail + body) will always add to 1.00

31. Example
Assume X = 650,  = 600,  = 100
What % or proportion did worse than X or p(X<650)
(1) Sketch
(2) find z-score
(3) look up proportion
z = ( ) / 100 = 0.5
Area that did worse is greater than 50%, so we look in which column?
PR = 69.15% proportion = .6915
So what’s the likelihood that we will draw a sample score of 650 or lower?

32. Example
Assume X = 400,  = 600,  = 100
What % did worse than X; p(x<400)
(1) Sketch
(2) find z-score
(3) look up proportion
z = ( ) / 100 = -2.0
Area that did worse is less than 50%, so we look in which column?
PR = 2.28%
So what is the likelihood of drawing a score of 400 from this population?

33. Example Assume X = 400 and X = 650,  = 600,  = 100
What % of cases fall between the scores?
(1) Sketch
(2) find z-scores
z = ( ) / 100 = -2.0
z = ( ) / 100 = 0.5
Which column should we use?
D and then add the result together…
Likelihood of obtain a score in this range?
Is there another way we could have found the PR?
PR for – 2.0 = 47.72
PR for .5 = 19.15
= PR = 66.87

34. Example Assume X = 700 and X = 800,  = 600,  = 100
What % of cases fall between the scores?
(1) Sketch
(2) find z-scores
(3) look up proportion
z = ( ) / 100 = +1.0
Z = ( ) / 100 = +2
PR for 1.0 = 34.13
PR for 2.0 = 47.72
47.32 – = 13.59
Column D and then subtract
Likelihood?

35. Example
Suppose a golf club takes only 3% of the population where  = 500k,  = 25k. You make 520k. Can you get in?
(1)Look in tail for
the z-score associated
With 0.03.
Z-score = 1.89
(2) Find the raw score
1.89 = (X - 500) / 25
1.89 * 25 = X - 500
= X
X = K

36. Example
Suppose you want to know the range of score that fall within the middle 30% of the distribution where  = 450,  = 40
Look in D for .15
z = -.38 and + .38
(2) Find the raw score
.38 = (X - 450) / 40
.38 * 40 = X - 450
= X
X = 465.2
-.38 = (X – 450) / 40
-.38 * 40 = X - 450
= X
X = 434.8

37. Class Problems: Groups of 3-4
Suppose we are interested in studying error on a simple task. Three people are asked to count the number of red cars driven on a busy road in one afternoon. The mean number of red cars reported was 975 with a standard deviation of 15. Assuming this is a normal distribution,
What % of counts fall between 960 & 990?
What % of counts fall above 995?
Is it reasonable for someone to have conscientiously counted 950?

38. Class Problems: Groups of 3-4
What score do you need to obtain to receive a PR of 95 on an exam where the  = 70,  = 20

39. Homework: Chapters 5 & 6 Chapter 5: Chapter 6:
3, 4, 5, 6, 11, 12, 19, 20, 23
Chapter 6:
2, 5, 7, 8, 9, 13, 15, 17, 18, 21, 22, 23
PLEASE NOTE: we did not cover every section in these chapters. You are only responsible for the sections we covered in class.