1. Chapter 5 Describing Data with z-scores and the Normal Curve Model
Do homework 4-1, 4-2, & 4-3

2. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation
This is the most useful and most commonly used of the measures of variability.
The standard deviation looks to find the average distance that scores are away from the mean.
Conversely, the standard deviation is the average amount of error we would expect if I used the sample mean to predict every score.
# Pancakes 8 7 9 3 5 4
SX = 36
-Example: Mike Brady took the kids to IHOP for an all you can eat pancake buffet, the data is the number of pancakes eaten by each kid in the Brady Bunch (Remember, there are six kids).
-It’s going to take a while to get to the standard deviation.

3. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation
The standard deviation looks to find the average distance that scores are away from the mean. X 8 7 9 3 5 4
SX = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
S(X-M)=0
-First we’ll compute the distance each score is from the mean, the deviation scores.
-Then we’ll average these deviations.
-Oh no! The deviations add to 0. How can I average 0?

4. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation
The standard deviation looks to find the average distance that scores are away from the mean. X 8 7 9 3 5 4
SX = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
S(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
S(X-M)2=28
-One way to solve the deviations summing to 0 problem is to square each one.
-When we square the deviations, they now sum to 28.

5. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation The Sums of Squares X 8 7 9 3 5 4
SX = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
S(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
S(X-M)2=28
The full name for this is the sum of the squared deviations from the mean.
-Along the way to computing the standard deviation, we will compute a few other useful statistics.
-In and of itself, the SS is sort of useless, but it will be an important thing to have for later.

6. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation The VarianceX 8 7 9 3 5 4
SX = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
S(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
S(X-M)2=28
The sample variance is the average of the squared deviations of the scores around the sample mean.
-In and of itself, the variance is sort of useless, but it will be an important thing to have for later.

7. Measures of Variability Toward a Useful Measure of Variability: The Standard Deviation Finally!!!X 8 7 9 3 5 4
SX = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
S(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
S(X-M)2=28
The standard deviation looks to find the average distance that scores are away from the mean.

8. Measures of Variability The Normal Curve and the Standard Deviation
If we know the mean and the standard deviation of normally distributed scores, we know a LOT of things.
-In this picture, the mean is 80 and the standard deviation is 5

9. The Normal Curve Model Why is it important to know about z-scores?
Because normally, raw scores are totally meaningless.
The best we can do is compare a raw score to other’s raw scores.
z-scores allow us to calculate a person’s relative standing in a distribution, relative to all other scores.
They allow us to compare two different scores.
We can compare apples and oranges!

10. The Normal Curve Model Understanding z-scores Converting Raw Scores to z-scores
z-scores take the mean and standard deviation of a distribution, and use this information to produce a numerical value to describe the location of individual raw scores in the population distribution.
or the sample distribution…..

11. The Normal Curve Model Understanding z-scores
z-scores produced from different sources can be compared directly. If you scored a 32 on an English test and a 73 on a Math can you compare the scores? Not directly, but if both raw scores were converted to z-scores, then we could.
An entire set of raw scores can be converted to z-scores. The resulting distribution will have the same shape as the original distribution, will have a mean of 0, and a standard deviation of 1.

12. The Normal Curve Model Understanding z-scores Converting z-scores to Raw Scores
z-scores can be converted back to raw scores.
or the sample distribution…..

13. The Normal Curve Model Understanding z-scores Creating a z-score Distribution
-Any normal distribution of raw scores can be converted to a distribution of z-scores.
-The mean of a z-score distribution is always 0.
-The SD of a z-score distribution is always 1.
-The curve is perfectly symmetrical.

14. The Normal Curve Model Understanding z-scores Computing Probability
Just like all probability distributions, we can compute percentiles in this distribution by looking at the portion of the curve falling to the left.
A z-distribution is a special case because it has been studied VERY much. We know ALL about it. We know the percentile of ALL
points in the distribution.
If we know a
persons z-score we
can compute their
percentile.
-Look in Table 1 in Appendix B.

15. The Normal Curve Model Understanding z-scores Converting from Raw Scores to Percentiles and Back Again
If we know X then:
If we know percentile then:

16. The Normal Curve Model It’s Like Comparing Apples and Oranges
I went to the cafeteria the other day with just $1. A piece of fruit costs exactly $1. I can buy just one piece of fruit. I am a real bargain shopper though. I definitely want to get the best value for my $1.
I found two pieces of fruit left, an orange that weighs 9 ounces and an apple that weighs 9 ounces
I want to know which one is the more outstanding fruit, which one is better value for my $1.

17. The Normal Curve Model It’s Like Comparing Apples and Oranges
50%
50%
-The average apple weighs 6 oz with a sd of 1.5 oz.
-It appears as though this is a pretty darn big apple!
3 oz
4.5 oz
6 oz
7.5 oz
9 oz
My apple

18. The Normal Curve Model It’s Like Comparing Apples and Oranges
-As far as oranges go, mine isn’t even average.
-Oranges weigh an average of 10 oz with a sd of 2 oz
6 oz
8 oz
10 oz
12 oz
14 oz
My orange

19. The Normal Curve Model It’s Like Comparing Apples and Oranges
-If we look up a z-score of +2.0, we find that the proportion of falls to the left, the apple’s weight falls at the 97.72nd percentile of all apples. The probability of finding an apple that large is only 2.28%
-If we look up a z-score of -.5 we find that a proportion of fall to the left. The oranges weight falls at the 30.85th percentile. The probability of finding an orange this large is 69.15%.
-The apple is definitely the better purchase.

20. Measures of Variability The Normal Curve and the Standard Deviation
If we know the mean and the standard deviation of normally distributed scores, we know a LOT of things.
-In this picture, the mean is 80 and the standard deviation is 5

21. The Normal Curve Model Why is it important to know about z-scores?
Because normally, raw scores are totally meaningless.
The best we can do is compare a raw score to other raw scores.
z-scores allow us to calculate a person’s relative standing in a distribution, relative to all other scores.
They allow us to compare two different scores.
We can compare apples and oranges!

22. The Normal Curve Model Understanding z-scores Converting from Raw Scores to Percentiles and Back Again
If we know X then:
If we know percentile then:

23. The Normal Curve Model Computing Probability of a Single Score
If exam scores are normally distributed with M = 56.8 and SX = 8.14, what is the probability of selecting one score from the population that is less than 62?
We would first convert the raw score to a z-score.
The question becomes, what is the probability of achieving a z-score lower than +.64.
We would look in the chart and find that fall between the mean and z, another falls below the mean, therefore such that we can say that the raw score of 62 is at the 73.89th percentile.
However, the question was probability of a score less than 62 which is 73.89%.

24. The Normal Curve Model Computing Probability of a Single Score
If IQ scores are normally distributed with m = 100 and s = 15, what is the probability of selecting one score from the population that is greater than 123?
We would first convert the raw score to a z-score.
The question becomes, what is the probability of achieving a z-score higher than
We would look in the chart and find that fall between the mean and z, such that we can say that the raw score of 123 is at the 93.7th percentile.
However, the question was probability, so we look at the tail and find .0630, so that the probability of a score of 123 or higher occurring is only 6.30%.

25. The Normal Curve Model Computing Probability of a Sample Mean
If IQ scores are normally distributed with m = 100 and s = 15, what is the probability of selecting a sample of four scores from the population that have a mean greater than 123?
To answer this, and other similar questions, we need to first understand the sampling distribution of means.

26. The Normal Curve Model The Sampling Distribution of Means
If we have a population of raw scores, we could draw a sample of size 4 from it.
In fact, we could draw many samples of size 4 from it (an infinite number if you want to do this for the rest of your life!).
If we draw many samples and compute the means from each sample…
then create a distribution of these means…
we have created a sampling distribution of means.

27. The Normal Curve Model The Sampling Distribution of Means98 94
104
118

28. The Normal Curve Model The Sampling Distribution of Means87 99
108
103

29. The Normal Curve Model The Sampling Distribution of Means97
109
106 52
-Even if I draw one REALLY extreme score, the other scores in the sample will probably bring the sample mean pretty close to the population average.

30. The Normal Curve Model The Sampling Distribution of Means
-If I did this same procedure over and over and over.
-The sample means would tend to cluster around the population mean.

31. The Normal Curve Model The Sampling Distribution of Means
The sampling distribution of means will have a mean equal to the m of the population of raw scores.

32. The Normal Curve Model The Sampling Distribution of Means
However…

33. The Normal Curve Model The Sampling Distribution of Means
Instead, the sampling distribution of the means will have a standard deviation (now called the standard error of the mean, or just standard error) that is directly related to the size of the s of the original population and the size of the samples in the following manner:

34. The Normal Curve Model The Sampling Distribution of Means
Therefore…

35. The Normal Curve Model The Sampling Distribution of Means The Central Limit Theorem
A sampling distribution is ALWAYS an approximately normal distribution.
It does not matter at all what shape the distribution of raw scores looked like.
The larger the sample size, the more normal the distribution of sample means becomes.

36. The Normal Curve Model The Sampling Distribution of Means The Central Limit Theorem
The mean of the sampling distribution is ALWAYS equal to the mean of the underlying raw score population from which we create the sampling distribution.

37. The Normal Curve Model The Sampling Distribution of Means The Central Limit Theorem
The central limit theorem states
“For any population with a mean of m and standard deviation sX, the distribution of sample means for sample size n will have a mean of m and a standard deviation of
and will approach a normal distribution as n approaches infinity.”
-Go to the following website which does a great graphical demonstration of the central limit theorem.

38. The Normal Curve Model Computing Probability of a Sample Mean
If IQ scores are normally distributed with m = 100 and s = 15, what is the probability of selecting a sample of four scores from the population that have a mean greater than 123?
First we need to convert the sample mean into a z-score.
Then we’ll need to look up the probability of the corresponding z-score.

39. The Normal Curve Model Computing Probability of a Sample Mean
First we’ll figure out the standard error.
Then we’ll calculate a z-score using a slightly different formula:
Then we’ll look this z-score up on the charts just like before.
-This z-score is at the 99.89th percentile. Therefore, the probability of selecting a sample of 4 scores that have a mean of 123 or higher is 0.11%

40. The Normal Curve Model Computing Probability of a Sample Mean
What is probability of buying a bag of a dozen oranges whose average weight is 9 oz or less?
Remember, oranges weigh m = 10 oz with s = 2 oz

41. The Normal Curve Model Computing Probability of a Sample Mean
First we’ll figure out the standard error.
Then we’ll calculate a z-score:
Then we’ll look this z-score up on the charts.
-A bag of oranges weighing 9 oz is at the 4.18th percentile.
-Therefore, the probability of buying a bag of oranges this small or smaller is only 4.18%.