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Chapter 5Describing Data with z-scores and the Normal Curve Model
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Measures of VariabilityToward 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
X = 36
66
36
N
XX
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Measures of VariabilityToward a Useful Measure of Variability:
The Standard Deviation
66
36
N
XX
X
8
7
9
3
5
4
X = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
(X-M)=0
The standard deviation looks to find the average distance that scores are away from the mean.
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Measures of VariabilityToward a Useful Measure of Variability:
The Standard Deviation
X
8
7
9
3
5
4
X = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
(X-M)=0
The standard deviation looks to find the average distance that scores are away from the mean.
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
(X-M)2=28
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Measures of VariabilityToward a Useful Measure of Variability:
The Standard DeviationThe Sums of Squares
66
36
N
XX
X
8
7
9
3
5
4
X = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
(X-M)2=28
282 XXSS
The full name for this is the sum of the squared deviations from the mean.
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Measures of VariabilityToward a Useful Measure of Variability:
The Standard DeviationThe Variance
X
8
7
9
3
5
4
X = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
(X-M)2=28
67.46
28)( 22
N
XX
N
SSS X
The sample variance is the average of the squared deviations of the scores around the sample mean.
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Measures of VariabilityToward a Useful Measure of Variability:
The Standard DeviationFinally!!!
X
8
7
9
3
5
4
X = 36
(X – M)
8 – 6 = +2
7 – 6 = +1
9 – 6 = +3
3 – 6 = -3
5 – 6 = -1
4 – 6 = -2
(X-M)=0
(X – M)2
22 = 4
12 = 1
32 = 9
(-3)2=9
(-1)2 = 1
(-2)2 = 4
(X-M)2=28
The standard deviation looks to find the average distance that scores are away from the mean.
16.267.42
2
sN
SS
N
XXSX
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Measures of VariabilityThe 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.
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The Normal Curve ModelWhy 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!
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The Normal Curve ModelUnderstanding 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….. X
-X=z
XS
Xz
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The Normal Curve ModelUnderstanding 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.
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The Normal Curve ModelUnderstanding z-scores
Converting z-scores to Raw Scores
z-scores can be converted back to raw scores.
or the sample distribution…..
Xz+=X
XSzXX
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110
5040
110
5060
z
z
-X=z
X
-3 -2 -1 1 2 30
The Normal Curve ModelUnderstanding z-scores
Creating a z-score Distribution
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The Normal Curve ModelUnderstanding z-scoresComputing 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 wecan compute theirpercentile.
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The Normal Curve ModelUnderstanding z-scores
Converting from Raw Scores to Percentiles and Back Again
If we know X then:
If we know percentile then:
PercentilescorezX chartinupLookS
XXz
X
XscorezPercentile zXXscorezingCorrespondFind
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The Normal Curve ModelIt’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.
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The Normal Curve ModelIt’s Like Comparing Apples and Oranges
3 oz
50% 50%
4.5 oz 6 oz 7.5 oz 9 oz
My apple
25.1
69
X
Xz
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The Normal Curve ModelIt’s Like Comparing Apples and Oranges
6 oz 8 oz 10 oz 12 oz 14 oz
My orange
5.0.2
910
X
Xz
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The Normal Curve ModelIt’s Like Comparing Apples and Oranges
AppleOrange
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Measures of VariabilityThe 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.
![Page 21: Chapter 5 Describing Data with z -scores and the Normal Curve Model](https://reader037.vdocument.in/reader037/viewer/2022102808/568130e7550346895d9702ad/html5/thumbnails/21.jpg)
The Normal Curve ModelWhy 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!
![Page 22: Chapter 5 Describing Data with z -scores and the Normal Curve Model](https://reader037.vdocument.in/reader037/viewer/2022102808/568130e7550346895d9702ad/html5/thumbnails/22.jpg)
The Normal Curve ModelUnderstanding z-scores
Converting from Raw Scores to Percentiles and Back Again
If we know X then:
If we know percentile then:
PercentilescorezX chartinupLookS
XXz
X
XscorezPercentile zXXscorezingCorrespondFind
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The Normal Curve ModelComputing 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 .2389 fall
between the mean and z, another .5000 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%.
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The Normal Curve ModelComputing Probability of a Single Score
If IQ scores are normally distributed with = 100 and = 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 +1.53. We would look in the chart and find that .4370 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%.
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The Normal Curve ModelComputing Probability of a Sample Mean
If IQ scores are normally distributed with = 100 and = 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.
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The Normal Curve ModelThe 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.
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The Normal Curve ModelThe Sampling Distribution of Means
f
70 85 100 115 130
98
118
94 104
5.1034
4144
1181049894
n
XX
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The Normal Curve ModelThe Sampling Distribution of Means
f
70 85 100 115 130
99
103
87108
25.994
3974
1081039987
n
XX
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The Normal Curve ModelThe Sampling Distribution of Means
f
70 85 100 115 130
97
106
52
10991
4
3644
1091069752
n
XX
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The Normal Curve ModelThe Sampling Distribution of Means
f
70 85 100 115 130
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The Normal Curve ModelThe Sampling Distribution of Means
The sampling distribution of means will have a mean equal to the of the population of raw scores.
XX
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The Normal Curve ModelThe Sampling Distribution of Means
However…
XX
XX
factin
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The Normal Curve ModelThe 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 of the original population and the size of the samples in the following manner:
nX
X
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The Normal Curve ModelThe Sampling Distribution of Means
Therefore…
5.74
15
nX
X
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The Normal Curve ModelThe 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.
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The Normal Curve ModelThe 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.
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The Normal Curve ModelThe Sampling Distribution of Means
The Central Limit Theorem
The central limit theorem states “For any population with a mean of and standard
deviation X, the distribution of sample means for sample size n will have a mean of and a standard deviation of
and will approach a normal distribution as n approaches infinity.”
nX
X
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The Normal Curve ModelComputing Probability of a Sample Mean
If IQ scores are normally distributed with = 100 and = 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.
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The Normal Curve ModelComputing 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.
07.35.7
100123
X
Xz
5.74
15
nX
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The Normal Curve ModelComputing 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 = 10 oz with = 2 oz
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The Normal Curve ModelComputing 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.
73.1577.
109
X
Xz
577.12
2
nX