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Z-SCORES
Location of Scores and
Standardized Distributions
68.26%
94.46%
99.73%
What is a normal distribution?
Many variables of interest to psychology are believed to be normally
distributed. Most inferential statistics are based on the assumption
the population of observations is normally distributed.
• Symmetric, bell-shaped curve
• No obvious skewness
• Single peak
• Described by μ and σ
What is a standard normal distribution?
-3 -2 -1 0 1 2 3
• Maintains the same shape as a
normal distribution
• However,
• μ = 0
• σ = 1
• Why is this
distribution
helpful?
If we can transform our data to this scale, we can
express an individual’s score in relation to two very
important distribution characteristics: μ and σ
• On an exam with
μ = 80 and σ = 5
• What does
x = 85 mean?
• To see where a score of X = 85 considering all individuals,
we must take into account both the M and the SD
• A score of 85 has a deviation score of 5
• If the standard deviation is 5, then a score of 85 is one standard
deviation above the mean
65 70 75 80 85 90 95
A Demonstration
score
z-scores
• The z-score specifies the precise location of each X value
within a distribution
• It is one number that establishes the relationship
between:
• The score
• The mean
• The standard deviation
• The numerical value specifies the distance from the mean
by counting the number of standard deviations
• The sign (+ or -) indicates whether the score is above the
mean (+) or below the mean (-)
-3 -2 -1 0 1 2 3
z
A Demonstration
On an exam with μ = 80 and σ = 5
• X = 90
• How many standard
deviations above
the mean?
• z-score: 2
• X = 75
• How many standard deviations below the mean?
• z-score: -1
65 70 75 80 85 90 95
Why Do We Use z-scores?
• To determine where one score is located in a distribution
with respect to all other scores
Is a z-score of -0.25 a typical score?
What about a z-score of 3.75?
• To standardize entire distributions, making them
comparable
Is a score of 85 on one test comparable to
a score of 90 on another?
Is a z-score of 0 on one test comparable to
a z-score of 0 on another?
The z-score
Formula
Accounts for:
• The deviation
score (X – µ)
• The standard
deviation (σ)



X
z
z =
85-85
5
=
0
5
= 0
For a distribution with a μ = 85 and σ = 5
What is the z-score for X = 85?
z =
90-85
5
=
5
5
=1
z =
77-85
5
=
-8
5
-1.6
What is the z-score for X = 90?
What is the z-score for X = 77?
What if I know the z-score but not the X?
For a distribution with μ = 60 and σ = 5:
• What score (X) corresponds to a z-score of -2.00?
X = (-2.00 × 5) + 60
X = -10 + 60
X = 50
Conceptually:
• The mean = 60 and the standard deviation = 5
• A score of 2 standard deviations below the mean would be:
60 – 5 – 5 = 50



X
z Þ X = zs +m
What if I know μ, X and z but not σ?
• For this distribution, we know:
μ = 65, X = 59, and z = -2.00
• What is σ?
• A z-score of -2.00 means what?
• How many points between
65 and 59?
• σ = 3
z-scores and Locations
USING Z-SCORES TO
STANDARDIZE A
DISTRIBUTION
z-scores as a Standardized Distribution
If we standardize every score, we have standardized our
distribution
• The distribution will be composed of scores that have been
transformed to create predetermined values for μ and σ
Properties of the Standardized
Distribution
• Shape
• Remains the same as the distribution with the original scores
• Does not move scores; just re-labels them
• Mean
• Will always have a mean of zero (0)
• Standard deviation
• Will always have a standard
deviation of one (1)
The Advantage of Standardized
Distributions
Two (or more) different distributions can be made the same
and compared to each other
• Distribution A: μ = 100, σ = 10
• Distribution B: μ = 40, σ = 6
• When these distributions are
transformed to z-scores,
both will have a
μ = 0 and σ = 1
28 34 40 46 52
μ
A Demonstration: Jesse’s Test Scores
• Math test score: X = 80
• Physics test score: X = 70
Which test did Jesse do better on?
Dunno. We need more information, yo.
We need to know the mean and standard
deviations for each distribution, yo.
Math: μ = 85, σ = 6 Physics: μ = 65, σ = 4



X
z z =
80-85
6
=
-5
6
= -0.833 z =
70-65
4
=
5
4
=1.250
Even though the raw score of 80 is greater than 70, Jesse
did comparatively better on the physics exam.
OTHER STANDARDIZED
DISTRIBUTIONS BASED ON Z-
SCORES
Other Standardized Distributions
• Some people find z-scores burdensome because they
consist of many decimal values and negative numbers
• Therefore, it is often more convenient to standardize a
distribution into numerical values that are simpler than z-
scores
• To create a simpler standardized distribution, you first
select the mean and standard deviation that you would
like for the new distribution
• Then, z-scores are used to identify each individual’s
position in the original distribution and to compute the
individual’s position in the new distribution.
Other Standardized Distributions
Can I transform my original values into a new distribution
with a specified μ and σ other than 0 and 1 while still
maintaining its properties?
Yes. We do this by transforming raw scores into z-scores,
and then transforming the z-scores into new X values with
specified parameters.
Here’s an
Example
For one student, I have the following
scores:
Test 1: X = 77 Test 2: X = 74
Has this student shown improvement?
Or is this evidence of decline?
I give two tests:
Test 1
μ = 75, σ = 7
Test 2
μ = 73, σ = 3
I want to make all of
my tests have a
μ = 70 and σ = 5 so
I can make direct
comparisons about
individual
improvement
Step 1: Transform raw scores into z-scores
For test 1: For test 2:
Step 2: Transform z-scores into new X-values
For test 1:
For test 2:
z =
77- 75
7
=
2
7
= 0.286 z =
74- 73
3
=
1
3
= 0.333
x = 0.286´5)( +70 =1.43+70 = 71.43
x = 0.333´5)( +70 =1.665+70 = 71.665
Individual position relative to all others is maintained
COMPUTING Z-SCORES
FOR A SAMPLE
Can z-scores be used for samples?
Yes.
• The definition of the z-score is the same for either a
sample or a population
• The formulas are also the same except that the sample
mean (M) and the standard deviation (s) are used in place
of the population mean (μ) and the standard deviation (σ)
• Using z-scores to standardize a sample has the same
effect as standardizing a population
• The mean will be 0 and the standard deviation will be 1
s
MX
z


Using z-scores to Detect Outliers
Outliers are extreme values that may skew our distribution
• z-scores are useful for determining if you have any
outliers
• i.e., any individuals answering differently from the majority of your
responses
• Transform all values into z-scores
• Since we are expressing scores in terms of standard deviations
from the mean, large z-scores (e.g. z=4.32) means that this score
is far away from the “typical” response
Remember these?
LOOKING AHEAD TO
INFERENTIAL STATISTICS
Inferential Statistics
• If we have the population parameters, z-scores can help
us determine if an individual treatment effect shows a
different response
• For a population of rats,
the average weight is 400 grams
with σ = 20
• We inject one rat with
growth hormone
• If rat weighs 418g
• Not much difference
from average untreated
rats
• If rat weighs 450g
• BIG difference!

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Z-SCORES: Standardizing Distributions

  • 1. Z-SCORES Location of Scores and Standardized Distributions
  • 2. 68.26% 94.46% 99.73% What is a normal distribution? Many variables of interest to psychology are believed to be normally distributed. Most inferential statistics are based on the assumption the population of observations is normally distributed. • Symmetric, bell-shaped curve • No obvious skewness • Single peak • Described by μ and σ
  • 3. What is a standard normal distribution? -3 -2 -1 0 1 2 3 • Maintains the same shape as a normal distribution • However, • μ = 0 • σ = 1 • Why is this distribution helpful? If we can transform our data to this scale, we can express an individual’s score in relation to two very important distribution characteristics: μ and σ
  • 4. • On an exam with μ = 80 and σ = 5 • What does x = 85 mean? • To see where a score of X = 85 considering all individuals, we must take into account both the M and the SD • A score of 85 has a deviation score of 5 • If the standard deviation is 5, then a score of 85 is one standard deviation above the mean 65 70 75 80 85 90 95 A Demonstration score
  • 5. z-scores • The z-score specifies the precise location of each X value within a distribution • It is one number that establishes the relationship between: • The score • The mean • The standard deviation • The numerical value specifies the distance from the mean by counting the number of standard deviations • The sign (+ or -) indicates whether the score is above the mean (+) or below the mean (-)
  • 6. -3 -2 -1 0 1 2 3 z A Demonstration On an exam with μ = 80 and σ = 5 • X = 90 • How many standard deviations above the mean? • z-score: 2 • X = 75 • How many standard deviations below the mean? • z-score: -1 65 70 75 80 85 90 95
  • 7. Why Do We Use z-scores? • To determine where one score is located in a distribution with respect to all other scores Is a z-score of -0.25 a typical score? What about a z-score of 3.75? • To standardize entire distributions, making them comparable Is a score of 85 on one test comparable to a score of 90 on another? Is a z-score of 0 on one test comparable to a z-score of 0 on another?
  • 8. The z-score Formula Accounts for: • The deviation score (X – µ) • The standard deviation (σ)    X z z = 85-85 5 = 0 5 = 0 For a distribution with a μ = 85 and σ = 5 What is the z-score for X = 85? z = 90-85 5 = 5 5 =1 z = 77-85 5 = -8 5 -1.6 What is the z-score for X = 90? What is the z-score for X = 77?
  • 9. What if I know the z-score but not the X? For a distribution with μ = 60 and σ = 5: • What score (X) corresponds to a z-score of -2.00? X = (-2.00 × 5) + 60 X = -10 + 60 X = 50 Conceptually: • The mean = 60 and the standard deviation = 5 • A score of 2 standard deviations below the mean would be: 60 – 5 – 5 = 50    X z Þ X = zs +m
  • 10. What if I know μ, X and z but not σ? • For this distribution, we know: μ = 65, X = 59, and z = -2.00 • What is σ? • A z-score of -2.00 means what? • How many points between 65 and 59? • σ = 3
  • 13. z-scores as a Standardized Distribution If we standardize every score, we have standardized our distribution • The distribution will be composed of scores that have been transformed to create predetermined values for μ and σ
  • 14. Properties of the Standardized Distribution • Shape • Remains the same as the distribution with the original scores • Does not move scores; just re-labels them • Mean • Will always have a mean of zero (0) • Standard deviation • Will always have a standard deviation of one (1)
  • 15. The Advantage of Standardized Distributions Two (or more) different distributions can be made the same and compared to each other • Distribution A: μ = 100, σ = 10 • Distribution B: μ = 40, σ = 6 • When these distributions are transformed to z-scores, both will have a μ = 0 and σ = 1 28 34 40 46 52 μ
  • 16. A Demonstration: Jesse’s Test Scores • Math test score: X = 80 • Physics test score: X = 70 Which test did Jesse do better on? Dunno. We need more information, yo. We need to know the mean and standard deviations for each distribution, yo. Math: μ = 85, σ = 6 Physics: μ = 65, σ = 4    X z z = 80-85 6 = -5 6 = -0.833 z = 70-65 4 = 5 4 =1.250 Even though the raw score of 80 is greater than 70, Jesse did comparatively better on the physics exam.
  • 18. Other Standardized Distributions • Some people find z-scores burdensome because they consist of many decimal values and negative numbers • Therefore, it is often more convenient to standardize a distribution into numerical values that are simpler than z- scores • To create a simpler standardized distribution, you first select the mean and standard deviation that you would like for the new distribution • Then, z-scores are used to identify each individual’s position in the original distribution and to compute the individual’s position in the new distribution.
  • 19. Other Standardized Distributions Can I transform my original values into a new distribution with a specified μ and σ other than 0 and 1 while still maintaining its properties? Yes. We do this by transforming raw scores into z-scores, and then transforming the z-scores into new X values with specified parameters.
  • 20. Here’s an Example For one student, I have the following scores: Test 1: X = 77 Test 2: X = 74 Has this student shown improvement? Or is this evidence of decline? I give two tests: Test 1 μ = 75, σ = 7 Test 2 μ = 73, σ = 3 I want to make all of my tests have a μ = 70 and σ = 5 so I can make direct comparisons about individual improvement Step 1: Transform raw scores into z-scores For test 1: For test 2: Step 2: Transform z-scores into new X-values For test 1: For test 2: z = 77- 75 7 = 2 7 = 0.286 z = 74- 73 3 = 1 3 = 0.333 x = 0.286´5)( +70 =1.43+70 = 71.43 x = 0.333´5)( +70 =1.665+70 = 71.665 Individual position relative to all others is maintained
  • 22. Can z-scores be used for samples? Yes. • The definition of the z-score is the same for either a sample or a population • The formulas are also the same except that the sample mean (M) and the standard deviation (s) are used in place of the population mean (μ) and the standard deviation (σ) • Using z-scores to standardize a sample has the same effect as standardizing a population • The mean will be 0 and the standard deviation will be 1 s MX z  
  • 23. Using z-scores to Detect Outliers Outliers are extreme values that may skew our distribution • z-scores are useful for determining if you have any outliers • i.e., any individuals answering differently from the majority of your responses • Transform all values into z-scores • Since we are expressing scores in terms of standard deviations from the mean, large z-scores (e.g. z=4.32) means that this score is far away from the “typical” response Remember these?
  • 25. Inferential Statistics • If we have the population parameters, z-scores can help us determine if an individual treatment effect shows a different response • For a population of rats, the average weight is 400 grams with σ = 20 • We inject one rat with growth hormone • If rat weighs 418g • Not much difference from average untreated rats • If rat weighs 450g • BIG difference!