Download Lesson 4 - West Virginia University

LESSON 5 Z-SCORES
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LOCATION IN A DISTRIBUTION
z-scores are used to describe the exact location of a score within
a distribution.
 The sign tells whether the score is above (+) or below (-) the
mean.
 The number tells the distance between the score and the
mean in terms of the standard deviation.
 Example: A score of +1 is one standard deviation above the
mean.
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WHY?
Converting a raw data set or score into a standardized
format indicates
 whether the raw score is below the mean or above the
mean.
 tells exactly how far above or below the mean the score
is.
 allows you to compare scores across entirely different
measurements.
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Z-SCORE FORMULA
The formula for converting a raw score from a
distribution to a z-score is:
z
X 

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OTHER FORMULA APPLICATIONS
Using the same formula you can:
 Find the raw score if you have its z-score, the mean, and the
standard deviation (or variance).
 Find the mean if you have a raw score, its z-score, and the
standard deviation (or variance).
 Find the standard deviation (or variance) if you have a raw
score, its z-score, and the mean.
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STANDARDIZING A DISTRIBUTION
If we convert every raw score in a distribution to a zscore, then we have standardized the distribution.
A standardized distribution has a number of advantages:
 The shape of the z-score distribution remains the same
as the original.
 The mean is always 0.
 The standard deviation is always 1.
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STANDARDIZED FACTS
The raw mean becomes 0 (any number subtracted from
itself = 0)
The raw standard deviation becomes 1 (any number
divided by itself = 1)
Every raw score X can be converted to a standardized zscore by using the z formula
z
X 

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MORE FACTS
All raw scores greater than the mean
standardize to z-scores greater than 0.
All raw scores less than the mean
standardize to z-scores less than 0.
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APPLICATIONS
Standardized distributions allow you to know the precise
location of every score in the distribution.
Standardized distributions can be used to compare two
or more dissimilar raw distributions.
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PICTURE THIS!
This is a normal distribution
 represents a total population
 bell shaped

 symmetric around 
 The further away from  a score is (either greater than or
less than), the lower the frequency at which that score
occurs.
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PICTURE THIS!
Because this distribution represents the exhaustive set
of all possible raw scores, the total proportion of possible
scores (represented by the area under the curve) is 1.00
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PROPORTIONS UNDER THE CURVE
The proportion of the area represented under the curve
that is described by any single score or any set of scores
is always between 0 and 1.
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PROPORTIONS UNDER THE CURVE
The proportion of those scores greater than
the mean is 0.5000

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PROPORTIONS UNDER THE CURVE
The proportion of those scores less
than the mean is 0.5000

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FROM RAW TO STANDARDIZED
When we standardize this normal distribution,
we simply exchange the X-axis from raw score
terms to standardized (z) score terms.
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+/- 3 STANDARD DEVIATIONS
Most z scores fall between -3 and +3. That is, within
3 standard deviations above and below the mean. A
very few will fall between -4 and -3 and between +3
and +4.
-3
-2
-1
0
+1 +2
+3
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FINDING A RAW SCORE
Given a z-score from a standardized distribution and both the
population mean and standard deviation (or the variance) for
that distribution, you can use this formula to find the raw score
associated with the given z-score.
X    z
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FINDING A RAW SCORE
Suppose we have a z-score of -2.70. If =100 and  =
20, then
X    z
 100  (2.70)( 20)
 100  (54.00)
 46
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FINDING A POPULATION MEAN
Given a raw score from a standardized distribution and its
associated z-score along with the population standard deviation
(or the variance) for that distribution, you can use the following
formula to find the raw mean () for that population.
  X  z
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FINDING A POPULATION STANDARD DEVIATION
Given a raw score from a standardized distribution
and its associated z-score along with the population
mean for that distribution, you can use the following
formula to find the raw standard deviation (or
variance) for that population.
X

z
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FINDING THE AREA UNDER A CURVE
We can use the standardized normal distribution to
answer many important questions about a distribution.
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SOME COMMON PROBLEMS
Finding the proportion (or %) of scores greater than X
 When X is greater than the mean.
 When X is less than the mean.
Finding the proportion (or %) of scores less than X.
 When X is greater than the mean.
 When X is less than the mean.
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MORE COMMON PROBLEMS
Finding the proportion (or %) of scores between two Xs:
 When one of the Xs is the mean.
 When both Xs are greater than the mean.
 When both Xs are less than the mean.
 When one X is greater than the mean and the other X is
less than the mean.
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PROPORTION GREATER THAN X
When X is greater than the mean:
tail

X
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PROPORTION GREATER THAN X
When X is greater than the mean:
 Convert the X value to a z-score.
 z = (X – ) /  > 0
 Find z in column (A).
 The proportion needed is in the tail, so use Column (C).
 For %, multiply by 100.
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PROPORTION GREATER THAN X
Example when X is greater than the mean
 Let  = 100,  = 20, and X=150
 Find z = (X – ) /  which is (150-100)/20
 So z = 50/20 = 2.5
 Double check—is z greater than 0? Yes.
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PROPORTION GREATER THAN X
 Find z=2.50 in Column (A)
 Find the correct proportion in the tail in Column (C). So the
area is .0062
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PROPORTION GREATER THAN X
When X is less than the mean:
body
X
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PROPORTION GREATER THAN X
When X is less than the mean:
 Convert the X value to a z-score.
 z = (X – ) /  < 0
 Find z in column A (ignore – sign).
 The proportion needed is in the body, so use Column
(B).
 For %, multiply by 100.
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PROPORTION GREATER THAN X
Example when X is less than the mean
 Let  = 100,  = 20, and X=85
 Find z = (X – ) /  which is (85-100)/20
 So z = -15/20 = -0.75
 Double check—is z less than 0? Yes.
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PROPORTION GREATER THAN X
 Find z=-0.75 in Column (A)
 Find the correct proportion in the body in Column (B). So the
area is .7734
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PROPORTION LESS THAN X
When X is greater than the mean:
body
X
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PROPORTION LESS THAN X
When X is greater than the mean:
 Convert the X value to a z-score.
 z = (X – ) /  > 0
 Find z in column A.
 The proportion needed is in the body, so use Column (B).
 For %, multiply by 100.
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PROPORTION LESS THAN X
When X is less than the mean:
tail
X
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PROPORTION LESS THAN X
When X is less than the mean:
 Convert the X value to a z-score.
 z = (X – ) /  < 0 (ignore – sign)
 Find z in column A.
 The proportion needed is in the tail, so use Column (C).
 For %, multiply by 100.
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PROPORTION BETWEEN TWO XS
When one of the Xs is the mean
X
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PROPORTION BETWEEN TWO XS
When one of the Xs is the mean
 Convert the X value to a z-score.
 z = (X – ) /  > 0 or z = (X – ) /  < 0
 Find z in column A (ignore sign).
 The proportion needed is in the “Proportion Between Mean
and z” column, which is Column (D).
 For %, multiply by 100.
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PROPORTION BETWEEN TWO XS
When both Xs are greater than the mean
X1
X2
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PROPORTION BETWEEN TWO XS
When both Xs are greater than the mean
 X1 < X2
 z1 = (X1 – ) /  > 0
 z2 = (X2 – ) /  > 0
 z1 < z2
 Find z1 in column (A). Use proportion in Column (D) as p1.
 Find z2 in column (A). Use proportion Column (D) as p2.
 Then…
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PROPORTION BETWEEN TWO XS
When both Xs are greater than the mean (continued)
 p = p 2 – p1
p2
=
p1
p
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PROPORTION BETWEEN TWO XS
Example when both Xs are greater than the mean
 Let  = 100,  = 20, X1=120, and X2=140
 Find z1 = (X1 – ) /  which is (120-100)/20
 So z1 = 20/20 = 1.00
 Find z2=(X2- )/  which is (140-100)/20
 So z2 = 40/20 = 2.00
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PROPORTION BETWEEN TWO XS
Example when both Xs are greater than the mean
-
 Then p1 = .3413 and p2 .
p2= .4772
=
p1= .3413
p=.1359
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PROPORTION BETWEEN TWO XS
When both Xs are less than the mean
X2 X1
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PROPORTION BETWEEN TWO XS
When both Xs are less than the mean
 X1 > X2
 z1 = (X1 – ) /  < 0
 z2 = (X2 – ) /  < 0
 z1 > z2
 Find z1 in column (A). Use proportion in the “Between mean
and z” column (D) as p1.
 Find z2 in column A. Use proportion in the “Between mean
and z” column (D) as p2.
 Then…
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PROPORTION BETWEEN TWO XS
When both Xs are less than the mean
(continued)
p = p2 – p1
p2
=
p1
p
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PROPORTION BETWEEN TWO XS
When one X is less than the mean and the other X is
greater than the mean
X1
X2
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PROPORTION BETWEEN TWO XS
When one X is less than the mean and the other X is greater
than the mean
 X1 < X2 ; X1 <  ; X2 > 
 z1 = (X1 – ) /  < 0
 z2 = (X2 – ) /  > 0
 z1 < z2 ; z1 < 0; z2 > 0
 Find z1 in column (A). Use proportion in the “Between mean and z”
column (D) as p1.
 Find z2 in column A. Use proportion in the “Between mean and z”
column (D) as p2.
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PROPORTION BETWEEN TWO XS
When one X is less than the mean and the other
X is greater than the mean (continued)
p = p2 + p1
+
p1
=
p2
p
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