Transcript Lesson 4 - West Virginia University

LESSON 5 Z-SCORES
1
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.
LESSON 5 Z-SCORES
2
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.
LESSON 5 Z-SCORES
3
Z-SCORE FORMULA
The formula for converting a raw score from a
distribution to a z-score is:
z
X 

LESSON 5 Z-SCORES
4
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.
LESSON 5 Z-SCORES
5
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.
LESSON 5 Z-SCORES
6
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 

LESSON 5 Z-SCORES
7
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.
LESSON 5 Z-SCORES
8
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.
LESSON 5 Z-SCORES
9
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.
LESSON 5 Z-SCORES
10
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
LESSON 5 Z-SCORES
11
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.
LESSON 5 Z-SCORES
12
PROPORTIONS UNDER THE CURVE
The proportion of those scores greater than
the mean is 0.5000

LESSON 5 Z-SCORES
13
PROPORTIONS UNDER THE CURVE
The proportion of those scores less
than the mean is 0.5000

LESSON 5 Z-SCORES
14
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.
LESSON 5 Z-SCORES
15
+/- 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
LESSON 5 Z-SCORES
16
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
LESSON 5 Z-SCORES
17
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
LESSON 5 Z-SCORES
18
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
LESSON 5 Z-SCORES
19
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
LESSON 5 Z-SCORES
20
FINDING THE AREA UNDER A CURVE
We can use the standardized normal distribution to
answer many important questions about a distribution.
LESSON 5 Z-SCORES
21
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.
LESSON 5 Z-SCORES
22
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.
LESSON 5 Z-SCORES
23
PROPORTION GREATER THAN X
When X is greater than the mean:
tail

X
LESSON 5 Z-SCORES
24
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.
LESSON 5 Z-SCORES
25
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.
LESSON 5 Z-SCORES
26
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
LESSON 5 Z-SCORES
27
PROPORTION GREATER THAN X
When X is less than the mean:
body
X
LESSON 5 Z-SCORES
28
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.
LESSON 5 Z-SCORES
29
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.
LESSON 5 Z-SCORES
30
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
LESSON 5 Z-SCORES
31
PROPORTION LESS THAN X
When X is greater than the mean:
body
X
LESSON 5 Z-SCORES
32
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.
LESSON 5 Z-SCORES
33
PROPORTION LESS THAN X
When X is less than the mean:
tail
X
LESSON 5 Z-SCORES
34
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.
LESSON 5 Z-SCORES
35
PROPORTION BETWEEN TWO XS
When one of the Xs is the mean
X
LESSON 5 Z-SCORES
36
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.
LESSON 5 Z-SCORES
37
PROPORTION BETWEEN TWO XS
When both Xs are greater than the mean
X1
X2
LESSON 5 Z-SCORES
38
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…
LESSON 5 Z-SCORES
39
PROPORTION BETWEEN TWO XS
When both Xs are greater than the mean (continued)
 p = p 2 – p1
p2
=
p1
p
LESSON 5 Z-SCORES
40
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
LESSON 5 Z-SCORES
41
PROPORTION BETWEEN TWO XS
Example when both Xs are greater than the mean
-
 Then p1 = .3413 and p2 .
p2= .4772
=
p1= .3413
p=.1359
LESSON 5 Z-SCORES
42
PROPORTION BETWEEN TWO XS
When both Xs are less than the mean
X2 X1
LESSON 5 Z-SCORES
43
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…
LESSON 5 Z-SCORES
44
PROPORTION BETWEEN TWO XS
When both Xs are less than the mean
(continued)
p = p2 – p1
p2
=
p1
p
LESSON 5 Z-SCORES
45
PROPORTION BETWEEN TWO XS
When one X is less than the mean and the other X is
greater than the mean
X1
X2
LESSON 5 Z-SCORES
46
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.
LESSON 5 Z-SCORES
47
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
LESSON 5 Z-SCORES
48