Measures of Dispersion and
Standard Scores
Deviation Review
• Deviation is the difference from a standard or
reference value (usually the mean).
• This is the starting point for determining both the
variance and the standard deviation of a set of
scores.
• We want to measure the dispersion of the scores
around the mean, so it makes sense to use the
deviation scores.
Activity #1 (Part 1)
• Measure to the nearest millimeter the writing
utensil you are using and write the
measurement on a piece of paper.
Sum of Squares (SS)
• Sum of squares = sum of squared deviations
• SS is used when calculating the variance and
the standard deviation
• If you are using a frequency distribution table:
Sum of Squares (SS)
• The population formulas are as follows:
Calculating SS
Score
Deviation
Deviation2
1
-3
9
3
-1
1
5
1
1
7
3
9
• Step 1: Find the mean.
• Step 2: Find the
deviation scores.
• Step 3: Square the
deviation scores.
• Step 4: Sum the squared
deviation scores.
Sum of Squares
Computational Formulas
• When you are trying to understand SS, use the
previous formulas, but when you are
computing SS, use these formulas (it’s faster).
Watch Out
• Know the difference between
.
and
•
means that you square Xs, then sum the
squared Xs.
•
means that you sum the Xs, then
square the sum of the Xs.
Calculating SS (the easy way)
X
X2
1
1
3
9
5
25
7
49
• Step 1: Sum the X column and
square the sum.
• Step 2: Square each score (X)
• Step 3: Sum the X2 column.
• Step 4: Plug the values into the
formula and solve.
Calculating SS with
Frequency Distribution Tables
X
f
fX
fX2
1
2
2
2
3
4
12
36
5
4
20
100
7
2
14
98
• Step 1: Create a fX column and
sum the values, then square
the sum.
• Step 2: Create a fX2 column by
multiplying fX(X) (do not
square fX).
• Step 3: Sum the fX2 column.
• Step 4: Plug the values into the
equation and solve (remember
N is the sum of f).
Population Variance (σ2)
• Variance = average of squared deviations
• Recall that SS is the sum of the squared
deviations, the numerator in the above
equation. So we can rewrite the equation as:
Sample Variance (s2)
• If we use the population formula for sample
data, we will probably underestimate the
variance (i.e., s2 will be smaller than σ2).
• To correct for this and get a better estimate of
the population variance, we change the
denominator to N-1.
Sample Variance (s2)
• When you use N-1 in the denominator, the result
is a larger estimate of the population variance.
• Why do we need to make our estimate larger?
• (Unadjusted) sample variance is always
less than or equal to population
variance.
Standard Deviation
• The standard deviation is the square root of
the variance.
Standard Deviation (Sample)
Calculation: Break it Down
Sum of Squares (SS)
Variance (s2)
Know These Equations
With these three equations you can understand and calculate
sample variance and standard deviation.
What Does it Look Like?
• Let’s look at an example (also see p. 103):
• The standard deviation is another unit of
measurement on the X axis.
What Should We Expect?
• For a small sample, we should expect the
standard deviation units to divide the sample
distribution into about 4 parts.
• For a large sample, we should expect closer to
six parts.
Activity #1 (Part 2)
• Given our writing utensil data, calculate (in the
following order):
– SS
– s2
–s
• Use either of the sample formulas and show all of
your work.
• Write your name on the paper.
Standard Scores (z scores)
• z score = the deviation of a raw score from the
mean in standard deviation units.
• The closer a score is to the mean, the smaller its z
score will be.
• A positive z-score indicates that the score is
above the mean, negative indicates that it is
below the mean. A z score of zero will always be
at the mean.
You Try
• What raw score would have a z score of -1?
82
72
62
52
You Try
• What raw score would have a z score of 2?
82
92
102
112
Calculating z scores
• Population:
• Sample:
• So you have to know the standard deviation
and the mean before you can find the z score.
Calculating Raw Scores
• If I know the z score but I don’t know the
corresponding raw score, using basic algebra I
can change the equation to solve for X.
Activity #2
• Given the above information, calculate the z
score values for the following raw scores: 75,
95, 100, 50, 125, and 80.
• Using the same sample information, calculate
the raw scores for the following z scores: 3.25,
-.25, 2, -1.75, and 2.5
Homework
• Study for Chapter 6 Quiz (know the equations
in red).
• Read Chapter 7
• Do Chapter 6 Homework