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```CRIM 483
Measuring Variability
Variability


Variability refers to the spread or dispersion of scores
Variability captures the degree to which scores within a dataset
differ from one another
– High variability=large distance between scores

Score set 7,6,3,3,1
– Low variability=small distance between scores

Score set 4,2,3,3,1
– No variability=no distance between scores

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Score set 4,4,4,4,4
Variability & mean are used together to describe the
characteristics of a distribution (sample) and show how
distributions differ from one another
There are 3 measures for variability: range, standard
deviation, and variance
Range


The range is the most general measure of
variability
The formula: r=h-l
– r=range
– h=highest score
– l=lowest score


Calculation of range provides a general estimate
of how wide or how much scores differ from one
another
Examples:
– Highest age=35, lowest age=21

35-21=14 years difference between age scores in sample
– Highest age=50, lowest age=15

50-15=35 years difference between age scores in sample
– Which sample has the greatest variability with regard to
age?
Standard Deviation

Standard deviation (SD)=average amount of
variability from the mean in the set of scores
(average distance from the mean)
– Standard deviation is used most often to measure
variability
– Reported (as a rule) in combination with means
– The greater the SD, the larger the distance between the
score and the mean

Formula to calculate the SD
s=√(∑(x-mean)2)/(n-1)

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s=standard deviation
∑=sigma (sum of)
x=individual score
Mean=mean of all scores
n=sample size
Clarification of Formula

Why not add up the deviations from the
mean?
– Sum of deviations from the mean is always
equal to zero (good way to check your work)

Why square the deviations?
summing to 0

Why take the square root?
with
Unbiased v. Biased Estimates

Unbiased
– You produce an unbiased estimate by dividing by (n-1) in the
SD formula
– Artificially forces the SD to be larger than it would be
otherwise
– Why? This produces a more conservative estimate that we can
feel more comfortable with–it is safer to overestimate than
underestimate

Biased
– You produce a biased estimate by dividing by (n) in the SD
formula
– Use biased estimate when you are merely describing your
sample and you have no intention of comparing it to the
population

Ultimately, the larger your sample size the less difference
there is between the unbiased and biased estimates (p. 40)
In Sum…
 Must
always compute the mean first
 SD play a critical role later when
comparing scores between groups
(e.g., do male and female attitudes
differ)
 Like means, SD are sensitive to
extreme scores
Variance
Final method of measuring variability is
variance
 Very similar to SD formula

– Formula to calculate the SD
s2= (∑(x-mean)2)/(n-1)
–
–
–
–
–


s2=standard deviation
∑=sigma (sum of)
x=individual score
Mean=mean of all scores
n=sample size
Variance is difficult to interpret and apply by itself
Variance has greater utility in the formulas of more
Standard Deviation v. Variance
Both measure variability, dispersion, or