MEASURES of SPREAD
(Standard Deviation and Variance)
The measures of spread or dispersion of a data set are
quantities that indicate how closely a set of data clusters
around its centre. There are several different measures of
spread.
A deviation is the difference between an individual value and the mean for the data.
The larger the size of the deviations, the greater the spread in the data.
values below
values above
(–) deviation
mean
(+) deviation
NOTE: The sum of all the deviations of a set of data will equal zero!!
For a SAMPLE:
For a POPULATION:
deviation = x i  x
deviation = x i  
The standard deviation is the square root of the mean of the squares of the
deviations.
For a SAMPLE:
For a POPULATION:
 fi (xi  x)2
 fi (xi  )2
s
n 1

N
NOTE: (n-1) is used to compensate for the fact that a sample tends to underestimate
the deviations of a population.
For grouped data, where fi is the frequency for the given interval and mi is the
midpoint of the interval, the standard deviation is calculated using:
For a SAMPLE:
For a POPULATION:
 fi (mi  ) 2
 fi (mi  x) 2

N
n 1
The variance is the mean of the squares of the deviations (which is equal to the
square of the standard deviation).
s
For a SAMPLE:
s
2
fi ( x i  x ) 2


n 1
For a POPULATION:

2
fi ( x i  ) 2


N
A z-score is the number of standard deviations that a datum is from the mean.
For a SAMPLE:
z
(–) z-score
Example 
For a POPULATION:
xi  x
s
values below
z
xi  

values above
mean
z=0
(+) z-score
Twenty students each took six shots with a basketball from the free
throw line. The number of baskets made by each student was
recorded. The results were as follows:
3
3
5
5
4
3
5
4
3
2
4
2
5
4
3
3
4
2
6
2
Determine the mean, standard deviation, variance, and
z-scores for the basketball data.
Data
(xi)
TOTALS:
Frequency
(fi)
Deviation
(xi - x )
(xi - x )2
fi(xi - x )2
Z-score
x x
z i
s