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Measures of Variation
• Range
• Standard Deviation
• Variance
The Range
the difference between the largest and smallest values of a
distribution
Find the range:
10, 13, 17, 17, 18
The range = largest minus smallest
= 18 minus 10 = 8
The standard deviation
a measure of the average variation of the data entries from
the mean
Standard deviation of a sample
1n
)xx(s
2
n = sample size
mean of the sample
To calculate standard deviation of a sample
• Calculate the mean of the sample.• Find the difference between each entry (x) and the
mean. These differences will add up to zero.• Square the deviations from the mean.• Sum the squares of the deviations from the
mean.• Divide the sum by (n 1) to get the variance.• Take the square root of the variance to get
the standard deviation.
The Variance
the square of the standard deviation
Variance of a Sample
1n)xx(
s2
2
Find the standard deviation and variance
x302622
2)x(x xx
4 04
16 016___3278 mean=
26
Sum = 0
1
)( 2
2
n
xxs = 32 2
=16
The variance
The standard deviation
s = 416
Find the mean, the standard deviation and
variance
Find the mean, the standard deviation and
variancex
4
5
5
7
4
2)x-(x xx
25
1
0
0
2
1
Find the mean, the standard deviation and
variance
Find the mean, the standard deviation and
variance
1
0
0
4
1 6mean = 5
The mean, the standard deviation and variance
Mean = 5
5.14
6Variance
22.15.1deviationdardtanS
Computation formula for sample standard
deviation:
n
xxSSwhere
1nSS
s
2
2
x
x
To find
Square the x values, then add.
2x
To find
Sum the x values, then square.
2)x(
Use the computing formulas to find s and s2
x
4
5
5
7
4
x2
16
25
25
49
1625 131
n = 5
(Sx) 2 = 25 2 = 625
Sx2 = 131
SSx = 131 – 625/5 = 6
s2 = 6/(5 –1) = 1.5
s = 1.22
Population Mean and Standard Deviation
population the in values data ofnumber N
deviation standard population
mean population
2
where
N
xx
N
x
COEFFICIENT OF VARIATION:
a measurement of the relative variability (or consistency) of data
100or100x
sCV
CV is used to compare variability or
consistency
A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound.
A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pounds.
Which (newborns or 3-month-olds) are more variable in weight?
To compare variability, compare Coefficient of Variation
For newborns:
For 3-month-olds:
CV = 16%
CV = 14%
Higher CV: more variable
Lower CV: more consistent
Use Coefficient of Variation
To compare two groups of data,
to answer:
Which is more consistent?
Which is more variable?
CHEBYSHEV'S THEOREM
For any set of data and for any number k,
greater than one, the proportion of the
data that lies within k standard deviations
of the mean is at least:
2k
11
CHEBYSHEV'S THEOREM for k = 2CHEBYSHEV'S THEOREM for k = 2
According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean?
At least
of the data falls within 2 standard deviations of the mean.
%7543
21
12
CHEBYSHEV'S THEOREM for k = 3CHEBYSHEV'S THEOREM for k = 3
According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean?
At least
of the data falls within 3 standard deviations of the mean.
%9.8898
31
12
CHEBYSHEV'S THEOREM for k =4CHEBYSHEV'S THEOREM for k =4
According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean?
At least
of the data falls within 4 standard deviations of the mean.
%8.931615
41
12
Using Chebyshev’s Theorem
A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6.
According to Chebyshev's Theorem, between what two values would at least 75% of the grades be?
Mean = 77 Standard deviation = 6
At least 75% of the grades would be in the interval:
s2xtos2x
77 – 2(6) to 77 + 2(6)
77 – 12 to 77 + 12
65 to 89
