measures of variation range standard deviation variance

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