Chapter 3
Descriptive Measures
Section 2
Measures of Variations
Slide 3-2
Measures of Variation
There are three common measures for the spread or
variability of a data set they are range, variance, and
standard deviation.
To describe the difference quantitatively, we use a
descriptive measure that indicates the amount of
variation, or spread, in the data set. These are referred
to as measures of variation or measures of spread.
The range rule of thumb:
A rough estimate of the standard deviation is
range
s
4
Slide 3-3
Variance:



is the average of the squares of the distance each value is
from the mean.
The symbol for the population variance is σ2. Greek lower
case letter sigma.
Symbol for sample variance is s2.
Sample Variance = s
2



Population Variance =  2 
X X

2
N 1
2


X



X = individual values
μ = population mean
N = population size
X = sample mean
n = sample size
N
4
Slide 3-4
The “data sets” have the same Mean, Median, and Mode
yet clearly differ!
Measures of Variation or Measures of Spread
Figure 3.3
Slide 3-5
Range of a Data Set
Range of a Data Set
The range of a data set is given by the formula
Range = Max – Min,
where Max and Min denote the maximum and minimum
observations, respectively.
Range:

distance between the highest value and the lowest value.

The symbol R is used for the range.

R = highest value – lowest value
Slide 3-6
Measures of Variation or Measures of Spread:
The Range
Team I has range 6 inches, Team II has range 17 inches.
Figure 3.4
Slide 3-7
Standard Deviation:

Measures variation by indicating how far, on average, the observations are
from the mean.

is the square root of the variance.

Symbol for the population standard deviation is σ (sigma). Symbol for the
sample standard deviation is s.

Variation and the Standard Deviation: The more variation that there is
in a data set the larger is its standard deviation

Rounding rule: The final answer should be rounded to one more decimal
place than the original data.
Sample Standard Deviation = s  s 
2
 X  X 
2
Population Standard Deviation =    
2
n 1
 X  
2
N
Slide 3-8
Sample Standard Deviation
Deviations from the Mean is how far each observation is
from the mean and is the first step in computing a sample
standard deviation.
Sum of Squared Deviation is the sum of the squared
deviations from the mean ∑(x1 - )2 and gives a measure
x
of the total deviations from the mean for all the
Slide 3-9
observation.
Sample Standard Deviation – standard deviation of a
sample. Take the square root of the sample variance.
2

(
x

x
)
2
Sample variance formula =s  s 
n 1
2
Why is the denominator (n – 1) rather than n?
Division by (n – 1) increases the value of the sample variance
so that it will more closely reflect the population variance.
Giving us an unbiased estimate for the population variance.
Shortcut or computational formulas for data obtained from samples:
Variance
s 
2

2
2
X

(
X
)

 /n
n 1
Standard Deviation

s
X
2

 ( X ) 2 / n

n 1
Slide 3-10
Variance and Standard Deviation

Variances and standard deviations can be used to determine
the spread of the data. If the variance or standard deviation is
large, the data are more dispersed. The information is useful
in comparing two (or more) data sets to determine which is
more (most) variable.

The measures of variance and standard deviation are used to
determine the consistency of a variable.

The variance and standard deviation are used to determine the
number of data values that fall within a specified interval in a
distribution.

The variance and standard deviation are used quite often in
inferential statistics.
Slide 3-11
Computing Formula for a
Sample Standard Deviation
Rounding rule: do not perform any rounding until the
computation is complete, otherwise, substantial round
off error can result.
Slide 3-12
Standard Deviation: the more variation, the larger the
standard deviation. Data set II has greater variation.
Table 3.10
Table 3.11
Slide 3-13
Data set II has greater variation and the visual clearly
shows that it is more spread out.
Figure 3.6
Data Set I
Figure 3.7
Data Set II
Slide 3-14
Three-Standard-Deviation Rule
Almost all the observations in any data set lie
within three standard deviations to either side of
the mean.
Slide 3-15
Example:
Find the variance and standard deviation.
Exam Scores: For 108 randomly selected college students,
the exam score frequency distribution was obtained.
(90+98)/
2 94
6 * 94
564
6*(94)2
53,016
Class
limits
90-98
99-107
108-116
117-125
126-134
f
Xm
Midpoint
6
94
22
103
43
112
28
121
9
130
108
f • Xm
f • (Xm)2
564
2266
4816
3388
1170
12,204
53,016
233,392
539,392
409,948
152,100
1,387,854
Slide 3-16
Example:
12,2042
1,387,854 
2
108  82.26 or 82.3
s
Variance = 
108  1
Standard Deviation = s  82.26  9.07 or 9.1
*** NOTE: 9.072 = 82.26 ***
Slide 3-17