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Slide 3-2
Chapter 3Descriptive Measures
Section 2 Measures of Variations
Slide 3-3
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
4s
Slide 3-44
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.
N
X
2
2 Population Variance =
2
2
1
X Xs
N
Sample Variance =
X = individual values
μ = population mean
N = population size
= sample mean
n = sample size
X
Slide 3-5
Figure 3.3
The “data sets” have the same Mean, Median, and Mode yet clearly differ!
Measures of Variation or Measures of Spread
Slide 3-6
Range of a Data SetRange 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 3-7
Figure 3.4
Measures of Variation or Measures of Spread:The Range
Team I has range 6 inches, Team II has range 17 inches.
Slide 3-8
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.
2
2 X
N
Population Standard Deviation =
Sample Standard Deviation = 2
2
1
X Xs s
n
Slide 3-9
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 of the total deviations from the mean for all the observation.
x
Slide 3-10
Sample Standard Deviation – standard deviation of a sample. Take the square root of the sample variance.
1
)( 22
2
n
xxss
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:
1
/)( 222
n
nXXs
Variance Standard Deviation
1
/)( 22
n
nXs X
Sample variance formula =
Slide 3-11
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 3-12
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 3-13
Table 3.10
Table 3.11
Standard Deviation: the more variation, the larger the standard deviation. Data set II has greater variation.
Slide 3-14
Figure 3.6
Data set II has greater variation and the visual clearly shows that it is more spread out.
Data Set I
Figure 3.7
Data Set II
Slide 3-15
Almost all the observations in any data set lie within three standard deviations to either side of the mean.
Three-Standard-Deviation Rule
Slide 3-16
Example:Find the variance and standard deviation.
Exam Scores: For 108 randomly selected college students, the exam score frequency distribution was obtained.
Class limits
f Xm Midpoint
f • Xm f • (Xm)2
90-98 6 94 564 53,016
99-107 22 103 2266 233,392
108-116 43 112 4816 539,392
117-125 28 121 3388 409,948
126-134 9 130 1170 152,100
108 12,204 1,387,854
(90+98)/2 94
6 * 94 564
6*(94)2 53,016
Slide 3-17
82.3or 26.821108
108204,12
854,387,12
2
s
9.1or 07.926.82 sStandard Deviation =
Variance =
*** NOTE: 9.072 = 82.26 ***
Example: