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Overview and
Descriptive Statistics
Copyright © Cengage Learning. All rights reserved.
1.4
Measures of Variability
Copyright © Cengage Learning. All rights reserved.
Measures of Variability
Center is just one characteristic of a data set. Different
datasets may have identical measures of center yet very
different other characteristics.
The dotplots of three samples with the same mean and
median, yet the extent of spread is different.
The three histograms also have the same mean.
Samples with identical measures of center but different amounts of variability
Figure 1.19
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Measures of Variability for
Sample Data
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Measures of Variability for Sample Data: Range
Simple measure: range, is the difference between the
largest and smallest sample values.
The range for sample 1 is much larger than sample 3, so
it has more variability.
A defect: it depends on the two most extreme observations
only and disregards the remaining n – 2 values.
Samples 1 and 2 have identical ranges, yet there is
much less variability or dispersion in the second sample
than in the first.
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Measures of Variability for Sample Data: Deviations
Our primary measures involve the deviations from the
mean,
, subtracting the mean from
each of the n sample observations.
+ or - depends on if the observation is > or < than the mean.
Small deviations, little variability;
Large deviations, a big variability.
Can we average them to get a single quantity? NO, because
One possibility is to use the average absolute deviation.
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Measures of Variability for Sample Data:
Variance and Standard Deviations
More conveniently, use the squared deviations
Rather than use the average squared deviation for several
reasons, we divide the sum of squared deviations by n – 1.
The sample variance, denoted by s2, is given by
The sample standard deviation, denoted by s, is the
(positive) square root:
Note that s2 and s are both nonnegative. The unit for s is
the same as the unit for each of the xis.
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Interpreting Sample Standard Deviations
If, for example, the observations are fuel efficiencies in
miles per gallon, them we might have
= 23, s = 2.0 mpg.
The size of a typical or representative deviation from the
sample mean of 23mpg is about 2pmg. So, many
observations fall around 21mpg and around 25 mpg.
If s = 3.0 for a second sample of cars of another type, a
typical deviation in this sample is roughly 1.5 times what it
is in the first sample, an indication of more variability in the
second sample.
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Example 17
Fuel efficiency of various vehicles, ww.fueleconomy.gov
Sxx = 314.106,
,
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Motivation for s2
Why divided by n-1 instead of n
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Motivation for s2, why divide by n-1
s2 is the sample variance, s is the sample standard deviation.
 2 is the population variance, and  is the population s.d.
When the population size N is finite,
which is the average of all squared deviations from the
population mean.
Just as is used to make inferences about the , we need to
define s2 in order to make inferences on  2 when  is needed.
However, the value of  is often unknown, so the sum of
squared deviations about must be used. But the xis tend to
be closer to than to , so to compensate for this, the divisor
n – 1 is used rather than n.
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Motivation for s2: why divide by n-1
If using a divisor n in the sample variance, the resulting
quantity would tend to underestimate  2 (too small), whereas
dividing by the slightly smaller n – 1 corrects this
underestimating.
It is customary to refer to s2 as being based on n – 1 degrees
of freedom (df). This terminology reflects the fact that
although s2 is based on the n deviations
these sum to 0, so specifying the
values of any n – 1 of them can determine the remaining one.
For example, if n = 4 and
then automatically
so only three of the
four values of
are freely determined (3 df).
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A Computing Formula for s2
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A Computing Formula for s2
Use statistical software (Minitab, SAS, etc)
Use Excel
Use a calculator with this function
If your regular scientific calculator does not have this
capability, there is an alternative formula for Sxx.
Both the defining formula and the computational formula for
s2 can be sensitive to rounding.
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Example 18
Recovery measurement of leg angle from knee surgery.
154 142 137 133 122 126 135 135 108 120 127 134 122
The sum of these 13 sample observations is
and the sum of their squares is
Thus the numerator of the sample variance is
From which,
s2 = 1579.0769/12 = 131.59
and s = 11.47.
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Two other properties of s2
Proposition
Let x1, x2, ……. , xn be a sample, c be any nonzero constant.
1. If a constant c is added to (or subtracted from) each data
value, the variance is unchanged.
If y1 = x1 + c, y2 = x2 + c, ….. , yn = xn + c, then
2. Multiplication of each xi by c results in s2 being multiplied
by a factor of c2.
If y1 = cx1, ….. , yn = cxn, then
where
is the sample variance of the x’s and
sample variance of the y’s.
is the
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Boxplots
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Boxplots
A boxplot, is to describe several data features: center,
spread, the extent and nature of any departure from
symmetry, and “outliers”.
The boxplot is based on the median and a measure of
variability called the fourth spread, which are not sensitive
to outliers.
Order the n observations ascendingly and separate the
smallest half from the largest half; the median is included in
both halves if n is odd. Then the lower (upper) fourth is
the median of the smallest (largest) half.
A measure of spread, the fourth spread fs, given by fs =
upper fourth – lower fourth
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Boxplots
The simplest boxplot is based on: smallest xi , lower fourth
median, upper fourth, largest xi
Draw a horizontal axis. Place a rectangle above it; the left
edge of the rectangle is at the lower fourth, and the right
edge is at the upper fourth (so box width = fs).
Place a vertical line segment inside the rectangle at the
median; the position of the median symbol relative to the
two edges conveys information about skewness in the
middle 50% of the data.
Draw “whiskers” out from either end of the rectangle to the
smallest and largest observations.
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Example 19
The five-number summary is as follows:
smallest 40, lower fourth 72.5, upper fourth 96.5 largest 125
The right edge of the box is much closer to the median
The box width (fs) is also reasonably large relative to the
range of the data
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Example 19
Figure 1.21 shows Minitab output from a request to
describe the corrosion data. Q1 and Q3 are the lower and
upper quartiles; these are similar to the fourths but are
calculated in a slightly different manner. SE Mean is
this will be an important quantity in our subsequent work
concerning inferences about .
Minitab description of the pit-depth data
Figure 1.21
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Boxplots That Show Outliers
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Boxplots That Show Outliers
A boxplot can reveal outliers..
Any observation farther than 1.5fs from the closest fourth is
an outlier. An outlier is extreme if it is more than 3fs from
the nearest fourth, and it is mild otherwise.
Let’s now modify our previous construction of a boxplot by
drawing a whisker out from each end of the box to the
smallest and largest observations that are not outliers.
Each mild outlier is represented by a closed circle and each
extreme outlier by an open circle. Some statistical
computer packages do not distinguish between mild and
extreme outliers.
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Boxplots That Show Outliers
Let’s now modify our previous construction of a boxplot by
drawing a whisker out from each end of the box to the
smallest and largest observations that are not outliers.
Each mild outlier is represented by a closed circle and each
extreme outlier by an open circle. Some statistical
computer packages do not distinguish between mild and
extreme outliers.
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Example 20
Among the pollutant loads in watersheds data of TN (total
nitrogen) loads (kg N/day) from a particular Chesapeake
Bay location, displayed here in increasing order.
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Example 20
Relevant summary quantities are
Subtracting 1.5fs from the lower 4th gives a negative
number, and none of the observations are negative, so
there are no outliers on the lower end of the data.
However,
upper 4th + 1.5fs = 351.015
upper 4th + 3fs = 534.24
Thus the four largest observations—563.92, 690.11,
826.54, and 1529.35—are extreme outliers, and 352.09,
371.47, 444.68, and 460.86 are mild outliers.
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Example 20
The whiskers in the boxplot in Figure 1.22 extend out to the
smallest observation, 9.69, on the low end and 312.45, the
largest observation that is not an outlier, on the upper end.
A boxplot of the nitrogen load data showing mild and extreme outliers
There is some positive skewness in the middle half of the
data (the median line is somewhat closer to the left edge of
the box than to the right edge) and a great deal of positive
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skewness overall.
Comparative Boxplots
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Comparative Boxplots
A comparative or side-by-side boxplot is a very effective
way of revealing similarities and differences between two or
more data sets consisting of observations on the same
variable—fuel efficiency observations for four different
types of automobiles, crop yields for three different
varieties, and so on.
We can use vertical Boxplots instead of horizontal in the
comparison.
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Example 21
Indoor radon from two houses, one having a child with
cancer
Both the mean and median suggest that the cancer sample
is centered to the right of the no-cancer sample.
The mean exaggerates the magnitude of this shift, largely
because of the outlier observation 210.
The s suggests more variability in the cancer sample, but this
impression is contradicted by the fourth spreads. Again, the
observation 210, an extreme outlier, is the culprit.
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Example 21
Figure 1.24 shows a
comparative boxplot
from the S-Plus
computer package.
A boxplot of the data in Example 1.21, from S-Plus
Figure 1.24
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Example 21
The no-cancer box is stretched out compared with the
cancer box (fs = 18 vs. fs = 11), and the positions of the
median lines in the two boxes show much more skewness
in the middle half of the no-cancer sample than the cancer
sample.
Outliers are represented by horizontal line segments, and
there is no distinction between mild and extreme outliers.
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