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Chapter 3 continued...
II. Measures of dispersion.
Measures of dispersion give us an idea of how
widely the data are spread around the mean. More
dispersion means more variability and uncertainty
and this is a great influence on our upcoming
discussions of probability.
A. Range
• range=(maximum value-minimum value)
• The range is highly influenced by extreme
values.
B. Interquartile range
• The IQR overcomes the extreme value
problem by dropping the upper and lower
values in the sample.
• Q3 is the value at the 75th percentile
• Q1 is the value at the 25th percentile
• IQR= Q3-Q1
C. Variance
The variance measures deviations about the
mean.
N
Population variance:
( x   )2
2 

i 1
Sample variance:
n
s2 
2
(
x

x
)
 i
i 1
n 1
i
N
Example of calculating a variance.
• Five airlines quote you
five different prices:
$175, 200, 220, 205, 295
Calculate the sample
mean and then the
sample variance of
these prices.
Did you find
x  219?
Did you find
2
s  2067.5?
If not, go back and check
your work. Make sure
you can do this!
D. Standard Deviation
• Sample:
• Population:
s  s2
  2
A standard deviation tells us, on average, how far
each observation lies from the mean. Larger
values mean the data exhibits more dispersion
about the mean.
E. Coefficient of variation.
The standard deviation can be misleading if you are
comparing two sets of data that are of different scale. For
example, take three years of national GDP and three years
of sales at Shipley’s, and the standard deviation for the
former will always be greater, simply because of the scale
of the data. This problem is alleviated by calculating the
CV because the CV measures the size of the standard
deviation as a fraction of the mean.
s
CV    100
x
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