**Unformatted text preview: **EM 521 Applied Statistics
Descriptive Statistics:
Numerical Measures - I
AS&W – Chapter 3 1 Descriptive Statistics: Numerical
Measures Several numerical measures that provide
additional alternatives for summarizing
data: • Measures of Location
• Measures of Variability 2 Measures of Location Mean Median Mode Percentiles Quartiles If the measures are computed
for data from a sample,
they are called sample statistics.
If the measures are computed
for data from a population,
they are called population parameters. A sample statistic is referred to
as the point estimator of the
corresponding population parameter. 3 Mean Provides a measure of central location for the data The mean of a data set is the average of
all the data values. The sample mean x is the point
estimator of the population mean . 4 Sample Mean x
For variable x x x Sum of the values
of the n observations i n
Number of
observations
in the sample Sample mean is a point estimator of the
population mean.
5 Population Mean For variable x x Sum of the values
of the N observations i N
Number of
observations in
the population 6 Sample Mean Example: Apartment Rents
Seventy efficiency apartments
were randomly sampled in
a small college town. The
monthly rent prices for
these apartments are listed
in ascending order on the next slide. 7 Sample Mean
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615 8 440
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615 Sample Mean
x 34,356 x=
=
=
i 70 n 425
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600 490.80
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615 Median The median of a data set is the value in the middle
when the data items are arranged in ascending ord Whenever a data set has extreme values, the med
is the preferred measure of central location. The median is the measure of location most often
reported for annual income and property value data A few extremely large incomes or property values
can inflate the mean. 10 Median For an odd number of observations:
26 18 27 12 14 27 19 7 observations
12 14 18 19 26 27 27 in ascending order
the median is the middle value.
Median = 19 11 Median For an even number of observations:
26 18 27 12 14 27 30 19 8 observations
12 14 18 19 26 27 27 30 in ascending order the median is the average of the middle two values.
Median = (19 + 26)/2 = 22.5 12 Median
Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
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575 430
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575 430
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580 435
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590 435
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600 435
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615 13 440
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615 Mode The mode of a data set is the value that occurs with
greatest frequency. The greatest frequency can occur at two or more
different values. If the data have exactly two modes, the data are
bimodal. If the data have more than two modes, the data are
multimodal. Important measure of location for qualitative data.
14 Mode
450 occurred most frequently (7 times)
Mode = 450
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575 430
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590 435
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600 435
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615 15 440
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615 Percentiles A percentile provides information about how the
data are spread over the interval from the smallest
value to the largest value. Admission test scores for colleges and universities
are frequently reported in terms of percentiles. 16 Percentiles The pth percentile of a data set is a value
such that at least p percent of the items
take on this value or less and at least (100
- p) percent of the items take on this value
or more. 17 Percentiles
Arrange the data in ascending order.
Compute index i, the position of the pth percentile.
i = (p/100)n
If i is not an integer, round up. The p th percentile
is the value in the i th position.
If i is an integer, the p th percentile is the average
of the values in positions i and i +1.
18 90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 585
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575 430
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575 430
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580 435
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600 435
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615 19 440
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615 90th Percentile
“At least 90%
of the items
take on a value
of 585 or less.” “At least 10%
of the items
take on a value
of 585 or more.” 63/70 = .9 or 90% 7/70 = .1 or 10% 425
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575 430
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575 430
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590 435
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600 435
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600 435
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615 20 440
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615 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile 21 Third Quartile
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
425
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575 430
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575 430
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580 435
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590 435
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600 435
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600 435
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600 435
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600 440
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615 22 440
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615 Measures of Variability It is often desirable to consider measures of variabil
(dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we
might consider not only the average delivery time f
each, but also the variability in delivery time for eac 23 Measures of Variability Range Interquartile Range Variance Standard Deviation Coefficient of Variation 24 Range The range of a data set is the difference between th
largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data
values. 25 Range
Range = largest value - smallest value
Range = 615 - 425 = 190
425
440
450
465
480
510
575 430
440
450
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485
515
575 430
440
450
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525
580 435
445
450
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525
590 435
445
450
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525
600 435
445
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600 435
445
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600 435
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600 440
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615 26 440
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615 Interquartile Range The interquartile range of a data set is the differenc
between the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values 27 Interquartile Range
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = 525 - 445 = 80
425
440
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480
510
575 430
440
450
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515
575 430
440
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525
580 435
445
450
472
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525
590 435
445
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525
600 435
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600 435
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600 435
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615 28 440
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615 Variance The variance is a measure of variability that utilizes
all the data. It is based on the difference between the value of
each observation (xi) and the mean
( for a sample
x for a population). 29 Variance The variance is the average of the squared
differences between each data value and the mean.
The variance is computed as follows:
2 2
(
x x
) i
s2 n 1 for a
sample 2
(
x ) i
2 N for a
population
30 Standard Deviation
The standard deviation of a data set is the positive
square root of the variance. It is measured in the same units as the data, making
it more easily interpreted than the variance. 31 Standard Deviation
The standard deviation is computed as follows: s s2 2 for a
sample for a
population 32 Coefficient of Variation
The coefficient of variation indicates how large the
standard deviation is in relation to the mean.
The coefficient of variation is computed as follows: s 100 %
x 100 % for a
sample for a
population 33 Variance, Standard Deviation,
And Coefficient of Variation Variance
2
(
x x
) i
s2 2,996.16
n 1 Standard Deviation the
standard
s s 2996.47 54.74
deviation is
about 11% Coefficient of Variation
of
of the
s
54.74 100
% 100 % 11.15% mean
x 490.80 2
2 34 ...

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- 10%, 50%, 90%, 11%, 11.15%