stat as tics and their use in hr 179
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8/4/2019 Stat as Tics and Their Use in Hr 179
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Slide
Chapter 3 Part ADescriptive Statistics: Numerical Methods
Measures of Location
Measures of Variability
x
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Slide
Measures of Location
Mean
Median
Mode
Percentiles
Quartiles
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Slide
Example: Apartment Rents
Given below is a sample of monthly rent values ($)
for one-bedroom apartments. The data is a sample of 70
apartments in a particular city. The data are presented
in ascending order.
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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Slide
Mean
The mean of a data set is the average of all the data
values. If the data are from a sample, the mean is denoted by
.
If the data are from a population, the mean is
denoted by m (mu).
xx
n
i
x
N
i
x
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Slide
Example: Apartment Rents
Mean
xx
n
i
34 356
70490 80
,.
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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Slide
Median
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.
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Median
The median of a data set is the value in the middle
when the data items are arranged in ascending order. For an odd number of observations, the median is the
middle value.
For an even number of observations, the median is
the average of the two middle values.
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Example: Apartment Rents
Median
Median = 50th percentile
i = ( p/100)n = (50/100)70 = 35.5Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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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.
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Example: Apartment Rents
Mode
450 occurred most frequently (7 times)
Mode = 450
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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Percentiles
A percentile provides information about how the
data are spread over the interval from the smallestvalue to the largest value.
Admission test scores for colleges and universitiesare frequently reported in terms of percentiles.
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The pth percentile of a data set is a value such that at
least p percent of the items take on this value or lessand at least (100 - p) percent of the items take on thisvalue or more.
•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.
Percentiles
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Example: Apartment Rents
90th Percentile
i = ( p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 585
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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14Slide
Quartiles
Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile
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15Slide
Example: Apartment Rents
Third Quartile
Third quartile = 75th percentile
i = ( p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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Measures of Variability
It is often desirable to consider measures of
variability (dispersion), as well as measures oflocation.
For example, in choosing supplier A or supplier B wemight consider not only the average delivery time for
each, but also the variability in delivery time for each.
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17Slide
Measures of Variability
Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation
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18Slide
Range
The range of a data set is the difference between the
largest and smallest data values. It is the simplest measure of variability.
It is very sensitive to the smallest and largest datavalues.
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19Slide
Example: Apartment Rents
Range
Range = largest value - smallest value
Range = 615 - 425 = 190
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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Interquartile Range
The interquartile range of a data set is the difference
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.
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21Slide
Example: Apartment Rents
Interquartile Range
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = 525 - 445 = 80
425 430 430 435 435 435 435 435 440 440
440 440 440 445 445 445 445 445 450 450
450 450 450 450 450 460 460 460 465 465
465 470 470 472 475 475 475 480 480 480
480 485 490 490 490 500 500 500 500 510
510 515 525 525 525 535 549 550 570 570
575 575 580 590 600 600 600 600 615 615
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22Slide
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 (x for a sample, for a population).
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23Slide
Variance
The variance is the average of the squared differences
between each data value and the mean. If the data set is a sample, the variance is denoted by
s2.
If the data set is a population, the variance is denoted
by 2.
s
xi
x
n
22
1
( )
2
2
( ) x
N
i
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24Slide
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 comparable, than the variance, to themean.
If the data set is a sample, the standard deviation isdenoted s.
If the data set is a population, the standard deviation
is denoted (sigma).
s s2
2
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25Slide
Coefficient of Variation
The coefficient of variation indicates how large the
standard deviation is in relation to the mean. If the data set is a sample, the coefficient of variation
is computed as follows:
If the data set is a population, the coefficient ofvariation is computed as follows:
s
x ( )100
( )100
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26Slide
Example: Apartment Rents
Variance
Standard Deviation
Coefficient of Variation
s xi x
n
2 2
12 996 16
( )
, .
s s 2 2996 47 54 7. .
s
x
100
54 74
490 80 100 1115
.
. .