Download Chapter 3 Part A

Chapter 3
Part A
Descriptive Statistics: Numerical Methods
Measures of Location
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
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
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
Example: Apartment Rents
Mean
x 
 xi
x
n
x


ii
N
 xi 34 , 356

 490 . 80
n
70
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.
Median for the Example:
Median = 50th percentile
i = (p/100)n = (50/100)70 = 35.5
Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
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.
Mode for the Example: 450 occurred most frequently (7 times)
Mode = 450
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.
•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.
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
Quartiles: Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile
Example: Apartment Rents
Third Quartile: Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
Measures of Variability
It is often desirable to consider measures of variability (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 for each, but also the variability in delivery time for each.
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 data values.
Example: Apartment Rents
Range = largest value - smallest value:
R an g e = 6 1 5 - 4 2 5 = 1 9 0
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.
Example: Apartment Rents
Interquartile Range:
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = 525 - 445 = 80
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, m for a population).
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.
2
 ( xi  x )
2
s 
If the data set is a population, the variance is denoted by s 2.
 22 
 (x
 )
N
ii
2
2
n 1
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 the mean.
If the data set is a sample, the standard deviation is denoted s.
If the data set is a population, the standard deviation is denoted s (sigma).
  2
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:
s
( 100 )
x
If the data set is a population, the coefficient of variation is computed as follows:
Example: Apartment Rents
Variance
s
2
Standard Deviation

(100)

( xi  x ) 2


 2 , 996 .16
n 1
s  s2  2996.47  54.74
Coefficient of Variation
s
54. 74
 100 
 100  11.15
x
490.80