Measure of Center
the value at the center or middle of a data set
Arithmetic Mean (Mean)
the measure of center obtained by adding the values and dividing the total by the number of values. Called an average.
Notation
(SUM)
x
n
N
(SUM) - denotes the sum of a set of values
x - is the variable usually used to represent the individual data values
n - represents the number of data values in a sample
N - represents the number of data values in a population
x-bar
mu
x - is pronounced "x-bar" and denotes the mean of a set of sample values
mu - is probounced "mu" and denotes the mean of all values in a population
Mean - Advantages & Disadvantages
Advantages
-Sample means drawn from the same population tend to vary less than other measures of center
- Takes every data value into account
Disadvantage
- Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant
Medium, x~
the middle value when the original data values
are arranged in order of increasing (or
decreasing) magnitude. Denoted by "x-tilde" & not affected by an extreme value - is a resistant measure of the center
Midrange
Sensitive to extremes
because it uses only the maximum and
minimum values, it is rarely used
Midrange
Redeeming Features
(1) very easy to compute
(2) reinforces that there are several ways to
define the center
(3) avoid confusion with median by defining
t
Round-off Rule for Measures of Center
Carry one more decimal place than is present in the original set of values
Which measure of center is the only one that
can be used with data at the nominal level of
measurement?
Mode
Which of the following measures of center is
not affected by outliers?
Median
Mode
the value that occurs with the greatest
frequency. Data set can have one, more than one, or no
mode.
Bimodal - two data values occur with the same greatest
frequency
Multimodal - more than two data values occur with the same greatest frequency
No Mode - n
Midrange
the value midway between the maximum and minimum
values in the original data set
Midrange = (maximum value + minimum value)/2
Sensitive to extremes because it uses only the maximum and minimum values, it is rarely used
The median of a data set is always/sometimes/never
(select one) one of the data points in a set of data.
Explain your answer with brief examples.
The general answer is that the median is sometimes in
a data set.
For example, the median of the data set 1, 2, 3, 4, 5 is 3.
The median of the data set 1, 2, 4, 5 is also 3.
Range
Range = (maximum value) - (minimum value)
It is very sensitive to extreme values; therefore, it is
not as useful as other measures of variation.
standard deviation
a set of sample values, denoted by s, is a measure of how much data values deviate away from the mean.
Standard Deviation - Important Properties
The standard deviation is a measure of variation
of all values from the mean.
The value of the standard deviation s is usually
positive (it is never negative).
The value of the standard deviation s can
increase dramatically with the inclusion of one or
mo