ST361: Ch2.1 Numerical Summary Measures of Center for Data
--------------------------------------------------------------------------------------------------------Topics:

Measures of Center: mean, median
(Note: Here we will cover the summary measure for DATA only. We will cover the
measures for DISTRIBUTIONS in the 4th and 5th weeks after we introduce the
concept of probability distribution.)
--------------------------------------------------------------------------------------------------------Measures of Center for Data:
(1) Mean

Mean x of n observations x1 , x 2 ,..., x n is
x
Ex. Sue wanted to study the systolic blood pressure (BP), x, of the NCSU freshmen; 7
freshmen were randomly selected and their BP values are
121, 110, 114, 100, 103, 130, 130 (Note:
x
i
 808 )
The sample mean of BP is
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(2) Median

Median is the middle value of the data such that there are same numbers of data
points above it and below it.

To get the Median ~
x of the n observations in the sample:
(1) Sort the data, from the smallest to the largest
(2) If n is odd, then ~
x = the middle value, i.e.,
 n 1
~
x = the 
 th data point
 2 
If n is even, then ~
x = the average of the middle two values, i.e.,
1
n
n 
~
x = [ the   th data point + the   1 th data point ]
2
2
2 
Ex. In the BP example, there are 7 observations: 121, 110, 114, 100, 103, 130, 130.
The sample median is:
Ex. In BP example, there is one more data point 105. Then the sample median
becomes:
2
Comment (1) : Mean vs. Median
1. Mean is sensitive to outliers (extreme values), while median is less affected by
outliers.
2. Mean is the __________ _ point of the data.
 A balance point is the point such that
=
Ex. Two data points
Ex. Three data points
Ex. A sample consists of 5 data points 1, 2, 3, 10, 14. The mean x =
Data point  x
Total distance to x =
Data point  x
Total distance to x =
Median is the ___________ point of a the distribution
That is,
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3. The relationship between mean and median depends on the ________ of the
distribution
a. For symmetric distribution, mean ___ median
b. For positively-skewed distribution, mean ____ median
c. For negatively-skewed distribution, mean ____ median
 In other word, from the relationship between mean and median, we can guess the
shape of the distribution
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Comment (2) : Change of Unit

Mean and median share the same unit as the measuring scale. The values changes
with the measuring unit.

When unit of measure changes from x to a  x  b , then
The new mean =
The new median =
Ex. Temperatures read in Fahrenheit, and the mean temperature is 82oF and median is
70oF. What are the mean ( xC ) and median ( xC ) if we switch to Centigrade? Note
that C  F  32 
5
.
9
5