3.1 Measure of Location:
Example 1:
Suppose the following data are the scores of 10 students in a quiz,
1, 3, 5, 7, 9, 2, 4, 6, 8, 10.
Some measures need to be used to provide information about the performance of the
10 students in this quiz.
(I) Mean:
n
Sample mean:
x
x
i 1
i
(sample statistic)
n
N
Population mean:

y
i 1
i
(population
N
parameter)
Basically, the mean can provide the information about the “center” of the data.
Intuitively, it can measure the rough “location” of the data.
Example 1 (continue):
x
1  3    10
 5.5
10
(II) Median:
The data are arranged in ascending (or descending) order. Then,
1. As the sample size is odd, the median is the middle value.
2. As the sample size is even, the median is the mean of the middle two numbers.
Example 1 (continue):
median 
56
 5.5
2
If the data are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Then,
median  6
1
Note: the median is less sensitive to the data with extreme values than
the mean. For example, in the previous data, suppose the last data has
been wrongly typed, the data become 1, 3, 5, 7, 9, 2, 4, 6, 8, 100. Then
the median is still 5.5 while the mean becomes 14.5.
(III) Mode:
The data value occurs with greatest frequency (not necessarily to be numerical).
Note: if the data have exactly two modes, we say that the data are
bimodal. If the data have more than two modes, we say that the data are
multimodal.
(IV) Percentile:
The pth percentile is a value such as at least p percent of the data have this value or
less and at least (100-p) percent of the data have this value or more.
Note: 50th percentile ＝ median!!
The procedure to calculate the pth percentile:
1. Arrange the data in ascending order.
 p 
2. Compute an index i, i  
n.
 100 
3. (a) If i is not an integer, round up, i.e., the next integer value greater than i denote
the position of the pth percentile.
(b) If i is an integer, the pth percentile is the average of the data values in positions
i and i+1.
Example 1 (continue):
Please find 40th percentile and 26th percentile for the previous data.
[Solution]
Step 1: the data in ascending order are
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Step 2:
For 40 th percentile,
 40 
i
10  4 .
 100 
2
For 26 th percentile,
 26 
i
10  2.6
 100 
Step 3:
40th percentile 
45
 4.5
2
and
26th percentile  3
(V) Quartiles:
When dividing data into 4 parts, the division points are referred to as the quartile!!
That is,
Q1  the first quartile or 25th percentile
Q2  the second quartile or 50th percentile
Q3  the third quartile or 75th percentile
Example 1 (continuous):
Find the first quartile and the third quartile for the previous example.
Step 2:
For the first quartile,
 25 
i
10  2.5 .
100


For the third quartile,
 75 
i
10  7.5
100


Step 3:
Q1  3
and
Q3  8
Online Exercise:
Exercise 3.1.1
Exercise 3.1.2
3