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QUARTILE DEVIATION
QUANTILES- are the extensions of the median concept because
they are values which divide a set of data into equal parts.
a.
b.
c.
d.
Median- divides the distribution into two equal parts.
Quartile- divides the distribution into four equal parts.
Decile- divides the distribution into ten equal parts.
Percentile- divides the distribution into one hundred equal
parts.
Quartiles are values in a given set of distribution that divide the data into
four equal parts. Each set of scores has three quartiles. These values can
be denoted by Q1, Q2, and Q3.
 First Quartile – Q1(lower quartile)- the middle number between the
smallest number and the median of the data set (25th Percentile).
 Second Quartile- Q2 – the median of the data that separates the
lower and upper quartile (50th Percentile).
 Third Quartile-Q3(upper quartile)- The middle value between the
median and the highest value of the data set (75th Percentile).
The difference between the upper and lower quartiles is called the
Interquartile Range.(IQR=Q3-Q1)
Quartile deviation or semi-interquartile range is one –half the
difference between the first and the third quartiles.(QD= Q3-Q1/2)
GETTING THE QUARTILE DEVIATION FROM UNGROUPED DATA
In getting the quartile deviation from ungrouped data, the following
steps are used in getting the quartiles:
0. Arrange the test scores from highest to lowest.
1. Assign serial numbers to each score. The first serial number is
assigned to the lowest test score, while the last serial number is
assigned to the highest test score.
2. Determine the first quartile (Q1). To be able to locate Q1’ divide N by
4. Use the obtained value in locating the serial number of the score
that falls under Q1
3. Determine the third quartile (Q3), by dividing 3N by 4. Locate the
serial number corresponding to the obtained answer. Opposite this
number is the test score corresponding to Q3.
4. Subtract Q1 from Q3, and divide the difference by 2.
Example
Score
17
18
24
28
30
31
32
40
Serial Number
1
2
3
4
5
6
7
8
N= 8
____N__= 8= 2
4
4
Q1= (17 + 18)
2
= 17.5
3N = 3(8)
4
4
=6
Q3= (30 + 31)
2
= 30.5
QD =(Q3-Q1)
2
=( 30.5 – 17.5)
2
= 6.5
GETTING THE QUARTILE DEVIATION FROM GROUPED DATA
In getting the quartile deviation from grouped data, the following steps
are used in getting the quartiles:
1. Cumulate the frequencies from the bottom to the top of the grouped
frequency distribution.
2. For the first quartile , use the formula Q3= L +3N – CF
_______________ i
F
Where
L= exact lower limit if the Q3 clss
3N/4= locator of the Q3 class
N = total number of scores
CF= Cumulative frequency below the Q3 class
i= class size/ interval
COMPUTATION OF THE
GROUPED TEST SCORES
QUARTILE
Classes
Frequency
46- 50
41- 45
36- 40
31- 35
26- 30
21- 25
16-20
11- 15
5
7
9
10
8
6
4
4
DEVIATION
FOR
Cumulative
Frequency
53
48
41
32
33
14
8
4
The computation procedure for determining the quartile deviation for
grouped test scores are reflected in the above.
For the First Quartile and Third Quartile
Definitions:




The lower half of a data set is the set of all values that are to the left
of the median value when the data has been put into increasing
order.
The upper half of a data set is the set of all values that are to the
right of the median value when the data has been put into increasing
order.
The first quartile, denoted by Q1 , is the median of the lower half of
the data set. This means that about 25% of the numbers in the data
set lie below Q1 and about 75% lie above Q1 .
The third quartile, denoted by Q3 , is the median of the upper half of
the data set. This means that about 75% of the numbers in the data
set lie below Q3 and about 25% lie above Q3 .
Example 1: Find the first and third quartiles of the data set {3, 7, 8, 5, 12,
14, 21, 13, 18}.
First, we write data in increasing order: 3, 5, 7, 8, 12, 13, 14, 18, 21.
As on the previous page, the median is 12.
Therefore, the lower half of the data is: {3, 5, 7, 8}.
The first quartile, Q1, is the median of {3, 5, 7, 8}.
Since there is an even number of values, we need the mean of the
middle two values to find the first quartile:
.
Similarly, the upper half of the data is: {13, 14, 18, 21}, so
.
Example 2: Find the first and third quartiles of the set {3, 7, 8, 5, 12, 14,
21, 15, 18, 14}.
Note that here we consider the two 14's to be distinct elements and
not representing the same item; consider this like you obtained a
score of 14 on two different quizzes.
First, we write the data in increasing order: 3, 5, 7, 8, 12, 14, 14, 15,
18, 21.
As before, the median is 13 (it is the mean of 12 and 14 — the pair
of middle entries).
Therefore, the lower half of the data is: {3, 5, 7, 8, 12}.
Notice that 12 is included in the lower half since it is below the
median value.
Then Q1 = 7 (there are five values in the lower half, so the middle
value is the median). Similarly, the upper half of the data is: {14, 14,
15, 18, 21}, so Q3 = 15.