Calculation of Quartiles, Deciles & Percentiles for Grouped Data

Calculation of Quartiles, Deciles & Percentiles from Grouped Data

• Quartiles

= Lower boundary of the quartile class
f= Frequency of quartile class
C= Cumulative frequency of the preceding quartile class
w= length of the quartile class
• Deciles

= Lower boundary of the decile class
f= Frequency of decile class
C= Cumulative frequency of the preceding decile class
w= length of the decile class

• Percentiles


= Lower boundary of the percentile class
f= Frequency of percentile class
C= Cumulative frequency of the preceding percentile class
w= length of the percentile class

Example 1: The following table shows the Price of 80 New Vehicles Sold Last Month at Toyota (in $ thousand) is given below.
Selling Price 15-17 18-20 21-23 24-26 27-29 30-32 33-35 Total
Freq 8 23 17 18 8 4 2 80
Calculate 3rd quartile, 3rd decile, 68th percentile for the previously given data.
Solution:

Table for necessary calculation

Selling Price Frequency Cumulative Frequency
15-17 8 8
18-20 23 31
21-23 17 48
24-26 18 66
27-29 8 74
30-32 4 78
33-34 2 80
Total 80

• 3rd Quartile
Here ¾N=60. So 3rd quartile lies in 24-26 class


Lower boundary of the class where median lies is ().

• 3rd Decile
Here 3N/10= 24. So 3rd quartile lies in 18-20 class.

Lower boundary of the class where median lies is ().

• 68th Percentile
Here 68N/100= 54.4. So 3rd quartile lies in 24-26 class.

Example 2:
For the following distribution find:
(i) the median.
(ii) the first quartile.
(iii) the third quartile.
(iv) the 10th percentile.
(v) the 6th decile.

Class f
3-6
6-9
9-12
12-15
15-18
4
10
12
9
5

Solution:

Class boundary f c.f
3-6
6-9
9-12
12-15
15-18
4
10
12
9
5
4
14
26
35
40
N=40

(i) median =½Nth observation
=20th observation, which lies in the class 9—12.
⇒ median class = 9—12. Therefore,


(ii)
Q1Nth observation
=10th observation, which lies in the class 6—9.
Q1 c1ass = 6—9. Therefore,

(iii) Q3N th observation
=30th observation, which lies in the Class 12—15.
Q3 class = 12—15. Therefore,

(iv)
P10=10N/100th observation
= 4th observation, which lies in the class 3—6.
P10 c1ass = 3—6. Therefore,

(v)
D6=6N/10th observation
= 24th observation, which lies in the Class 9—12.
D6 C1ass = 9—12. Therefore,

Example 3:
The following is the frequency distribution of 30 automobiles tested for fuel efiiciency.

Kilometer/litre Frequency
8-12
13-17
18-22
23-27
28-32
3
5
15
5
2

Construct a cumulative percentile graph and hence find:
(a) the median.
(b) the percentile rank of 17.5.
(c) the lower quartile.
(d) the 7th decile.
(e) the 10th percentile.
Solution:


(a) median:10 kilometer/litre.
(b) percentile rank of 17.5=28%.
(c) Q1=16.6 kilometer/litre.
(d) D7=22 kilometer/litre.
(e) P10=12 kilometer/litre.

Percentiles:
Percentiles are the values of the variable that divide a set of observations into 100 equal parts. Each set of observations has 99 percentiles and are denoted by P1, P2, . . . , P99.

The kth percentile, Pk is a value such that k% of the observations are smaller than or equal to Pk and (100-k)% of the observations are larger than Pk. For example, if a value is located at the 80th percentile, it means that 80% of the values that fall below the value and 20% of the values fall above it.

Notes:
(i) A percentile is a value in the data set.
(ii) A percentile rank of a given value is a percent that indicates the percentage of data is smaller than the value.

Deciles:
Deciles are the values of the variable that divide a set of observations into 10 equal parts. Each set of observations has 9 deciles are denoted by D1, D2, …, D9.

The first decile D1 is a value in the data set that 10% of the values fall below D1 and 90% of the values fall above D1.

Similarly, the second decile D2 is a value in the data set that 20% of the values fall below D2 and 80% of the values fall above D2 and so on.

Note: The first decile and tenth percentile are the same i.e. D1=P10.
Sirnilarly, D2=P20, …, D3=P30, D9=P90.

Quartiles:
Quartiles are the values of the variable that divide a set of observations into 4 equal parts. Each set of observations has 3 quartiles and they are denoted by Q1, Q2 and Q3.

The first quartile Q1 is a value in the data set that 25% of the values fall below Q1 and 75% of the values fall above Q1.

The second quartile Q2 is a value in the data set that 50% of the values fall below Q2 and 50% of the values fall above Q2.

The third quartile Q3 is a value in the data set that 75% of the values fall below Q3 and 25% of the values fall above Q3.

Note: Q1=P25, Q2=P50, Q3=P75.

Median: The 50th percentile, 5th decile and second quartile of a distribution are equal to the same value and are referred to as the median. That is
median=Q2=D5=P50.

Percentile graph:
The percentile graph is similar to ogive. To construct a percentile graph the upper bound- aries are plotted on the x—axis and cumulative percentage are plotted on the y—axis. Using a percentile graph one can find the approximate percentile rank for a given value and find the approximate value of data for a given percentile rank.
Let’s read post Drawing Box-and-Whisker Plots (Boxplots).

RELATED POSTs

Leave a Reply

Your email address will not be published.