Calculation of Quartiles, Deciles & Percentiles from Grouped Data
• Quartiles

ℓ= 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

ℓ= 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

ℓ= 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 (ℓ).

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

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

• 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)
Q₁=¼Nth observation
=10th observation, which lies in the class 6—9.
⇒Q₁ c1ass = 6—9. Therefore,

(iii) Q₃=¾N 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, P_k is a value such that k% of the observations are smaller than or equal to P_k and (100-k)% of the observations are larger than P_k. 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).