Quartiles, Deciles & Percentiles Lecture 10

Introduction

To 

Quartiles, Deciles & Percentiles 

Lecture 10 

Quartiles

A set of three values that split a sorted dataset into four equal parts, each part having an equal number of observations. These partitioning values are represented by Q1, Q2, and Q3. 

In discrete data, quartiles can be calculated as:

In grouped data, quartiles can be obtained as:

Where:

L1 and L2 are the lower and upper class boundaries, and c is the cumulative frequency of the preceding class.

Example 3.35: Calculate lower and upper quartiles for the following data.

4

9

7

6

15

11

18

24

29

25

27

16

Solution: Arrange the data in ascending order of magnitude and assign ranks.

Rank	1	2	3	4	5	6	7	8	9	10	11	12
Data	4	6	7	9	11	15	16	18	24	25	27	29

The third item is 7.

So

Q1 = 7 

The 9^th item is 24.

So

Q3 = 24

Example 3.36: Find the quartiles of the marks obtained by first years in the annual examination.

497

495

480

465

440

490

443

398

390

365

400

412

432

416

389

Solution:

Rank	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Data	365	389	390	398	400	412	416	432	440	443	465	480	480	495	497

Round up 3.75 to the 4th item.

So

Q1 = 398

 

Round up 11.25 to the 12th item.

So

Q3 = 480

Example 3.37: Find the lower and upper quartiles for the following frequency distribution.

Class	40 – 50	50- 60	60–70	70–80	80–90	90 - 100	100–110
Frequency	20	17	44	57	68	55	34

Solution:

Class	Frequency	Cumulative frequency
40 – 50	20	20
50- 60	17	37
60–70	44	81
70–80	57	138
80–90	68	206
90–100	55	261
100 - 110	34	295
	295

For Q1, we divided the total observation by 4.

We select the class that lies against 73.75; 73.75 is not available in the cumulative frequency column. We select the class lies against 81, which is 60–70.

L1 = 60, L2 = 70, f = 44, c = 37

For Q3, we divided the 3 times of total observation by 4.

Which is lies against 90–100,

L1 = 90, L2 = 100, f = 55, c = 206

DECILES

A set of nine values that split a sorted dataset into ten equal parts, each part having an equal number of observations. These partitioning values are represented by Di.

Where:

L1 and L2 are the lower and upper class boundaries, and c is the cumulative frequency of the preceding class.

PERCENTILES

A set of ninty-nine values that split a sorted dataset into hundred equal parts, each part having an equal number of observations. These partitioning values are represented by Pi.

Where:

L1 and L2 are the lower and upper class boundaries, and c is the cumulative frequency of the preceding class.

Example 3.38: Find the 5^th, 9^th deciles, and 50^th, 75^th percentiles for the following frequency distribution.

Group	10 – 20	20- 30	30 – 40	40 – 50	50- 60	60–70	70 - 80
No, of person	20	35	78	60	75	84	98
Group	80–90	90 - 100	100 - 110	110 - 120
No, of person	125	87	99	102

Solution:

Group	Frequency	Cumulative frequency
10 – 20	20	20
20- 30	35	55
30 – 40	78	133
40 – 50	60	193
50- 60	75	268
60–70	84	352
70–80	98	450
80–90	125	575
90–100	87	662
100–110	99	761
110 – 120	102	863
	863

For 5^th decile:

Which lies against 70-80 group, so our  L1 = 70, L2 = 80, f = 98, c = 352

For 9^th decile:

Which lies against 110-120 group, so our  L1 = 110, L2 = 120, f = 102, c = 761

For 50^th percentiles:

Which lies against 70-80 group, so our  L1 = 70, L2 = 80, f = 98, c = 352

For 75^th percentiles:

Which lies against 90-100 group, so our  L1 = 90, L2 = 100, f = 87, c = 575

• Consult: MCQ's & Practice Questions on Measure of Central Tendency

• Read: Measure of Dispersion