Here we will learn how to find the quartiles for arrayed data.

**Step I:** Arrange the grouped data in ascending order and from
a frequency table.

**Step II:** Prepare a cumulative-frequency table of the data.

**Step III:** (i) For Q_{1}: Select the cumulative
frequency that is just greater than \(\frac{N}{4}\), where N is the total
number of observations. The variate whose cumulative frequency is the selected
cumulative frequency, is Q_{1}.

(ii) For Q_{3}: Select the cumulative frequency that is just greater than \(\frac{3N}{4}\), where N is the total number of observations. The variate whose cumulative frequency is the selected cumulative frequency, is Q_{3}.

**Note:** In case \(\frac{N}{4}\) or \(\frac{3N}{4}\) is equal to the cumulative frequency of the variate, take the mean of the variate and the next variate.

Solved Examples on Find the Quartiles for Arrayed Data:

**1.** Find the lower and upper quartiles of the following
distribution.

**Variate**

2

4

6

8

10

**Frequency**

3

2

5

4

2

**Solution:**

The cumulative-frequency table of the data is as below.

2 4 6 8 10 |
3 2 5 4 2 N = 16 |
3 5 10 14 16 |

Here, \(\frac{N}{4}\) = \(\frac{16}{4}\) = 4.

The cumulative frequency just greater than 4 is 5.

The variate whose cumulative frequency is 5 is 4.

So, Q_{1} = 4.

Next, \(\frac{3N}{4}\) = \(\frac{3 × 16}{4}\) = \(\frac{48}{4}\) = 12.

The cumulative frequency just greater than 12 is 14.

The variate whose cumulative frequency is 14 is 8.

**2.** Marks obtained by 70 students in an examination are given below.

Find the upper quartile.

**Marks**

25

50

35

65

45

70

**Number of Students**

6

15

12

10

18

9

**Solution:**

Arrange the data in ascending order, the cumulative-frequency table is constructed as below.

**Marks**

25

35

45

50

65

70

**Frequency**

6

12

18

15

10

9

**Cumulative Frequency**

6

18

36

51

61

70

Here, \(\frac{N}{4}\) = \(\frac{70}{4}\) = \(\frac{35}{2}\) = 17.5.

Cumulative frequency just greater than 17.5 is 18.

The variate whose cumulative frequency is 18, is 35.

So, Q_{1} = 35.

Again, \(\frac{3N}{4}\) = \(\frac{3 × 70}{4}\) = \(\frac{105}{4}\) = 52.5.

Cumulative frequency just greater than 52.5 is 61.

The variate whose cumulative frequency is 61, is 65.

Therefore, Q_{3} = 65.

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