The three quartiles are numerical values that divide the given data into four equal parts. The lower quartile, the median, and the upper quartile divide the data at the one-fourth, one-half, and three-fourths values respectively. They are denoted by the symbols Q_{1}, Q_{2}, and Q_{3}. In this article, we explain how to calculate the value of the quartile for grouped frequency distributions.

**Formula for Calculating Quartiles for Grouped Data:**

The quartiles can be calculated using the formula, Q_i = L +\frac{h}{f}\left(\frac{iN}{4} - c.f \right)

The class containing Q_{i} is the class whose cumulative frequency is just greater than (iN/4). The notations are:

N = Total Frequency.

h = class size.

L = lower limit of the class containing Q_{i}.

f = frequency of the class containing Q_{i}.

c.f = cumulative frequency of the class interval preceding the interval containing Q_{i}.

**How to find Quartiles for Grouped Data:**

- Find the value of N, by adding up all the frequencies.
- Calculate the less than cumulative frequencies for all class intervals.
- Calculate the value of (iN/4) in order to find the i
^{th}quartile (i = 1, 2 or 3). - The quartile is contained in the class whose cumulative frequency is just greater than (iN/4).
- Calculate the value of the i
^{th}quartile by using the formula given above.

**Example 1:**

Calculate the upper quartile Q_{3} for the following grouped frequency distribution:

Class Interval | Frequency |

0-10 | 4 |

10-20 | 7 |

20-30 | 2 |

30-40 | 5 |

40-50 | 12 |

** Solution**: Since we want to calculate Q

_{3}we have that i=3 in the above formula.

We first calculate the less than cumulative frequencies by adding up the current and preceding frequencies of a class interval.

Class Interval | Frequency | Cumulative Frequency |

0-10 | 4 | 4 |

10-20 | 7 | 4+7 = 11 |

20-30 | 2 | 4+7+2 = 13 |

30-40 | 5 | 4+7+2+5 = 18 |

40-50 | 2 | 4+7+2+5+2 = 20 |

N = 20 |

Now, (iN/4) = (3*20/4) = 15.

The value of cumulative frequency just greater than 15 is 18. Therefore, Q_{3} lies in the class interval 30-40. Here, L = 30, f = 5, h =10 and c.f = 13. Substituting all these values in the above formula we get, Q_3 = L +\frac{h}{f}\left(\frac{3N}{4} - c.f \right)Q_3 = 30 +\frac{10}{5}\left(\frac{3*20}{4} - 13 \right)Q_3 = 30 +2*(15-13) = 30 + 4 =34

**Example 2:**

Calculate the lower quartile Q_{1} for the following grouped frequency distribution:

Class Interval | Frequency |

0-5 | 5 |

5-10 | 10 |

10-15 | 7 |

15-20 | 3 |

20-25 | 3 |

** Solution**: Since we want to calculate Q

_{1}we have that i=1 in the above formula.

Class Interval | Frequency | Cumulative Frequency |

0-5 | 5 | 5 |

5-10 | 10 | 15 |

10-15 | 7 | 22 |

15-20 | 3 | 25 |

20-25 | 3 | 28 |

N = 28 |

Now, (iN/4) = (1*28/4) = 7.

The value of cumulative frequency just greater than 7 is 15. Therefore, Q_{3} lies in the class interval 5-10. Here, L = 5, f = 10, h =5 and c.f = 5. Substituting all these values in the above formula we get, Q_1 = L +\frac{h}{f}\left(\frac{N}{4} - c.f \right)Q_1 = 5 +\frac{5}{10}\left(\frac{28}{4} - 5 \right)Q_1 = 5 +\frac{1}{2}*(7-5) = 5+1 =6

Therefore the value of the lower quartile is 6.

**Calculating Quartiles Graphically:**

- Draw a cumulative frequency curve by plotting the values of observations or class intervals along the horizontal scale (i.e. x-axis) and cumulative frequency along the vertical scale (i.e., y-axis).
- Determine the quartile value, that is, the value of (in/4)
^{th}observation (i = 1, 2, 3), where n is the total number of observations in the data set. - Locate this value on the y-axis and from this point draw a line parallel to the x-axis meeting the ogive at a point, say P. Draw a perpendicular on the x-axis from P and it meets the x-axis at the point, say M. Then M is the required value of the quartiles.

**Example:**

Consider the data given in the form of a grouped frequency distribution as follows:

Class Interval | Frequency |

Below 875 | 69 |

875 – 950 | 167 |

950 –1025 | 207 |

1025 –1100 | 65 |

1100 – 1175 | 58 |

1175 – 1250 | 24 |

1250 – 1325 | 10 |

We first calculate the cumulative frequencies by adding each frequency value with the preceding frequencies.

Class Interval | Frequency | Cumulative Frequency |

Below 875 | 69 | 69 |

875 – 950 | 167 | 236 |

950 –1025 | 207 | 443 |

1025 –1100 | 65 | 508 |

1100 – 1175 | 58 | 566 |

1175 – 1250 | 24 | 590 |

1250 – 1325 | 10 | 600 |

N = 600 |

Let us calculate the value of the second quartile, that is, the median. We have that,

Second Quartile = (iN/4)th term = (2*600/4)th term = 300^{th} term.

We now construct the cumulative frequency curve (ogive) and locate the second quartile as follows.

Therefore the value of the second quartile is 973.2