# How to Calculate Quartiles? Formulae with Examples

Quartiles are the values that divide a list of numbers into quarters. In Statistics, “Quartile’ is one of the values of a variable that divides the distribution of the variable in to four groups having equal frequencies. Thus, Quartile divides the list of variables into following four categories:

• 25% of the values are less than or equal to Q1 (or 25% of the values are ≤ Q1)
• 25% of the values are between Q1 and Q2 (50% of the values are ≤ Q2)
• 25% of the values are between Q2 and Q3 (75% of the values are ≤ Q3)
• 25% of the values are greater than or equal to Q3

The Q1or lower quartile separates the bottom one-fourth of the data from the upper three-fourths of the data, the Q2 or middle quartile separates the bottom half from the top half, and the Q3 or upper quartile separates the bottom three-fourths of the data from the upper one-fourth of the data.

To understand quartiles, we must understand quantiles or percentiles because the calculation of quartile depends on the definition of percentile. The first Quartile (usually written as Q1) is the 25th percentile, the second Quartile (Q2), or 50th percentile is Median Value and the third Quartile (Q3) is similarly is the 75th percentile. Median represents the value of any set of observation where 50% of the values are either less and 50% is higher than the Median. It is the value at the center of a set of observation or it is the middle value of the data set.

## How lower Quartile is calculated

• Arrange the number of observation in ascending order
• If there is odd number (say n+1) in the set, one in the center [or (n+1) /2] is the median. If the number of the set is even, take two digits in the middle, add them together and divide the total by 2 to find the median.
• When the number of data in the set minus 1 is evenly divisible by 4, the lower quartile will be the median of the data set’s median and all the data to its left (the series should have been arranged in complete ascending order earlier). If the series is not divisible by 4, the lower quartile is simply the median of the numbers to the left of the data set’s median.

## An  QuaritleExample

We will use one example to make things clear. There is a set of twenty data in the following:

1, 2, 2, 3, 4, 6, 6, 7, 7, 7, 8, 11, 12, 15, 15, 15, 17, 17, 18, 20

Since the total number of data in the set is 20 which is even, we will first find out the mean. In the given series, median (Q2) is the mean of the tenth and eleventh values:

1, 2, 2, 3, 4, 6, 6, 7, 7, 7, M 8, 11, 12, 15, 15, 15, 17, 17, 18, 20

Therefore, Median is (7 + 8)/2 = 7.50 = Q2

Now we will consider the bottom half of the series to find out first Quartile (Q1):

The Median of this set will be between 5th and 6th place.

1, 2, 2, 3, 4, M 6, 6, 7, 7, 7

There the Median will be (4 + 6) / 2 = 5 = Q1

Similarly, we can find the third Quartile (Q3):

The median of the top half of the original data set is calculated in the same manner.

8, 11, 12, 15, 15, M 15, 17, 17, 18, 20

Median = (15+15) / 2 = 15 = Q3

We can find quartiles by using the following formulas also:

First quarter (Q1)

Q1 = X to the position [1 (n + 1)] / 4

Second quarter (Q2)

Q2 = X to the position [2 (n + 1)] / 4

Third quarter (Q3)

Q3 = X to the position [3 (n + 1)] / 4

[when n = number of terms in the series)

(a)   Application of the above formula when (n + 1)/4 is a whole number.

Example: 20, 12, 14, 43. 15, 47, 67

Answer: The series is arranged in ascending order as 12, 14, 15, 20,43,47,67

N or Number of Term is 7

By applying the above formula,

First Quartile or Q1=1 (n + 1)] / 4 = [1 (7 + 1)] / 4 = 2nd position

Second Quartile or Q2= [2 (n + 1)] / 4 = [2 (7 + 1)] / 4 = 4th position

Third Quartile or Q3 = [3 (n + 1)] / 4 = [3 (7 + 1)] / 4 = 6th position

Therefore, the series with their Quartile value is 12, 14 (Q1), 15, 20 (Q2), 43, 47(Q3), 67

(b)   Application of the above formula when (n + 1)/4 is not a whole number.

Example: 20, 12, 14, 42. 18, 47, 50, 67

Answer: The series is arranged in ascending order as 12, 14, 18, 20,42,47,50, 67

N or Number of Term is 8

First Quartile or Q1=1 (n + 1)] / 4 = [1 (8 + 1)] / 4 = 2.25th position

Second Quartile or Q2= [2 (n + 1)] / 4 = [2 (8 + 1)] / 4 = 4.5th position

Third Quartile or Q3 = [3 (n + 1)] / 4 = [3 (8 + 1)] / 4 = 6.75th position

The series with three Quartiles will be 12, 14, Q1, 18, 20, Q2, 42, 47, Q3, 50, 67

This time for fractional value of position, we have to interpolate the value in the following way to find the value of three Quartiles:

Q1 = 14 + {0.25 X (18 -14)} = 14 + 1= 15

** Repeat this process for Q2 and Q3

Q2 = 20 + {0.50 X (42 -20)} = 20 +11= 31

Q3 = 47 + {0.75 X (50 -47)} = 47 + 1 = 48