Interquartile Range Formula

The interquartile range (IQR), typically demonstrates the middle 50% of a data set. In order to calculate it, you need to first arrange your data points in order from the lowest to the greatest, then identify your 1^st and 3^rd quartile positions by using the IQR formula  (N+1)/4 and 3 × (N+1)/4 respectively, where N represents the number of points in the data set. Lastly, subtract the 1^st quartile from the 3^rd quartile to find out the interquartile range for the data set. Simply, an IQR in maths is a computation of variability, based on dividing a data set into quartiles.

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Different Interquartile Formulas

The interquartile range can be calculated using different formulas. The values that divide each part are called the 1^st, 2^nd and the 3^rd quartiles; and they are signified by Q1, Q2, and Q3, respectively.

Q1 is the “centermost” value in the 1^st half of the rank-arranged set.

Q2 is the value of the median in the data set.

Q3 is the “centermost” value in the 2^nd half of the rank-arranged set.

With this equation, the formula for the interquartile range is as below:-

IQR = Q3 − Q1

Where,

IQR = Interquartile range

Q1 = 1st quartile

Q3 = 3rd quartile

Further, Q1 can also be calculated by using the following formula

Q1= \[\left \{ \frac{\left ( n+1 \right )}{4} \right \}^{th}\] term

Similarly, Q3 can also be calculated by using the following formula:

Q3 = \[\left \{ 3\frac{\left ( n+1 \right )}{4} \right \}^{th}\] term

In these circumstances, if the values are not the whole number, we need to take the average of the elements at the immediate neighbouring integer positions as shown in the figure below:

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IQR as a Test for Normal Distribution

We can also use the IQR formula with the mean and standard deviation in order to test whether or not a population experiences a normal distribution. The formula to find out if or not a population is normally distributed is as mentioned below:

Q1 = μ – (.675)σ

Q3 = μ + (.675)σ

Here, Q1 refers to the 1^st quartile, Q3 is the 3^rd quartile, σ represents the standard deviation and μ is the mean. For the purpose of stating whether a population is normally distributed, simplify for the values of Q1 and Q3 and then compare the outcomes. If the outcomes of the values of 1^st or 3^rd quartiles calculated this way and the previous way match exactly, then the population is normally distributed.

Advantages and Disadvantages of IQR

The interquartile range carries an exceptional advantage of being able to determine and eradicate deviation on both ends of a data set. IQR is a more effective tool for data analysis than the mean or median of a data set. An interquartile range also makes for an outstanding measure of variation in situations of skewed data distribution. The method of calculating IQR can be functional for grouped data sets, so long as you use a cumulative frequency distribution to order your data points.                                          

The IQR formula for grouped data is just the same as the non-grouped data, with the interquartile range being equal to the value of the 1^st quartile subtracted from the value of the 3^rd quartile.                                                                                                             

On the other hand, it has some disadvantages in comparison to standard deviation. These comprise less susceptibility to several extreme scores and a sampling consistency that is not as powerful as standard deviation.

Solved Examples for Interquartile Range Formula

Example1: Solve the interquartile formula for an odd set of numbers:-

1, 51, 9, 21, 27, 11, 13, 15, 5, 33, 19

Solution:

Simply follow a step-by-step process and easily solve the interquartile range equation.

Step 1: Arrange all the numbers in order.

1, 5, 9, 11, 13, 15, 19, 21, 27, 33, 51

Step 2: Find out the median.

(1, 5, 9, 11, 13) 15 (19, 21, 27, 33, 51)

Step 3: Put the parentheses around the numbers above and below the median.

Not compulsorily statistical, but it makes Q1 and Q3 easier to identify.

(1, 5, 9, 11, 13) 15 (19, 21, 27, 33, 51)

Step 4: Find Q1 and Q3

Assume Q1 as a median in the lower half of the data set and think of Q3 as a median for the upper half of data, with which you get

(1, 5, 9, 11, 13) 15 (19, 21, 27, 33, 51)

Q1 = 9

Q3 = 27

Step 5: Subtract Q1 from Q3 to determine the interquartile range.

27 – 9 = 18

Hence, IQR = 18

Example2: Find out the interquartile range for the first 10 numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Solution:

The total number of terms n = 10.

So, the data set is divided into two parts: 2, 4, 6, 8, 10 and 12, 14, 16, 18, 20

Q1 = Median of first part = 6

Q3 = Median of second part = 16

Now, use the formula for interquartile range i.e., IQR = Q3 – Q1

IQR = 16 – 6 = 10

Conclusion

So, by now you must have understood the meaning and significance of the interquartile range. Moving forward you shall be able to apply the formulas and find out the interquartile range for any given set of data and also comment on the normality of the distribution of the data by observing the results.