Interquartile Range Formula

Interquartile Range Calculation

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 lowest to greatest, then identify your 1st and 3rd 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 1st quartile from the 3rd quartile to find out the interquartile range for the data set. Simply, an iqr in math is a computation of variability, based on dividing a data set into quartiles.

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Different Interquartile Formula

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

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

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

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

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



IQR = Interquartile range

Q1 = 1st quartile

Q3 = 3rd quartile

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

Q1=(n+14)th term, whereas

Q3 can also be calculated by using the following formula:

Q3 = (3(n+1)th / 4 th term

In these circumstances, if the values are not the whole number, we need to round them up to the nearest integer.

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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:

Q1 – (σ z1) + X

Q3 – (σ z3) + X

Where Q1 refers to the 1st quartile, Q3 is the 3rd quartile, σ represents the standard deviation, z refers to the z-score (standard score) and X is the mean. For the purpose of stating whether a population is normally distributed, simplify both the equations and then compare the outcomes. If there is a substantial difference between the outcomes and the 1st or 3rd quartiles, then the population is NOT 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 with non-grouped data, with interquartile range being equal to the value of the 1st quartile subtracted from the value of the 3rd quartile.                                                                                                             

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

Solved Examples For Interquartile Range Formula


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

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


Simply follow a step-by=step process and easily solve 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


Find out the interquartile range for 1st ten odd numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20?


Total number of terms n = 10.

Median = (n2)the term+(n2+1) th term2

Median = 9+11 / 2 = 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

FAQ (Frequently Asked Questions)

Q1. What is Interquartile Range and Its Uses?

The interquartile range, commonly abbreviated as the IQR, illustrate the range from the 25th percentile to the 75th percentile of any given data set. The IQR can be used to identify what the average range of performance on a test would be. IQR can also be useful to:

  • determine how much money the average employee in an organization makes monthly

  • spot the dispersion range instead of only a single number

  • Observe where most people's scores on a specific test fall

Q2. What is Meant By Dispersion?

The statistical calculation which enables us to tell how far apart your data is spread. There are a number of ways that are available to compute dispersion, but two of the best are considered to be the range and the average deviation. The range is the difference between the lowest and highest value of your statistics. While the average deviation sees your mean and how each data point is differentiated from the mean.   As an example, think of your data's lowest number is 2 and the highest is 7. Subtracting the lowest value from the highest value, you will get the range, which equals 5.