how to calculate interquartile range

Question

The interquartile range (IQR) is a statistic used to measure the spread of data by focusing on the middle 50% of the values.

Answer 1

Here's how to calculate the IQR:

Steps:

1. Order your data set: Arrange your data points from the smallest value to the largest value.
2. Find the median (Q2):
   • If you have an odd number of data points, the median is the middle value.
   • If you have an even number of data points, the median is the average of the two middle values.
3. Find the first quartile (Q1):
   • This represents the middle value of the lower half of your data set.
   • If you have an odd number of data points, Q1 is the value exactly halfway between the first element (smallest value) and the median (Q2).
   • If you have an even number of data points, Q1 is the median of the lower half of the data set (excluding the median itself).
4. Find the third quartile (Q3):
   • This represents the middle value of the upper half of your data set.
   • If you have an odd number of data points, Q3 is the value exactly halfway between the median (Q2) and the last element (largest value).
   • If you have an even number of data points, Q3 is the median of the upper half of the data set (excluding the median itself).
5. Calculate the IQR:
   • IQR = Q3 - Q1

Formula:

The IQR can also be calculated using the formula:

IQR = Q3 - Q1

where:

• Q3 is the third quartile (upper quartile)
• Q1 is the first quartile (lower quartile)

Example:

Let's say you have the following data set:

{2, 5, 7, 9, 11, 13, 15, 17}

1. Ordered data set: {2, 5, 7, 9, 11, 13, 15, 17}
2. Median (Q2): Since we have an even number of data points, the median is the average of the two middle values ((9 + 11) / 2) = 10.
3. First quartile (Q1): Q1 is the median of the lower half (excluding the median): {2, 5, 7, 9}. So, Q1 = 7.
4. Third quartile (Q3): Q3 is the median of the upper half (excluding the median): {11, 13, 15, 17}. So, Q3 = 13.
5. IQR: IQR = Q3 - Q1 = 13 - 7 = 6.

Therefore, the interquartile range (IQR) for this data set is 6.

Interpretation:

The IQR tells you that the middle 50% of the data points fall within a range of 6 units (from 7 to 13). In other words, the IQR is a measure of variability within the central portion of your data set.