Calculating and understanding the Interquartile Range (IQR) is an important skill for Year 10 students studying math. The IQR helps us see how data is spread out, especially the middle half of the data. This can be super helpful when we're looking at different sets of data and want to compare them.

What is the Interquartile Range (IQR)?

The Interquartile Range is the difference between two parts of a data set: the upper quartile (Q3) and the lower quartile (Q1). It tells us how spread out the middle 50% of our data is around the average.

To make it easier to understand, let’s think about how to find the IQR step by step:

1. Order the Data: First, put your data in order from the smallest number to the largest.

2. Find the Median (Q2): The median is the middle number. If there is an odd number of values, it’s the exact middle one. If there is an even number, it’s the average of the two middle numbers.

3. Determine Q1 and Q3:

   • Q1 (First Quartile): This is the median of the lower half of the data (the values lower than the overall median).
   • Q3 (Third Quartile): This is the median of the upper half of the data (the values higher than the overall median).

4. Calculate the IQR: You can find the IQR with this simple formula: $\text{IQR} = Q3 - Q1$

Example Calculation

Let’s look at an example using exam scores:

30, 45, 44, 55, 60, 75, 80, 90

Step 1: Order the data (it's already ordered here).

Step 2: Find the median (Q2).

• There are 8 numbers, which is even. So, we find the average of the 4th and 5th numbers: $Q2 = \frac{55 + 60}{2} = 57.5$

Step 3: Find Q1 and Q3.

• For the lower half: 30, 44, 45, 55
  • The median (Q1) is: $Q1 = \frac{44 + 45}{2} = 44.5$
• For the upper half: 60, 75, 80, 90
  • The median (Q3) is: $Q3 = \frac{75 + 80}{2} = 77.5$

Step 4: Calculate the IQR. $\text{IQR} = 77.5 - 44.5 = 33$

Interpreting the IQR

Now that we’ve calculated the IQR, what does it mean? An IQR of 33 shows how far apart the middle 50% of the exam scores are.

If the IQR is larger, it means the scores are more spread out. If it’s smaller, it means the scores are closer together.

It’s also important to compare IQRs across different groups. If one class has an IQR of 25 and another has an IQR of 40, the second class’s scores are more spread out, which tells us something different about their performance.

Understanding the IQR helps us see data more clearly. It allows us to make better decisions based on how the data points are grouped together.