A researcher wants to know what people in Melbourne think about football. She asks ten people to rate their attitude towards football on a scale from 0 (don't like football at all) to 10 (like football a lot). The ratings are as follows: 10, 9, 6, 9, 5, 5, 1, 5, 10, 6 What is the interquartile range (IQR) of this data?

Question

A researcher wants to know what people in Melbourne think about football. She asks ten people to rate their attitude towards football on a scale from 0 (don't like football at all) to 10 (like football a lot). The ratings are as follows: 10, 9, 6, 9, 5, 5, 1, 5, 10, 6  What is the interquartile range (IQR) of this data?

BlackTom AI · Accepted Answer

We begin by organizing the data in ascending order to make the quartiles easier to identify. The given ratings are: 10, 9, 6, 9, 5, 5, 1, 5, 10, 6. When sorted, this becomes: 1, 5, 5, 5, 6, 6, 9, 9, 10, 10.

Next, determine the quartiles. Since there are 10 data points, the lower half consists of the first 5 values: 1, 5, 5, 5, 6. The median of this lower half is the third value, which is 5. Therefore, Q1 = 5.

The upper half consists of the last 5 values: 6, 9, 9, 10, 10. The median of this upper half is the third value, which is 9. Therefore, Q3 = 9.

The interquartile range (IQR) is the difference between Q3 and Q1: IQR = Q3 − Q1 = 9 − 5 = 4.

Now, examine the answer options. Option a lists 6 and 6, which does not match either quartile. Option b lists 6 and 9, which would imply Q1 = 6 and Q3 = 9, not matching the calculated Q1. Option c lists 5 and 6, which again does not align with the quartile values found. Option d lists 5 and 9, which exactly correspond to Q1 and Q3 values derived from the data, aligning with the standard method for computing quartiles in a dataset of size 10.

Thus, the pair 5 and 9 represents the first and third quartiles identified from the data, which is why this option is the correct choice among the provided alternatives. The IQR itself, computed as 4, is a separate summary measure that follows from these quartile positions.

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