For a set of scores, will the interquartile range always be less than the range? Explain your answer with an example.

I know the range runs from the lowest to the highest score. What would be a good example to show this?

1+1=

To determine whether the interquartile range will always be less than the range for a set of scores, we need to understand what each of these measures represents.

The range is simply the difference between the highest and lowest scores in a data set. It provides information about the spread or variability of the data.

The interquartile range, on the other hand, measures the spread of the middle 50% of the scores. It is calculated by finding the difference between the upper quartile (75th percentile) and the lower quartile (25th percentile) of the data.

Now, let's consider an example to illustrate this concept. Suppose we have the following set of scores: 10, 15, 20, 25, 30, 35, 40, 45, 50.

First, we can sort these scores in ascending order: 10, 15, 20, 25, 30, 35, 40, 45, 50.

The lowest score in this data set is 10, and the highest score is 50. Therefore, the range is 50 - 10 = 40.

Next, we need to find the quartiles to calculate the interquartile range. The lower quartile (Q1) represents the score below which 25% of the data falls. In our example, Q1 is the median of the lower half of the data, which is (15 + 20) / 2 = 17.5.

Similarly, the upper quartile (Q3) represents the score below which 75% of the data falls. Q3 is the median of the upper half of the data, which is (35 + 40) / 2 = 37.5.

To calculate the interquartile range, we subtract Q1 from Q3: 37.5 - 17.5 = 20.

In this example, we can clearly see that the interquartile range (20) is less than the range (40).

However, it's important to note that this does not always hold true for every data set. There may be scenarios where the interquartile range is equal to or greater than the range. It ultimately depends on the distribution and spread of the data.