For a set of scores, will the interquartile range always be

less than the range? Explain your answer with an example.

Check this explanation of interquartile range.

http://www.mathwords.com/i/interquartile_range.htm

What do you think?

Wow! This is hard!

IQR=less than median - more than median
Range= highest-lowest

2 5 6 9 12

Range=12-2=10
median=6
so
IQR=10.5-3.5=7

So I'm thinking yes?

The interquartile range is the difference between the first and the third quartiles, and is always less than or equal to the range.

In normally distributed data, it is less than the range.
Example 1:
48,60,65,80,85,92,99
The quartiles are shown in bold, namely Q1=60, Q2=80, and Q3=92.
The interquartile range is therefore 92-60=32.
The range is 99-48=51

Example 2:
65,65,65,70,75,89,89
The interquartile range is 89-65=24
The range is also 89-65=24.
This is an extreme example to illustrate that it is possible to have the interquartile range equal to the range, but highly unlikely in real life.

Thanks for the help!

To determine whether the interquartile range will always be less than the range for a set of scores, let's first understand what each measure represents.

1. Range: The range is calculated by subtracting the smallest value in the set from the largest value. It represents the spread or variability of the entire data set.

2. Interquartile Range (IQR): The IQR is a measure of statistical dispersion. It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). The first quartile represents the value below which 25% of the data falls, while the third quartile represents the value below which 75% of the data falls.

Now, if we have a set of scores where all the data points are the same, then the range and the interquartile range will be equal. This is because the largest and smallest values will be identical, and also Q1 and Q3 will be the same since 25% and 75% of data points are the same.

Example:
Consider the set of scores: {5, 5, 5, 5, 5}
Range = largest value - smallest value = 5 - 5 = 0
IQR = Q3 - Q1 = 5 - 5 = 0

In this example, both the range and interquartile range are zero, confirming that they are equal.

However, in most cases, especially with varied data sets, the range and interquartile range will not be equal. The range takes into account the extreme values, whereas the interquartile range focuses on the middle 50% of the data.

For example:
Consider the set of scores: {1, 2, 3, 6, 9}
Range = largest value - smallest value = 9 - 1 = 8
IQR = Q3 - Q1 = 6 - 2 = 4

In this example, the range is 8, while the interquartile range is 4. Therefore, the interquartile range is significantly less than the range.

In conclusion, the interquartile range will not always be less than the range. It depends on the spread and distribution of the data set.