For a set of scores, will the interquartile range always be
less than the range? Explain your answer with an example.
use range 3.5
No, but the interquartile range will be always less than or equal to the range.
For example:
10 10 14 16 18 20 20
Here the range is 20-10=10
The interquartile range is also 20-10=10.
For practical sized distributions, it is quite unlikely that equality holds, but it is a possibility.
To determine whether the interquartile range will always be less than the range for a set of scores, let's first understand what each of these measures represents.
The range is the difference between the highest and lowest values in a dataset. It gives us an idea of the spread or variability of the data. To calculate the range, subtract the minimum value from the maximum value.
The interquartile range (IQR) represents the spread of the middle 50% of the data. It is calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1). Quartiles divide the data into four equal parts, where Q1 separates the lowest 25% of scores from the upper 75% and Q3 separates the lowest 75% from the upper 25%.
Now, let's see an example to understand if the IQR will always be less than the range:
Consider the following set of scores:
{2, 5, 7, 8, 9, 12, 15}
To find the range, we subtract the minimum value (2) from the maximum value (15):
Range = 15 - 2 = 13
To find the quartiles, we first need to order the data in ascending order:
2, 5, 7, 8, 9, 12, 15
Q1 is the median of the lower half, which is (2, 5, 7). Q1 = 5.
Q3 is the median of the upper half, which is (9, 12, 15). Q3 = 12.
Now, let's calculate the IQR by subtracting Q1 from Q3:
IQR = Q3 - Q1 = 12 - 5 = 7
In this example, the range is 13, and the IQR is 7. Therefore, we can conclude that the interquartile range (IQR) is indeed less than the range for this set of scores.
However, it is important to note that this may not always be the case for all datasets. Depending on the distribution of the data, there may be instances where the IQR is equal to or greater than the range.