Question

You have been tracking how much you spend on gas every month for several months and found that on average you spend about $60 per month. If you had to guess, what would you expect the IQR (Interquartile Range) of your gas spending to be? Would $1, $10 or $100 be most logical? Give statistical reasoning to back up your answer.

Answer 1

Here, the variable of interest is spending on gas every month.

The average amount spent every month is $60. Without loss of generality, we can assume that amount spent per months in normally distributed. So, the median amount spent per month will be approximately $60.

The Inter-quartile range (IQR) is defined as (Q3 - Q1)/2.

Now, the variable of interest is expected to have less variation.

If we assume IQR = 1, then, (Q3 - Q1) = 2

Since, Q2 = 60 and Q3 - Q2 = Q2 - Q1 for normal distribution i.e. any symmetric distribution, so, Q3 = 61, Q1 = 59.

This is quite unlikely as the variation is very less.

If we assune IQR = 100, then, (Q3 - Q1) = 200

Since, Q2 = 60 and Q3 - Q2 = Q2 - Q1 for normal distribution i.e. any symmetric distribution, so, Q3 = 160, Q1 = - 40, which is not possible as the 1st quantile can not be negative as the varaible takes only positive values.

If we assume, IQR = 10, then, (Q3 - Q1) = 20

Since, Q2 = 60, by same logic as above, Q3 = 70 and Q1 = 50, which seems more reasonable.

So, the value of IQR being $10 is most likely.