Minimum variance for a given IQR in a sample of 7

Question

for a sample of 7 elements it is given that its inter-quartile range $Q3-Q1$ equals $4$. I need to find the minimum value the variance can take. No other information about the sample is given.

Honestly, I don't know how to start. By definition we know that we will have $7-1=6$ in the denominator, but I have no idea how to use the given fact about IQR and find the solution.

Answer 1

Hints:

• You want all the values to be as close to the mean as possible so that the sum of the squares of their differences from the mean is as small as possible
• You want the second highest value to be $4$ more than the sixth highest value
• The sum of squares of the differences from the mean of the second highest and the sixth highest will be minimised when they are symmetric about the mean
• The highest value must be at least as big as the second highest and the seventh highest no higher than the sixth highest
• The other values can be equal to the mean, with their squares of the differences from the mean being $0$