Question

How is it possible Quartile 1, Quartile 2, and Quartile 3 to all be numbers that are not listed in your data set? Can you give an example for this?

Answer 1

Consider a set of four values. Let's go for the set {1,2,3,4}

The middle-most number is a tie between 2 and 3. The midpoint of which is (2+3)/2 = 5/2 = 2.5

median = 2.5

The median is another name for quartile 2 (aka Q2)

This value of 2.5 is not in the set {1,2,3,4}

To find the quartiles, we split the data into two halves: lower and upper

• L = lower set = values smaller than median
• U = upper set = values larger than median

Then,

• L = {1,2}
• U = {3,4}

The median of L and U is 1.5 and 3.5 respectively.

This then leads to Q1 = 1.5 and Q3 = 3.5

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Here's the five number summary of the set {1,2,3,4}

• min = 1
• Q1 = 1.5
• Q2 = median = 2.5
• Q3 = 3.5
• max = 4

We see that Q1 = 1.5, Q2 = 2.5, and Q3 = 3.5 are values not found in the set {1,2,3,4}

The min and max on the other hand will always be found in the set. This is because we simply look for the smallest and largest items. We don't do any kind of math to get the min and max. The most work we'd need to do is sort the values from smallest to largest.