The statistics below provides a summary of IQ scores of 100 children
Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110
About 50 of the children in this sample have IQ scores that are
A. less than 84
B. less than 110
C. between 84 and 110
D. between 64 and 130
E. more than 100
I'm not sure how to start solving the problem. How would I start?
Let's look at each of the possibilities:
A: less than 84. Taking a look at our summary statistics, we see that 84 is actually the First Quartile or $Q_{1}$, which is by definition the 25th percentile. This means that 25% of our data points lie below 84. Since we have 100 data points, we expect 25 students to have IQ below 84.
B: less than 110. Again, we look at our statistics. We note that 110 is actually $Q_{3}$ or the 75th percentile. So we expect 75 students to have IQ less than 110.
C: between 84 and 110. This one looks promising. If 75% of IQs lie below 110 and 25% of IQs lie below 84, then we must have 50% of IQs between 84 and 110.
D: between 64 and 130. Hmm, not sure where those numbers are coming from. Would you rule this one out?
E: more than 100. This is a good looking answer, but there is a problem with it. If the distribution of scores were perfectly symmetric, then we would expect 50 students to lie above the mean and 50 below. However, I can see from the information given that the distribution is not symmetric but skewed since $(Q_{3} - Q_{2}) \neq (Q_{2}-Q_{1})$ where $Q_{2}$ is the 50th percentile, aka the median.
Basically, the question relies on you knowing the definition of first and third quartiles.
The answer I would choose is C.