# Boxplots and bar graphs

I'm studying for the GRE and came across several questions that I was unable to answer in a practice booklet, even after looking at the answer and trying to work backwards, and searching google and other sites for helpful hints. I think I am missing a fundamental understanding or useful heuristic for solving many of these problems; any advice would be greatly appreciated as my exam is Monday (Aug 1st).

1. Eight hundred insects were weighed, and the resulting measurements, in milligrams, are summarized in the boxplot below.

If the 80th percentile of the measurements is 130 milligrams, about how many measurements are between 126 milligrams and 130 milligrams?

I calculated the range (41), the quartiles(Q1=114, Q2=118, Q3=126), and the IQR (12), but I'm confused about the question. If the 80th percentile (so 80% of the measurements?) is at 130, then 640 are within this percentile. I'm not sure if this is true and even if it is, where to go from here. Each quartile is 25% of the data, correct? So from 126 to 146 must contain 25%, or 200 measurements? (1-.8)(200) = 40, but conceptually I'm lacking what that means.

2. This question refers to the following graph:

(a) In 2003 the family used a total of 49 percent of its gross annual income for two of the categories listed. What was the total amount of the family’s income used for those same categories in 2004 ?

My confusion lies with the fact that I can't seem to find any combination of two categories that add up to 49%. Plus, it seems that the total % expenditure in each year is 101%. The chart is not 100% accurately drawn, but even being very liberal in measuring there seems to be a discrepancy.
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For the first one, 5% of the data lies within 126 to 130.. so .05*800=40. – mathmath8128 Jul 29 '11 at 19:51
For the second one, just eyeballing it seems that the family used 49% of its income in savings and mortgage taxes... they used (27+12)=39 so 39%*45,000$=17550$ of their income in those categories in 2004... – mathmath8128 Jul 29 '11 at 19:57
I agree with Abraham. To clarify, there's 25% in savings and 24% in mortgage expenditures. (A trick for finding them: the two numbers sum to 49, so one of them is at least 49/2 = 24.5, so you really only need to look at the large expenditures.) – Michael Lugo Jul 29 '11 at 20:10

For the first problem you’ve shown that you have the tools that you need; you just haven’t recognized how they apply. To say that the $80$th percentile is at $130$ mg is to say that $80$% of the measurements are at or below $130$ mg -- to the left of the $130$ mg mark in the picture. $Q3$ gives you the $75$th percentile: $75$% of the measurements are at or below $126$ mg, or to the left of the $126$ mg mark in the picture. The measurements lying between $126$ and $130$ mg are inside the leftmost $80$% but outside the leftmost $75$%, so they make up $5$% of the total, and $5$% of $800$ measurements is $40$ measurements.
Alternatively, you could work from your observation that $640$ of the measurements are at or below $130$ mg. (That’s not $640$ measurements within the $80$th percentile, however: it’s $640$ measurements within the first $80$ percentiles altogether.) In exactly the same way you can calculate that $75$%, or $600$, of the $800$ measurements are at or below $126$ mg. Thus, $640-600 = 40$ measurements must lie between $126$ and $130$ mg.
In the second question, Savings ($25$%) plus Mortgage, Insurance, & Property Taxes ($24$%) amounted to $49$% in 2003; this doesn’t seem to require any generosity of interpretation of the graph. In 2004 those items come to $12+27=39$% of $\$45,000$, or$\$17,550$.