SAT Math: Statistics and Histograms

Last week’s post looked at Statistics and Frequency Tables. Today’s let’s look at a similar problem where the data is instead presented to us in a histogram. If you didn’t read last week’s post about frequency tables, you might want to start there first.

Let’s look at three problems based on data in a histogram.

The next three problems are based on the histogram below.

The library at the local college is trying to determine how frequently the library is used by students. They surveyed 25 students asking them “How many times did you use the library this week?” The results are summarized in the histogram below.

1.  What is the mean number of times the students used the library?

(A)  2.12
(B)  2.50

(C)  2.52
(D)  2.61
(E)  2.74

2.  What is the median number of times the students used the library?

(A)  1.5
(B)  2
(C)  2.5
(D)  3
(E)  3.5

3.  What is the mode of the data presented in the table?

(A)  1 only
(B)  2 only
(C)  3 only
(D)  Both 4 and 5
(E)  There is no mode for the data in the table.

To find the mean, we need to know the total number of times that the library was used. We can find that by multiplying the number of uses by the number of students for each bar of the histogram. For instance, there were 6 students who used the library 3 times this week, so that’s a total of 6(3) = 18 uses for that bar. The calculation would look like this:

0(2) + 1(4) + 2(7) + 3(6) + 4(3) + 5(3)
= 0 + 4 + 14 + 18 + 12 + 15
= 63 total uses of the library

Now divide the number of uses by the number of students: 63/25 = 2.52. The mean number of uses is 2.52 and the correct answer to the first question is Choice C.

To find the median, you could list all of the numbers in order and find the middle number taking into consideration that many of the numbers occur more than once. While it’s possible to do that for this problem, with 25 numbers it might take a while. Is there an easier way? If you divide 25 (the number of students) by 2, you get 12.5. Rounding that up to 13 tells you that the 13th number is the middle number or median.

(Note: If there are an even number of data points, you still divide by 2. Take that term and the next one and average them to get the median. For instance, suppose there are 40 data points. Dividing by 2 gets you 20. The median is the average of the 20th and 21st terms.)

So which term is the 13th in our list? Begin adding from the left-most bar of the histogram. There are 2 students who never visited the library and another 4 who visited once. That’s 6 students. Next, we see that 7 students visited the library twice. That brings us up to 13 students, so the 13th number in the list would be a 2. The median is 2 and the correct answer is Choice B.

The mode is the number that appears most frequently. What answer did the students give more than any other? From the histogram we can see that more students went to the library twice than any other number of times. The mode is 2 and the correct answer to the third question is Choice B.

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