What is the median in this gre question?

Question

This is gre preparation question of data interpretation, Distribution of test score among students(Score range -> total % of students)

0-65 -> 16
65-69 -> 37
70-79 -> 25
80-89 -> 14
90-100 -> 8

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Question is that

Which of the following point ranges includes the median reading test score for ninth grade students in School District X for 1993 ?

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From my understanding, median is the middle value of the dataset

1,2,3,4,5 median=3
1,2,3,4,5,6 median=3.5

From this logic answers is 0-65 as 16 comes in middle (8,14,16,25,37), However that is not right, Correct answer is 65-69.

Can anyone explain to me what I am doing wrong?

Answer 1

The right column shows the percentage of students for each range. Adding the percentages, starting from the top (or bottom for that matter), the range where you cross the 50 % limit will be the your answer.

What you're doing wrong is that you use the percentages themselves as the dataset. The dataset is the scores, which have been represented by ranges.

Answer 2

You are completely wrong. The median can be calculated as follows, write down the observations in a increasing order including multiplicities, and then choose the middlemost value, you wrote down the frequency of the corresponding classes and chose the median of the frequencies, not the median of the given data.

You need to write down the cumulative frequencies and choose the class whose cumulative frequency just crosses $N/2$, where $N$ is the total number of observations.

$$\begin{array}{ccc} \text{Classes}&\text{Frequencies}&\text{Cumulative Frequencies}\\ 0-65 & 16 &16\\ 65-69 & 37 &16+37=53\\ 70-79 & 25 &53+25=78\\ 80-89 & 14 &78+14=92\\ 90-100 &8&92+8=100 \end{array}$$ Here $N/2=50$. That is the $50^th$ value is the median. The class $65-69$ has cumulative frequency $53>50$ and the class $0-65$ has cumulative frequency $16<50$. So the median, that is the $50^{th}$ observation lies in the class $65-69$.