Median of grouped data To find the median of a distribution obtained for $n$ observations, we first determine the class into which the median must fall. Then, if there are $j$ values in this class and $k$ values below it, the median is located $\frac{(n / 2)-k}{j}$ of the way into this class, and to obtain the median we multiply this fraction by the class interval and add the result to the lower boundary of the class into which the median must fall. This method is based on the assumption that the observations in each class are "spread uniformly" throughout the class interval, and this is why we count $\frac{n}{2}$ of the observations instead of $\frac{n+1}{2}$ as on page 35 .

To illustrate, let us refer to the nanopillar height data on page 25 and the frequency distribution on page 26. Since $n=50$, it can be seen that the median must fall in class $(285,325]$, which contains $j=23$ observations. The class has width 40 and there are $k=3+11=14$ values below it, so the median is

\[285+\frac{25-14}{23} \times 40=264.13\]

(a) Use the distribution obtained in Exercise 2.10 on page 32 to find the median of the grouped nanopillar diameters.

(b) Use the distribution obtained in Exercise 2.12 on page 33 to find the median of the grouped particle sizes.

Data From Exercise 2.10

Data From Exercise 2.12

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2.10 To continually increase the speed of computers, elec- trical engineers are working on ever-decreasing scales.