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Elementary Statistics Supplementary Notes

David W. Lyons

Section 1.2 Histograms

Subsection 1.2.1 Examples

Here are some basic problem types about histograms.
  1. The base of a block in a histogram sits on the interval from 15 to 40 in the horizontal axis, and has height \(2.3\text{.}\) How much data lies in the range from 15 to 40?
  2. The base of a block in a histogram sits on the interval from 15 to 40 in the horizontal axis, and has height \(2.3\text{,}\) and the data is evenly spread within the block. How much data lies in the range from 15 to 20?
  3. 26 percent of data values in a data set called \(X\) lie in the range \(34.7\) to \(40.8\text{.}\) What is the height of the block in a histogram for \(X\) whose base sits on the interval from \(34.7\) to \(40.8\text{?}\)
The solutions use the equation \(\text{ area } = \text{ (base)(height)}\) and the fact that the area of a block in a histogram is the percent of data in the range along the base of the block. Here are the solutions.
  1. \begin{align*} \text{ area } \amp = \text{ (base)(height)}\\ \amp = (40-15)(2.3)\\ \amp = 57.5\% \end{align*}
  2. \begin{align*} \text{ area } \amp = \text{ (base)(height)}\\ \amp = (20-15)(2.3)\\ \amp = 11.5\% \end{align*}
  3. \begin{align*} \text{ height } \amp = \frac{\text{area}}{\text{base}}\\ \amp = \frac{26}{40.8-34.7}\\ \amp \approx 4.26\text{ percent per unit} \end{align*}

Subsection 1.2.2 Percentile

The percentile rank of a data value \(A\) is the percent of all data whose value is less than or equal to \(A\text{.}\) If the percent of data less than or equal to \(A\) is \(p\text{,}\) then \(A\) has \(p\)-th percentile. In picture form, the percentile rank of a data value \(A\) is the area contained in the histogram to the left of a vertical line drawn through the point \(A\) on the horizontal axis. The text introduces percentile and percentile rank in the context of normally distributed data (Chapter 5, pp.90–91), but the terms percentile and percentile rank apply to all types of data, whether it is normally distributed or not.

Subsection 1.2.3 Histogram practice problems

Exercises Exercises

1.
Each row in the table below gives three of the four values \(A\text{,}\) \(B\text{,}\) \(p\text{,}\) and height for a rectangle in a histogram on the class interval from \(A\) to \(B\text{,}\) where \(p\) percent of the data lies in the class interval. Find the missing entry in each row.
\begin{align*} A \amp\spacer\amp B \amp\spacer\amp p \amp\spacer\amp \text{height} \\ \rule{.3in}{.1ex} \amp \amp \rule{.3in}{.1ex} \amp \amp \rule{.3in}{.1ex} \amp \amp \rule{.5in}{.1ex}\\ 25 \amp\spacer\amp 43 \amp\spacer\amp 15\% \amp\spacer\amp \\ 42 \amp\spacer\amp 61 \amp\spacer\amp \amp\spacer\amp 3.5\\ 15 \amp\spacer\amp \amp\spacer\amp 18.4\% \amp\spacer\amp 4.6\\ \amp\spacer\amp 90 \amp\spacer\amp 18\% \amp\spacer\amp 6.0 \end{align*}
Answer.
\begin{align*} A \amp\spacer\amp B \amp\spacer\amp p \amp\spacer\amp \text{height} \\ \rule{.3in}{.1ex} \amp \amp \rule{.3in}{.1ex} \amp \amp \rule{.3in}{.1ex} \amp \amp \rule{.5in}{.1ex}\\ 25 \amp\spacer\amp 43 \amp\spacer\amp 15\% \amp\spacer\amp .83\\ 42 \amp\spacer\amp 61 \amp\spacer\amp 66.5\% \amp\spacer\amp 3.5\\ 15 \amp\spacer\amp 19 \amp\spacer\amp 18.4\% \amp\spacer\amp 4.6\\ 87 \amp\spacer\amp 90 \amp\spacer\amp 18\% \amp\spacer\amp 6.0 \end{align*}
2. Percentiles and histograms.
The distribution of data called \(X\) is given in Table 1.2.1 below. The data within each class interval is evenly distributed.
Table 1.2.1. Distribution table for data \(X\)
class interval percent of data
0—10 \(20\%\)
10—30 \(50\%\)
30—50 \(30\%\)
Each row in the table below gives either the \(X\) value or the percentile rank for an \(X\) value. Find the missing entry in each row. Hint: draw the histogram first. Find the heights of the three blocks. Then use "area equals base times height".
\begin{align*} X \amp\spacer\amp \text{percentile rank}\\ \rule{.3in}{.1ex} \amp \amp \rule{1.2in}{.1ex}\\ 8.0 \amp\spacer\amp \\ 25.0 \amp\spacer\amp\\ 42.0 \amp\spacer\amp \\ \amp\spacer\amp 40\text{th percentile}\\ \amp\spacer\amp 80\text{th percentile} \end{align*}
Answer.
Using "height equals area divided by base", the heights of the three blocks are \(2.0=20/(10-0)\text{,}\) \(2.5=50/(30-10)\text{,}\) and \(1.5=30/(50-30)\text{,}\) respectively. The percentile rank for \(X=8.0\) is the total area to the left of \(X=8.0\text{,}\) which is \(8\cdot 2 = 16\%\text{.}\) The \(X\) value with 80th percentile must be somewhere between 30 and 50 (because 30 is given to be 70th percentile). Solve \((X-30)\cdot 1.5 = 10\) to get \(X\approx 36.7\text{.}\)
\begin{align*} X \amp\spacer\amp \text{percentile rank}\\ \rule{.3in}{.1ex} \amp \amp \rule{1.2in}{.1ex}\\ 8.0 \amp\spacer\amp 16\text{th percentile}\\ 25.0 \amp\spacer\amp 58\text{th percentile}\\ 42.0 \amp\spacer\amp 88\text{rd percentile}\\ 18.0 \amp\spacer\amp 40\text{th percentile}\\ 36.7 \amp\spacer\amp 80\text{th percentile} \end{align*}
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