Graphs of Data

Main idea

Before computing anything, draw the data. A good graph shows you the shape, the center and the spread at a glance, and the right kind of graph depends on whether you have numbers, categories, or something changing over time.

Histogram class width

\text{width} \approx \frac{{\htmlData{sym=0}{x_{\max}}} - {\htmlData{sym=1}{x_{\min}}}}{\htmlData{sym=2}{{k}}}

What the symbols mean

$x_{\max}$
Largest value
The top of the data's range.
$x_{\min}$
Smallest value
The bottom of the data's range.
$k$
Number of classes
How many bars; round the width up so every value fits.

Explanation

A distribution is simply the pattern of a set of numbers: which values show up, and how often. A list of 24 race times tells you almost nothing at a glance, but a picture of their distribution shows right away where most of them sit and whether a few stragglers trail off. The class-width formula $\frac{x_{\max} - x_{\min}}{k}$ is just the range of the data (largest minus smallest) chopped into $k$ equal pieces (one per bar). Nothing scary here; this lesson is mostly about learning to read pictures.

Stem-and-leaf plots split each number into a stem (leading digits) and a leaf (last digit). Think of it as a histogram turned on its side that still shows every individual value. Stems $3, 4, 5$ with leaves $3 \mid 2\,5\,8$ , $4 \mid 1\,1\,6$ , $5 \mid 0$ show the data $32, 35, 38, 41, 41, 46, 50$ .

Histograms group numbers into equal-width classes and draw a bar for each class's frequency. Picking the number of bars is a judgment call. Too few bars lump everything together and hide the shape; too many leave each bar with only one or two values, so the picture turns noisy. There's a histogram further down where you can try different widths yourself.

Shape vocabulary:

Symmetric: the two sides mirror each other.
Skewed right: a long tail toward large values (incomes, house prices).
Skewed left: a long tail toward small values (scores on an easy exam).
Uniform: roughly flat. Bimodal: two peaks.

Bar graphs are for categories: the bars are separate and their order can be rearranged. Line graphs show a variable over time.

A common mistake. A bar graph and a histogram look alike, so it's tempting to treat them the same. They aren't. A histogram's bars touch because its classes cover a number line with no gaps, and their order is fixed. A bar graph's categories (red, blue, green) have no natural order, so the bars stand apart and you can shuffle them freely.

See it

w

4.00

These are the same 24 values drawn with bar width w. At w = 1 most bars hold just one value and the picture is mostly noise; at w = 12 everything collapses into a handful of blocks. Around w = 4 to 6 the shape comes through: a hump in the 20s and 30s and a long tail out to 66, so the data are skewed right.

Worked examples

Example 1

Find the median of the stem-and-leaf data $3 \mid 2\,5\,8$ , $4 \mid 1\,1\,6$ , $5 \mid 0$ .

1
Reading each stem with its leaves gives the values already in order: $32, 35, 38, 41, 41, 46, 50$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

In the stem-and-leaf plot above, how many values are greater than 40?

Problem 2+5 XP

Find the median of the stem-and-leaf data.

Problem 3+5 XP

Which graph suits counts of students' favorite colors?

Problem 4+5 XP

A histogram has a long tail to the right. Its shape is:

Problem 5+5 XP

Data run from 12 to 85. What class width gives 5 classes?

Problem 6+5 XP

24 runners' 5K times range from 18.5 to 41.2 minutes. You want a histogram with 6 classes of equal width, rounded up to a whole number of minutes. What width should you use?

Problem 7+5 XP

A histogram has bars centered at 10, 20 and 30 with frequencies 2, 5 and 3. Estimate the mean.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A data set has mean 52, median 47 and standard deviation 10. Find Pearson's skewness coefficient, $\frac{3(\text{mean} - \text{median})}{s}$ . Which way is the distribution skewed?

Where it's used

Newsrooms, dashboards and lab reports live on these graphs, and so do misleading charts: truncated axes and uneven bins are the usual tricks.

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