Histograms and the Shape of Data: Quick Revision

A brief note on Histograms and the Shape of Data. Read it before a test. Try each question at the end before you open its answer.

Histograms

• Bins: equal pieces of the number line. An edge value goes into the bin above.
• Bars: touch each other. The height is the frequency, or the relative frequency (count divided by the total).
• Bin width: narrow bins look ragged, wide bins hide detail. About $\sqrt{n}$ bins is a starting point.
• Unequal bins: height = frequency density = $\dfrac{\text{frequency}}{\text{width}}$, so area stands for the count.

Shapes

• Symmetric (mound-shaped when it has one hill), skewed right (tail to the right), skewed left (tail to the left).
• Unimodal (one peak), bimodal (two peaks, often two groups mixed), uniform (flat).
• Unusual features: clusters, gaps and outliers. Ask why they are there.

Shape and the two averages

• The mean follows a long tail; the median hardly moves.
• Usually: skewed right gives mean > median, skewed left gives mean < median.
• Report the median for skewed data, the mean for symmetric data.

A worked example

A histogram of 40 reading times has bins 0 to 10, 10 to 20, 20 to 30 and 30 to 60 minutes with counts 8, 16, 10 and 6.

1. The densities. $0.8$, $1.6$, $1$ and $0.2$ readers per minute.
2. The shape. The bars rise to 10 to 20 minutes and then fall slowly through a wide last bin: skewed right.
3. The averages. Expect the mean to be above the median.

Practice with new numbers

A program to try

The program finds the densities of the worked example and says which bin is the most crowded.

Mistakes to avoid

• Gaps between bars of a histogram.
• Naming skew after the peak. Name it after the tail.
• Heights by frequency with unequal bins. Use frequency density.
• The mean as typical for skewed data.

Check yourself

1. Bins are 0 to 4, 4 to 8 and 8 to 12 cm. Where does a value of 8 cm go?

2. A bin from 10 to 25 s holds 9 values. Find its frequency density.

3. The mean of a data set is 31 and the median is 44. Which shape is likely?

4. A histogram shows two peaks with a deep valley between them. What should you ask?

Sources

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), sections on histograms and on skewness and the mean, median and mode.