Math Core

A list of numbers hides its story. A good picture shows at a glance where the data are centered, how spread out they are, what shape they make and whether anything unusual is going on. This lesson covers the three standard displays for one numerical variable (dot plots, histograms and box plots) and how to read the shape of a distribution from them.

Dot plots

A dot plot (or line plot) puts one mark above a number line for every value. It shows every single data point, so it works best for small data sets with a limited number of different values.

Worked example: Goals per game

A soccer team recorded the number of goals it scored in each of its $18$ games.

Goals scored per game

Find the median and the mean. What do you notice?

Median. With $18$ values, the median is the mean of the 9th and 10th values. Counting from the left, values 1–4 are $1$ s and values 5–10 are $2$ s. So the 9th and 10th values are both $2$ , and the median is 2 goals.

Mean. Multiply each value by how many times it occurs:

\bar{x} = \frac{1 \cdot 4 + 2 \cdot 6 + 3 \cdot 3 + 4 \cdot 2 + 5 \cdot 1 + 6 \cdot 1 + 8 \cdot 1}{18} = \frac{52}{18} \approx 2.89 \text{ goals}.

The mean is almost a whole goal higher than the median. Most games had $1$ to $3$ goals, but a few high-scoring games stretch out to the right and pull the mean toward them.

The shape of a distribution

The pattern of a data set, meaning which values occur and how often, is called its distribution. The goals data pile up on the left and trail off to the right. That trailing part is called a tail.

Definition

Shapes of distributions

Symmetric: the left and right sides are roughly mirror images. Many values sit near the middle.
Skewed right: a long tail stretches to the right, toward larger values.
Skewed left: a long tail stretches to the left, toward smaller values.
Uniform: all values occur about equally often; the display looks flat.
Bimodal: there are two separate peaks, which often means two different groups were mixed together.

The direction of a skew is the direction of the tail, not the direction of the peak. The goals data have their peak on the left and their tail on the right, so they are skewed right.

Shape tells you mean versus median

The mean is pulled toward the tail, and the median is not.

Skewed right: mean greater than median.
Skewed left: mean less than median.
Symmetric: mean and median are about equal.

That's why the median (with the IQR) is the better summary for skewed data.

Histograms

For large data sets or data with many different values, a dot plot gets crowded. A histogram groups the values into bins (intervals) of equal width and draws a bar for each bin. The height of the bar is the number of values in that bin, its frequency. The bars touch because the bins cover the number line with no gaps.

A value that lands on the edge between two bins goes in the bin to its right. In the histogram below, the first bar covers commutes from $0$ minutes up to (but not including) $10$ minutes, and a $10$ -minute commute is counted in the second bar.

Worked example: Reading a histogram

Commute times of 30 employees (minutes)

Describe the shape.
What percent of the employees commute $30$ minutes or more?
Which bin contains the median?

Solutions.

The bars peak on the left and trail off to the right, so the distribution is skewed right.
The bins from $30$ up are $5 + 3 + 2 = 10$ employees. $\dfrac{10}{30} \approx 0.333$ , so about 33.3%.
With $30$ values, the median is between the 15th and 16th values. The first bar holds values 1–3, the second holds 4–12, and the third holds 13–20. So both the 15th and 16th values are in the 20 to 30 minute bin.

A histogram tells you how many values fall in each bin, but not the exact values. You can find the bin that holds the median, but not the median itself.

Box plots and outliers

A box plot is drawn from the five-number summary: minimum, $Q_1$ , median, $Q_3$ and maximum. The box runs from $Q_1$ to $Q_3$ with a line at the median, and whiskers reach out to the smallest and largest values. Each of the four sections holds about a quarter of the data.

In high school statistics you'll usually see a modified box plot, which shows outliers. Use the $1.5 \times \text{IQR}$ rule from the last lesson: each whisker stops at the most extreme value that is not an outlier, and each outlier is drawn as a separate dot.

Worked example: A modified box plot

Twelve people ran a 5 km fun run. Their times, in minutes, were

22, \ 25, \ 27, \ 28, \ 30, \ 31, \ 33, \ 34, \ 36, \ 38, \ 40, \ 58.

Make a modified box plot.

Five-number summary. The median is $\dfrac{31 + 33}{2} = 32$ . The lower half is $22, 25, 27, 28, 30, 31$ , so $Q_1 = \dfrac{27 + 28}{2} = 27.5$ . The upper half is $33, 34, 36, 38, 40, 58$ , so $Q_3 = \dfrac{36 + 38}{2} = 37$ .

Outliers. $\text{IQR} = 37 - 27.5 = 9.5$ and $1.5 \cdot 9.5 = 14.25$ . The fences are $27.5 - 14.25 = 13.25$ and $37 + 14.25 = 51.25$ . The time $58$ is above $51.25$ , so it is an outlier. Nothing is below $13.25$ .

Draw it. The box runs from $27.5$ to $37$ with a line at $32$ . The left whisker reaches $22$ . The right whisker stops at $40$ , the largest value that is not an outlier, and $58$ gets its own dot.

Finishing times for a 5 km fun run (minutes)

Common mistake

A box plot does not show how many values there are, and a longer section does not mean more values. Every section holds about $25\%$ of the data; a long section just means those values are spread out. A box plot also hides some shape details: a bimodal data set can have a perfectly ordinary-looking box plot.

Comparing distributions

To compare two groups, draw their displays on the same scale and discuss four things: shape, center, spread and outliers. Use the median and IQR if either group is skewed or has outliers, and the mean and standard deviation if both are roughly symmetric.

Worked example: Two cities in April

The box plots show the daily high temperatures in two cities during April.

Daily high temperatures in April (°F)

Compare the two distributions.

Center. City Q's median ( $66^\circ$ ) is higher than City P's ( $62^\circ$ ), so City Q was typically a little warmer.
Spread. City P's IQR is $70 - 55 = 15^\circ$ and City Q's is $69 - 63 = 6^\circ$ . City P's temperatures varied much more from day to day. The ranges agree: $35^\circ$ for P versus $18^\circ$ for Q.
Overlap. City P had both the coldest day ( $45^\circ$ ) and the hottest day ( $80^\circ$ ). Even though City Q was usually warmer, you couldn't count on it being warmer on any given day.

Tip

Choosing a display: use a dot plot for a small data set where you want to see every value; a histogram for a large data set where the overall shape matters; and box plots to compare several groups side by side or to flag outliers.

Practice

Practice 1

The dot plot shows the scores of $20$ students on a $10$ -point quiz. How would you describe its shape?

Quiz scores

Practice 2

Use the quiz-score dot plot from the previous problem. What is the mean score?