D. Data · Data, Statistics and Mathematical Modelling

Histograms, Stem-and-Leaf Plots and the Shape of Data

D1.2 A1About 70 minutes

Learning goals

I can represent one-variable data with a stem-and-leaf plot, a frequency table, a histogram and a box plot.
I can describe the shape of a distribution as symmetric, skewed right, skewed left, bimodal or uniform.
I can explain how the shape affects the mean and median.
I can choose an appropriate representation for a purpose and justify my choice.

Words to know

distribution
The pattern of how often each value (or interval of values) occurs in a data set.

stem-and-leaf plot
A display that splits each value into a stem (leading digits) and a leaf (last digit).

histogram
A graph of continuous data grouped into equal intervals, with touching bars.

symmetric
A distribution whose two sides are roughly mirror images.

skewed right
A distribution with a long tail toward the larger values (mean usually greater than median).

skewed left
A distribution with a long tail toward the smaller values (mean usually less than median).

bimodal
A distribution with two peaks.

One data set, many pictures

A community centre held a 5 km fun run in the fall. Thirty runners finished. The organizers want to know: What is a typical time? How spread out are the times? Were there any unusual times?

You could answer with a dot plot, a stem-and-leaf plot, a histogram or a box plot. Each one shows something the others hide. In this lesson you will represent the same data in several ways, describe the shape of a distribution, and choose the best representation for a purpose.

The data (finishing times in minutes, in order)

22, 24, 25, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 40, 41, 43, 46, 52, 58

Summary statistics: n = 30, sum = 1 009, mean ≈ 33.6 min, median = (31 + 32) ÷ 2 = 31.5 min, mode = 30 min, range = 58 − 22 = 36 min.

Five-number summary: 22, 28, 31.5, 37, 58 (Q1 is the 8th value and Q3 is the 23rd). IQR = 9. Upper fence = 37 + 13.5 = 50.5, so 52 and 58 are outliers.

Worked example

Example 1: Stem-and-leaf plot

Make a stem-and-leaf plot of the run times.

Use the tens digit as the stem and the ones digit as the leaf.
2 | 2 4 5 6 7 7 8 8 9 9
3 | 0 0 0 1 1 2 2 3 4 5 5 6 7 8
4 | 0 1 3 6
5 | 2 8
Add a key: 2 | 2 means 22 minutes. Check the count: 10 + 14 + 4 + 2 = 30. ✓

The stem-and-leaf plot keeps every value and also shows the shape: most times are in the 20s and 30s, with a few long times in the 40s and 50s.

Worked example

Example 2: Frequency table and histogram

Group the times into intervals of 5 minutes and draw a histogram.

Choose equal intervals: 20 to 25, 25 to 30, 30 to 35, and so on up to 55 to 60. Each interval includes its left end but not its right end (so 25 goes in 25 to 30).
Count: 20–25: 2; 25–30: 8; 30–35: 9; 35–40: 5; 40–45: 3; 45–50: 1; 50–55: 1; 55–60: 1. Total 30. ✓
Draw bars with no gaps (time is continuous). The height of each bar is the frequency.

The histogram shows a peak at 30 to 35 minutes and a long tail to the right.

A histogram of the 30 run times. The bars touch because time is continuous. The tallest bar is 30 to 35 minutes, and the bars trail off to the right.

The same data as a modified box plot. The outliers 52 and 58 are dots, and the right whisker stops at 46, the longest time that is not an outlier. The right whisker is longer than the left one.

The shape of a distribution

The distribution of a data set is the pattern of how often each value occurs. Describe its shape with these words:

Symmetric: the left and right sides are roughly mirror images. The mean and median are close.
Skewed right (positively skewed): a long tail to the right (a few large values). The mean is pulled above the median.
Skewed left (negatively skewed): a long tail to the left (a few small values). The mean is pulled below the median.
Bimodal: two peaks. This often means the data contains two different groups (for example, walkers and runners in the same race).
Uniform: all bars about the same height.

The run times are skewed right: the mean (33.6) is greater than the median (31.5).

Four common shapes. The side with the long, thin tail names the skew. In a skewed distribution, the mean is pulled toward the tail.

Representation	Good for	Limitations
Dot plot	Small data sets; seeing every value and clusters	Hard to draw with many values or a wide range
Stem-and-leaf plot	Small to medium sets; keeps every value and shows shape; easy to find the median	Messy with large data sets or decimals
Histogram	Large sets of continuous data; shows shape clearly	Hides individual values; the shape changes with the interval width
Box plot	Summarizing; comparing groups; showing outliers	Hides shape details such as two peaks; does not show n
Bar graph or circle graph	Categorical data	Not for continuous numerical data

Choosing a representation for one-variable data.

Using technology

In Desmos, type a list such as T=[22,24,25,...] and then histogram(T,5) or boxplot(T). In a spreadsheet, select the data and insert a histogram or box-and-whisker chart. Technology lets you try different interval widths quickly. An interval that is too wide hides the shape; one that is too narrow makes the graph spiky.

Just for fun

Funny but true: statisticians really do call a long stretch of data on one side of a graph a tail. So a skewed-right histogram is, technically, wagging its tail.

Hands-on activity

Representation challenge

You need: a family recipe book or several food packages (cereal boxes, cans), paper, a pencil, and a device with Desmos or a spreadsheet if you have one.

Choose one numerical variable, for example the number of grams of sugar per serving on 20 food packages, or the cooking time in minutes of 20 recipes.
Record the 20 values. Find the mean, median and five-number summary.
Make three representations: a stem-and-leaf plot, a histogram (choose an interval width) and a box plot.
Describe the shape (symmetric, skewed right, skewed left, bimodal).
Which representation would you use on a poster for your family? Explain in two sentences.

What to look for: a key on the stem-and-leaf plot, equal intervals with no gaps on the histogram, a correct box plot, and a shape description that agrees with how the mean compares to the median.

Your turn

A stem-and-leaf plot shows: 1 | 5 8 8; 2 | 0 3 4 4 7; 3 | 1 6 (key 1 | 5 = 15). List the data and find the median.

For the data in question 1, find the mode and the range.

A data set has mean 52 and median 45. What shape is it likely to have? Explain.

Which representation would you choose for the heights of all 1 200 students in a high school? Why?

Why do the bars of a histogram touch, but the bars of a bar graph do not?

The ages of people at a family skating party form a bimodal distribution. Suggest a reason.

More practice: warm-up

In a stem-and-leaf plot with key 4 | 7 = 47, what value does 6 | 2 represent?

Name the shape: a long tail to the left.

In a symmetric distribution, how do the mean and median compare?

Which graph is best for favourite Canadian sports teams?

A histogram interval is 10 to 20. In which interval does the value 20 go if intervals include their left end?

More practice: on your own

Make a frequency table with intervals 0 to 10, 10 to 20, 20 to 30, 30 to 40 for: 3, 8, 12, 14, 15, 18, 19, 21, 22, 25, 27, 33.

Describe the shape of the distribution in the last question.

Household incomes in Canada are skewed right. Which measure, mean or median, do news reports usually prefer for a "typical" income? Why?

Exam marks in a very easy test are clustered between 85 and 100 with a few low marks. Name the shape and say how the mean compares with the median.

A box plot hides one important feature that a histogram shows. What is it? Give an example.

Challenge

The run-time data has mean ≈ 33.6 min and median 31.5 min. If the two outliers (52 and 58) are removed, find the new mean and median.

Explain how changing the interval width of a histogram can change what you think about the data. Use the run times as an example.

A teacher says, "The mean test mark was 68 and the median was 74." Sketch or describe a possible histogram and explain what probably happened.

Check your understanding

Quiz

In a stem-and-leaf plot with key 3 | 4 = 34, how many values are shown by 5 | 0 2 2 9?

A distribution has a long tail to the right. Which is most likely?

Which representation is best for showing the shape of 5 000 continuous values?

A histogram has two clear peaks. What is this shape called?

Which feature can a box plot show that a stem-and-leaf plot does not show directly?

Why must histogram intervals be equal in width?

For parents and teachers

Big idea: D1.2 says students analyse one-variable data "in various ways". This lesson connects the representations: each shows something different, and good analysts choose deliberately.

How to help: Look at a histogram in the news or a sports website together and ask, "What shape is it? Where is the tail? So is the mean bigger or smaller than the median?"

Common difficulties:

Naming the skew after the peak instead of the tail (the tail names it).
Putting gaps between histogram bars, or using unequal intervals.
Forgetting the key on a stem-and-leaf plot.
Putting a boundary value (like 25) in the wrong interval. Agree on a rule and use it consistently.