Graphical Representations for One Quantitative Variable

The concept in 60 seconds

A quantitative graph keeps numerical values in their natural order. A $10$-minute journey belongs to the left of a $30$-minute journey on a horizontal time axis. The distance between positions must agree with the numerical scale.

Dotplots and stem-and-leaf plots can preserve individual recorded values. A histogram preserves the interval counts, but usually does not tell you every exact value inside an interval.

Before constructing any graph, identify what was measured, the units, and the number of observations. Then check that the finished graph accounts for that same number of observations.

How long did these students take to reach school?

Use Cedar High, the fictional school from the earlier lessons. For this lesson, use a new, small teaching sample of $20$ students. Each student records their one-way home-to-school travel time on the Tuesday being studied, rounded to the nearest minute.

Here are all $20$ recorded times, sorted from smallest to largest:

Travel time, minutes:
$8$, $10$, $12$, $14$, $14$, $15$, $16$, $18$, $18$, $18$,
$20$, $22$, $24$, $25$, $26$, $28$, $30$, $32$, $35$, $42$

The observational units are the $20$ sampled students. The quantitative variable is each student’s travel time. The sample size is $n = 20$. A travel duration is an underlying continuous measurement, even though these recorded values have been rounded to whole minutes.

The data are invented for this lesson. This $20$-student sample has its own total; use $20$ when calculating its relative frequencies.

Dotplot, stem-and-leaf plot or histogram?

Quantitative values cannot be rearranged like unordered category labels. Keep numerical positions and stems in increasing order. A histogram’s intervals represent amounts, not arbitrary category names.

Use a title and units such as minutes, centimetres or books. “Number of students” is a frequency label; “Travel time in minutes” names the variable being studied. A graph’s count axis does not mean that a second characteristic was measured on each student.

How to construct dotplots and stem-and-leaf plots

Construct a dotplot

1. Label the variable and units. Use “One-way travel time (minutes)” for the Cedar High sample.
2. Draw a consistent numerical scale. Equal differences in time must occupy equal distances on the axis.
3. Place $1$ dot for each observation. Put the $8$-minute observation at $8$, the $10$-minute observation at $10$, and continue through the list.
4. Stack repeated values. Put $2$ dots at $14$ and $3$ at $18$. Keep a consistent vertical spacing.
5. Count the dots. There should be $20$, matching the number of recorded students.

In this plot, a stack height tells you how many students share a recorded time. The $3$ dots at $18$ mean $3$ recorded $18$-minute journeys, not a journey lasting $3$ minutes.

The measurement axis can start near the smallest observed value; it does not have to start at zero. It must use a clearly labeled, consistent scale. If values are rounded or grouped to a plotting resolution, state that resolution. Here the recorded precision is $1$ minute.

Construct a stem-and-leaf plot

For these whole-minute values, use the tens digit as the stem and the ones digit as the leaf. For example, $2\mid5$ represents $25$ minutes. Write a key so the reader knows the place values and units.

1. Choose a stem and leaf unit. Here a stem represents tens of minutes and a leaf represents $1$ minute.
2. List the stems in increasing order. Use $0$, $1$, $2$, $3$ and $4$ for this dataset.
3. Record one leaf per observation. Every recorded time contributes one leaf, including repeats.
4. Sort the leaves within each stem. Keep them in increasing order from left to right.
5. Include the key and check the total. There must be $20$ leaves.

The row $1\mid0\;2\;4\;4\;5\;6\;8\;8\;8$ represents $10$, $12$, $14$, $14$, $15$, $16$, $18$, $18$ and $18$ minutes. It contains $9$ observations. Each leaf is $1$ recorded value, not just one distinct value.

When a stem has too many leaves

A split stem can make a crowded plot easier to read. For example, repeat stem $1$ twice: the first row holds leaves $0\text{–}4$, and the second holds leaves $5\text{–}9$. For this sample, those rows would be $1\mid0\;2\;4\;4$ and $1\mid5\;6\;8\;8\;8$. Clearly state the split rule and use it consistently throughout the display.

Keep empty stems where needed to preserve the numerical sequence. For values $12$, $14$ and $35$, a stem-and-leaf plot should include an empty stem $2$ between stems $1$ and $3$. An empty stem shows that no values occupy that range.

How to construct a histogram

A histogram counts observations inside numerical intervals. Unlike a dotplot, it groups several possible values into each bin.

Set a boundary rule before you count

Use $10$-minute bins for Cedar High. Define each bin to include its lower boundary and exclude its upper boundary: $10\le \text{time} < 20$ means at least $10$ minutes but less than $20$ minutes.

The frequencies total $1 + 9 + 6 + 3 + 1 = 20$. The percentages total $100\%$. The bins cover every observed time, from $8$ through $42$ minutes, without double-counting a boundary value.

For a vertical histogram, bin boundaries are on the horizontal axis and frequencies or relative frequencies are on the vertical axis. Start the frequency axis at zero and use equal numerical intervals. Adjacent nonempty bins have touching bars. A bin with $0$ observations remains as an empty interval on the numerical scale.

The $20\text{–}30$ bar has height $6$. That means $6$ students had recorded times of at least $20$ but less than $30$ minutes. It does not mean $6$ students each took $20$ minutes, or that the travel time was $6$ minutes.

Frequency versus relative frequency histograms

A relative frequency histogram uses the same bin boundaries, but changes the bar heights to proportions or percentages. For these $5$ bins, the proportions are $0.05$, $0.45$, $0.30$, $0.15$ and $0.05$, or $5\%$, $45\%$, $30\%$, $15\%$ and $5\%$.

With the same dataset and bins, converting every height from a count to a relative frequency changes the vertical scale, not the pattern across intervals. Clearly label whether the height represents students, a proportion or a percentage.

Bin width changes the picture, not the observations

Now group the same $20$ times into $5$-minute bins instead of $10$-minute bins:

For example, the $9$ times in $10\le \text{time} < 20$ divide into $4$ in $10\le \text{time} < 15$ and $5$ in $15\le \text{time} < 20$. Narrower bins reveal finer detail. Wider bins combine more values and can hide detail.

There is no single bin width that always gives the best picture. Choose one that makes the data readable, check nearby reasonable choices, and keep the actual observations unchanged. More bars do not mean more students were measured.

Worked examples