Topic 1.5 Notes – Graphical Representations for One Quantitative Variable

What a Graph of One Quantitative Variable Shows

A quantitative variable is a numerical measurement or count, like quiz scores, heights, or commute times. A graph of one quantitative variable puts those values from smallest to largest and shows either exact values or grouped intervals.

The three displays here are:

• Dotplot
• Stem-and-leaf plot (stemplot)
• Histogram

The big difference is detail:

• Dotplots and stemplots keep the individual observations visible.
• Histograms group values into intervals, so you lose exact values but see the overall pattern more clearly.

A complete graph needs:

• the variable identified by title or context
• a labeled axis
• units if they exist
• an evenly spaced numerical scale
• a frequency or relative frequency scale when needed
• a key for a stemplot

You also need to account for every observation exactly once. That’s the core correctness check on AP Stats.

For vertical scales:

• Frequency = number of observations
• Relative frequency = proportion of observations

$$\text{relative frequency}=\frac{\text{frequency}}{n}$$

Relative frequencies should add to $1$ or $100\%$, aside from rounding.

Dotplots, Stemplots, and Histograms

These three displays can show the same quantitative data in different ways. In the example below, the dotplot keeps each exact value, the stemplot organizes those values by place value, and the histogram groups them into intervals.

Same data in three displays

Dotplots

A dotplot uses one dot for one observation at its exact value.

• If a value repeats, the dots stack
• The height of the stack shows the frequency of that exact value
• Best for small data sets where exact values matter

In the example, the repeated values stack above the same number. That’s why dotplots are great when a question asks for repeats, minimum, maximum, or exact values.

Stem-and-leaf plots

A stemplot splits each value into a stem and a leaf.

• Example: with whole numbers, $2|4$ might mean 24
• Stems go in increasing order
• Leaves also go in increasing order
• Repeats show up as repeated leaves

You must include a key, because $2|4$ could mean 24, 2.4, or 240 depending on the data.

The stemplot in the figure uses stems 2, 3, and 4 with a key that tells you $2|2=22$. Stemplots are best when the data set is small to moderate and the place values split nicely.

Histograms

A histogram groups data into bins or intervals.

• Each bar stands for a whole interval, not one exact value
• Bar height shows frequency or relative frequency in that interval
• Bars touch because the intervals are numerical and continuous

On the histogram here, you can see the overall shape quickly, but you can’t recover every exact data value from the bars. Histograms are best for larger or more crowded data sets.

How to Construct Each Graph

Dotplot

1. Draw a numerical axis and label it.
2. Use a scale covering the full data range.
3. Place one dot per observation.
4. Stack repeated values.
5. Count dots and make sure the total is $n$.

Stem-and-leaf plot

1. Choose a consistent stem-leaf split.
2. Write stems in order.
3. Include empty stems if needed so the scale stays honest.
4. Add one leaf per observation.
5. Put leaves in increasing order.
6. Include a key and units.
7. Count leaves to make sure the total is $n$.

If one stem gets crowded, you can split stems consistently, such as 0 to 4 on one row and 5 to 9 on another.

Histogram

1. Choose ordered, nonoverlapping bins.
2. Make sure every value goes into exactly one bin.
3. Use equal-width bins in AP Stats.
4. Count frequency or compute relative frequency for each bin.
5. Draw touching bars in order.
6. Label the vertical axis correctly.
7. Make boundary rules clear, like lower included and upper excluded.

Choosing the Right Display and Reading It Correctly

Use a dotplot when you want exact values and the data set is small. Use a stemplot when you want exact values in a compact ordered display. Use a histogram when there are too many values for individual observations to be useful.

What each graph answers:

• Dotplot or stemplot
  exact values, repeats, smallest value, largest value
• Histogram
  how many or what proportion fall in an interval

For histograms, always read the whole setup before interpreting bars:

• bin width
• starting point or boundaries
• frequency vs. relative frequency

Changing the bin width or starting point can make the same data look pretty different. A frequency histogram and a relative-frequency histogram with the same bins have the same shape. Only the vertical scale changes.

Common Mistakes and Easy Points to Lose

A lot of missed points here are tiny construction errors.

This comparison is a good quick reminder of one of the easiest mix-ups on the exam.

• Histogram vs. bar chart
  Histograms are for quantitative data and use numerical intervals. Bar charts are for categorical data and usually have separated bars.
• A histogram bar shows all values in an interval, not one exact value.
• Leaving out empty stems or zero-frequency bins can hide a gap.
• Uneven numerical scales make the graph wrong.
• No stemplot key = incomplete graph.
• Every observation must appear exactly once.
• Don’t mix up frequency and relative frequency on the vertical axis.