Displaying And Describing Numerical Data

Q: What is a dot plot?

A dot plot shows each data value as a dot above its position on a number line. Identical values stack vertically. Dot plots are best for small datasets of discrete numerical values.

Q: What is a stem-and-leaf plot?

A stem-and-leaf plot splits each data value into a stem (the leading digit or digits) and a leaf (the last digit). All the values are preserved — nothing is lost. It works well for medium-sized datasets where you still want to see each value.

Q: When should I use a histogram?

Use a histogram for continuous data or for large datasets where listing every value would be cluttered. Group the values into class intervals and draw bars whose heights match the frequencies. Histogram bars TOUCH because the data is continuous.

Q: What is a modal class?

For grouped data, the modal class is the class interval with the highest frequency. It's an interval like '10 to 14', not a single number. The true mode can't be pinpointed exactly from grouped data.

Q: How do I find the class width and midpoint?

The class width is the span of one interval. For the interval 150 to 154 (in whole cm), the width is 5. The midpoint is the average of the two endpoints: (150 + 154) divided by 2 equals 152.

Q: What is an outlier?

An outlier is a value far from the rest of the data. In a dot plot it shows up as a single dot well separated from the main cluster. In a histogram it sits in a far-off interval with very low frequency.

Theory

Choose the display by dataset type: dot plots for small discrete data, stem-and-leaf for medium datasets where you want to keep every value, and histograms (with grouped frequency tables) for continuous or large datasets. The modal class is the interval with the highest frequency. Class width, midpoint, and outliers are key quantities to recognise.

For numerical data, the type of display depends on the data values. Small discrete datasets use dot plots; medium-sized datasets often use stem-and-leaf plots; large or continuous datasets are displayed with histograms.

A dot plot shows each data value as a dot above its position on a number line. Identical values stack vertically. The mode is the tallest stack.

A stem-and-leaf plot splits each value into a stem (leading digit(s)) and a leaf (last digit). All values are preserved — nothing is lost. The row with the most leaves is the modal stem class.

A histogram displays grouped numerical data: values are bundled into class intervals, and each bar's height is the frequency for that interval. Bars touch because the data is continuous.

For grouped data:

Class width = the span of one interval (e.g. interval $150$ – $154$ has width $5$ in whole cm).
Midpoint = average of the two endpoints ( $150$ + $154$ ) ÷ 2 = $152$ .
Modal class = the interval with the highest frequency.

An outlier is a value far from the rest — visible as an isolated dot or an unusually low-frequency far-off bar.

The first diagram is a dot plot of "pets per household". The second is a histogram of "sales calls per day", with the modal class highlighted in orange.

Each dot is one household. The tallest stack (at 1 pet) is the mode.

Bars touch (numerical, grouped). The modal class is

10

–

14

calls.

The only formulas are for grouped data.

Class width

For the interval $a$ – $b$ in whole units, the width is

$width = b - a + 1$

width = b - a + 1

For continuous intervals like " $150 \leq h < 155$ ", the width is simply $155 - 150 = 5$ .

Class midpoint

$midpoint = \frac{a + b}{2}$

midpoint = \frac{a + b}{2}

When to use which display

Dataset	Best display
Small, discrete (whole numbers)	Dot plot
Medium, want to keep all values	Stem-and-leaf plot
Large or continuous	Grouped frequency table + histogram

Modal class isn't a single number. For grouped data, the mode is the entire interval (e.g. "10–14"). You can't pin down the exact mode without the raw values.

Drawing a dot plot

Draw a horizontal number line spanning the data's range.
Above each value, stack one dot per occurrence.
Label the axis and identify the mode (the tallest stack).

Building a stem-and-leaf plot

Decide what's stem and what's leaf — usually the last digit is the leaf, everything before it is the stem.
List each unique stem in a column. For each data value, write its leaf in the row for its stem, in increasing order.
Identify the smallest value, largest value, modal stem class, and count the total leaves.

Building a grouped frequency table and histogram

Choose class intervals of equal width that cover the data.
Tally how many values fall in each class.
Draw the histogram with bars whose heights equal the frequencies — bars touch.
State the modal class (interval with the highest frequency), class width, and midpoints as needed.

EXAMPLE 1 — READ A DOT PLOT

For the pets dot plot above, what is the mode and how many families were surveyed?

SOLUTION

The tallest stack is at $x = 1$ , so the mode is $1$ pet. To find the total, add the dots in every stack.

Total	$=$	$3 + 5 + 3 + 1$
	$=$	$12 families$

Answer: mode = $1$ pet; $12$ families were surveyed.

mode = 1, total = 12

EXAMPLE 2 — READ A STEM-AND-LEAF PLOT

A stem-and-leaf plot has rows:

3 | 5 8

,

4 | 2 4 7

,

5 | 0 1 3 5 8

,

6 | 2 4 6 9

,

7 | 1 5

. Find the minimum, maximum, and total number of values.

SOLUTION

The smallest value has the smallest stem and the smallest leaf in that row. The largest has the largest stem and largest leaf. The count is the total number of leaves.

Min	$=$	$35$
Max	$=$	$75$
Count	$=$	$2 + 3 + 5 + 4 + 2 = 16$

Answer: min $= 35$ , max $= 75$ , $16$ values in total.

min = 35, max = 75, n = 16

EXAMPLE 3 — MODAL CLASS

A grouped table for sales calls per day:

0

–

4

:

3

;

5

–

9

:

8

;

10

–

14

:

12

;

15

–

19

:

5

;

20

–

24

:

2

. State the modal class.

SOLUTION

The class with the highest frequency ( $12$ ) is the modal class.

Modal class

=

10

–

14

calls

Answer: the modal class is $10$ – $14$ calls per day.

modal class = 10 – 14

EXAMPLE 4 — CLASS WIDTH AND MIDPOINT

For the class

150

–

154

cm (heights recorded to the nearest whole cm), find the class width and the midpoint.

SOLUTION

For whole-unit intervals, width = top − bottom + 1. The midpoint is the average of the endpoints.

Width	$=$	$154 - 150 + 1 = 5 cm$
Midpoint	$=$	$\frac{150 + 154}{2} = 152 cm$

Answer: class width $= 5$ cm, midpoint $= 152$ cm.

width = 5, midpoint = 152

Common pitfalls

Bar charts have gaps; histograms don't. Bar charts (categorical) have gaps between bars. Histograms (numerical, continuous-style grouping) have bars touching. Drawing them the wrong way around loses easy marks.

The modal class is an interval, not a number. "

10

–

14

" is an interval — that's the answer for a grouped distribution. You can't pin down the exact mode without the raw values.

Class width depends on whether the data is discrete or continuous. For whole-number intervals like

150

–

154

, the width is

5

(using

154 - 150 + 1

). For continuous intervals like

150 \leq h < 155

, the width is also

5

. Both give the same midpoint of

152.5

or

152

, depending on convention.

Outliers stand out — but check. A single dot far from the cluster might be an outlier, or it might be a recording error. Note its position and (in real work) investigate before deleting.

Stem-and-leaf plots need ordered leaves. Within each row, leaves should be in increasing order. An unsorted leaf list still preserves the values but loses some readability — and might cost marks.

Frequently asked questions

What is a dot plot?

A dot plot shows each data value as a dot above its position on a number line. Identical values stack vertically. Dot plots are best for small datasets of discrete numerical values.

What is a stem-and-leaf plot?

A stem-and-leaf plot splits each data value into a stem (the leading digit or digits) and a leaf (the last digit). All the values are preserved — nothing is lost. It works well for medium-sized datasets where you still want to see each value.

When should I use a histogram?

Use a histogram for continuous data or for large datasets where listing every value would be cluttered. Group the values into class intervals and draw bars whose heights match the frequencies. Histogram bars TOUCH because the data is continuous.

What is a modal class?

For grouped data, the modal class is the class interval with the highest frequency. It's an interval like '10 to 14', not a single number. The true mode can't be pinpointed exactly from grouped data.

How do I find the class width and midpoint?

The class width is the span of one interval. For the interval 150 to 154 (in whole cm), the width is 5. The midpoint is the average of the two endpoints: (150 + 154) divided by 2 equals 152.

What is an outlier?

An outlier is a value far from the rest of the data. In a dot plot it shows up as a single dot well separated from the main cluster. In a histogram it sits in a far-off interval with very low frequency.

Video Lessons

Practice Questions

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Practice Questions

Displaying And Describing Numerical Data

📖 Theory

Class width

Class midpoint

When to use which display

Drawing a dot plot

Building a stem-and-leaf plot

Building a grouped frequency table and histogram

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions