Line Plots with Fractions

One dot per measurement

A line plot puts a dot above each value on a number line. Repeated values stack up, so the tallest column is the most common measurement.

Twelve pencils are measured to the nearest quarter-inch:

$$2, \; 2\tfrac{1}{4}, \; 2\tfrac{1}{4}, \; 2\tfrac{1}{2}, \; 2\tfrac{1}{2}, \; 2\tfrac{1}{2}, \; 2\tfrac{3}{4}, \; 2\tfrac{3}{4}, \; 3, \; 3, \; 3, \; 3\tfrac{1}{2}$$

Reading it: the most common length is a tie between $2\tfrac{1}{2}$ and $3$ inches, with three pencils each, and one pencil is noticeably longer than the rest.

Choosing the scale

Mark the line in the smallest unit that appears in the data.

Data uses	Mark the line in
halves only	halves
halves and quarters	quarters
anything in eighths	eighths

Marking in halves when a quarter appears leaves that measurement with nowhere to sit. Marking in eighths when nothing needs them clutters the line with empty positions.

Every position gets a mark even if no dot goes above it. A gap in the data is information, and it only shows if the place for it is drawn.

Finding the total

Multiply each value by the number of dots above it, then add.

For the pencils:

Length	Dots	Subtotal
$2$	$1$	$2$
$2\tfrac{1}{4}$	$2$	$4\tfrac{1}{2}$
$2\tfrac{1}{2}$	$3$	$7\tfrac{1}{2}$
$2\tfrac{3}{4}$	$2$	$5\tfrac{1}{2}$
$3$	$3$	$9$
$3\tfrac{1}{2}$	$1$	$3\tfrac{1}{2}$

$$2 + 4\tfrac{1}{2} + 7\tfrac{1}{2} + 5\tfrac{1}{2} + 9 + 3\tfrac{1}{2} = 32$$

Thirty-two inches of pencil in total.

Multiplying by the count is what saves the work. Adding twelve values one at a time gives the same answer with four times the chances to slip.

Sharing out equally

A classic question: if all the pencil were shared equally, how long would each be?

That is the total divided by how many:

$$32 \div 12 = 2\tfrac{2}{3} \text{ inches}$$

This is the mean, arrived at the way the name suggests — level everything out and see where it settles.

Note the answer is not one of the measured values, and does not have to be. No pencil is $2\tfrac{2}{3}$ inches long; that is what they would all be if the material were redistributed.

Why a line plot suits measurement data

Measurements repeat. Several pencils come out at exactly $2\tfrac{1}{2}$ inches, and a list of twelve numbers hides that while a stack of three dots shows it.

The display also keeps every value. Nothing is grouped away, so a lone long pencil stays visible as a lone dot rather than disappearing into a bar. That is the difference from a histogram, which groups values into intervals and is the right choice once there are too many to dot.

Roughly thirty measurements is where a line plot stops being readable. Below that, it shows more than any summary would.

Worked examples

Common mistakes

Practice problems