Statistical Charts and Displays

A statistical display turns a list of numbers into a picture. Which display to use depends on the question: which category is biggest, what share each takes, how a measurement is spread, or how a quantity changed over time.

What kind of data do you have?

Decide the type of data first, because it rules out most of the displays. Categorical data are labels rather than amounts, such as eye color or bus route.

Numerical data are counts or measurements, so arithmetic on them is allowed.

Numerical data split again. A count is discrete: a family has 2 children or 3, never 2.4. A measurement is continuous, so 168 cm stands for every height that rounds to 168. That difference decides between a vertical line chart and a histogram. See Categorical and Numerical Data.

How do you read a chart before you can read numbers?

In a pictogram each symbol stands for a fixed number, and the key states that number. Count the symbols first, then multiply by the key. See Reading Pictograms.

A bar chart replaces the symbols with a scale up the side. The scale does not always climb in ones, so check whether it goes up in twos or fives first. See Reading Bar Charts.

A table has no picture at all. Find the row, find the column, and take the number where they cross. See Reading Tables.

A line graph plots one reading per point and joins them in order, because the quantity between two readings exists: a temperature at 11:30 is real. See Line Graphs.

When the measurements are fractions, mark the number line in halves and fourths and put one dot above each measurement. See Line Plots in Halves and Fourths and the fractions guide.

Add the values and divide by how many there are: that is the mean, what each item would get if the total were shared out evenly. See The Mean.

Which chart should you use?

Match the display to the question, not to the data alone.

A bar chart is the standard display for categories; grouped bar charts put two bars side by side per category. See Bar Charts.

In a pie chart each slice's angle is its share of a full turn: multiply the fraction by 360, and the angles must total 360°. See Pie Charts.

A dot diagram puts one dot above each value, so a stack of dots is a count. See Dot Diagrams. A vertical line chart uses one line per value, with no width, because a score of 2 is a single number and not a range. See Vertical Line Charts.

A stem-and-leaf display is a chart and a table at once: 47 splits into a stem of 4 and a leaf of 7, so the rows show the shape and the median can be read off exactly. See Stem and Leaf.

Counting before charting

A tally counts observations in gates of five, and the frequency column records each row's count. Group the values when they spread too wide, so each row counts a whole interval. See Tally and Frequency Tables.

A two-way table classifies the same people by two questions at once, such as year group against method of travel. Each cell counts one combination. See Two-Way Tables.

A frequency tree splits a total into groups and splits each group again. Following one branch and ignoring the rest is what conditioning means, which the probability guide picks up. See Frequency Trees.

What makes a histogram different?

In a histogram the frequency is the area of a bar, not its height. With equal class widths, area and height are proportional, so the chart looks like a bar chart with the gaps closed. See Histograms.

Once the class widths differ, height alone stops being a fair comparison.

Divide each frequency by its class width. The result is the frequency density, a rate in items per unit, and plotting it as the height makes each bar's area equal its frequency. See Frequency Density.

To find the frequency in part of a class, take that fraction of the bar's area. A class from 20 to 40 minutes holds 30 people, so its frequency density is 30 ÷ 20 = 1.5 people per minute. The stretch from 30 to 40 minutes is 10 minutes wide, so it holds about 10 × 1.5 = 15 people. This assumes the values are spread evenly inside the class, which is why the answer is an estimate.

How do charts mislead?

A chart can be accurate in every number and still leave a false impression. See Misleading Graphs.

• A truncated vertical axis. Starting the scale at 94 instead of 0 makes a difference of 3 look like a doubling. Read the axis before trusting the shape.
• Unequal intervals on the horizontal axis. Plotting 1990, 2000, 2005 and 2010 at equal spacing distorts every trend across them.
• Scaling by area. Drawing a bag twice as tall and twice as wide multiplies the drawn area by four, not two.
• A pictogram with symbols of different sizes. If one row's symbol is larger, the key has stopped working.
• A pie chart drawn in 3D. Perspective makes the front slices look larger than the back ones.

The mistakes worth naming

• Reading a histogram's height as a frequency. With unequal widths the height is a density, so multiply by the class width first.
• Leaving gaps in a histogram. Continuous data has no gap between 20–30 and 30–40, so the bars must touch.
• Ignoring the key on a pictogram. Counting symbols instead of quantities divides every answer by the key.
• Joining the tops of bars in a category chart. There is nothing between "walk" and "bus" to join.
• Comparing slices across two pie charts. Equal slices of different totals are different quantities, and neither pie shows its total.

Where this leads next

A chart shows the shape of the data, and measuring it comes next. The averages and spread guide gives the median, the quartiles, the standard deviation and box plots. With two measurements per person, the display becomes a scatter plot: see scatter, correlation and regression. Reading a scale is covered in units, money and time.

Learn this properly in the app

Math Challenge teaches each display as an illustrated lesson with a drawn chart, a worked reading and practice questions.