Statistical Charts and Displays
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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?
Start by deciding what type of data you have, because the type rules out most displays. Categorical data are labels rather than amounts, such as eye color or bus route, so they can be counted but not averaged. Numerical data are counts, which are discrete, or measurements, which are continuous. Suppose a survey records the favorite fruit and the number of pets of 20 students. Fruit is categorical, so a bar chart or pie chart fits, and counts of 8, 7 and 5 make three bars. Pets are discrete numbers, so a vertical line chart fits, and a mean number of pets makes sense. A display should only show what the data allow; a mean of eye colors means nothing. Numbers are sometimes labels, such as bus route 12 or a jersey number, and then they are categorical. Check a count table against the total: 8 + 7 + 5 = 20.
Numerical data are counts or measurements, so arithmetic on them is allowed.
A family has 2 children or 3, never 2.4, while a height of 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.
Now you
Is "shoe size" a category or a measurement?
Is "eye color" a category or a measurement?
How do you read a chart before you can read numbers?
A pictogram can be read before any numbers are written down, because each symbol stands for a fixed number and the key states that number. Count the symbols, then multiply by the key. See Reading Pictograms. If one apple symbol stands for 4 children and the Monday row shows 6 apples, then 6 × 4 = 24 children chose apples on Monday. A row of 9 symbols beside a row of 6 shows 3 more symbols, or 3 × 4 = 12 more children. Every symbol is worth the same amount, so a row's total is its count times that amount. The steps change when a symbol is cut: half an apple stands for half the key, 2 children, and a quarter of one stands for 1. Read the key before anything else, since the same picture could stand for 1, 2 or 10. Check a row by adding one key at a time: 4 + 4 + 4 + 4 + 4 + 4 = 24.
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.
Now you
How many votes for blue and red together?
How many more votes for red than blue?
Which chart should you use?
Choose a chart by the question it must answer and the type of data you have. Bar charts compare categories, pie charts show how parts make up a whole, dot diagrams suit small numerical sets, line graphs show change over time, and histograms show spread across a continuous range. Suppose 40 people name their favorite sport: 20 choose soccer, 10 tennis, and 10 swimming. A bar chart makes the comparison easy, while a pie chart shows soccer as half the group, a slice of degrees. Each chart is built for one kind of reading: lengths for comparing, angles for shares, positions over time for trends. A pie chart needs parts that make a whole, so it cannot show data where one person may pick two answers. Continuous measurements such as times need grouping into classes first. Add the angles of a pie chart to confirm it: 180 + 90 + 90 = 360.
the box spans Q₁ to Q₃, IQR = 13, and holds ten of the twenty values however narrow it gets: a narrow box is density, not fewer points
Pack the middle ten together, then push the top value past the fence
| Display | Suits | Answers |
|---|---|---|
| Bar chart | Categories | Which is biggest, and by how much |
| Pie chart | Categories that form a whole | What share each takes |
| Vertical line chart | Discrete counts | The shape of a distribution of whole numbers |
| Dot diagram | Small numeric sets | Clusters, gaps and outliers |
| Stem and leaf | Small numeric sets | Shape, while keeping every original value |
| Histogram | Continuous measurements | How the values are spread across a range |
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. 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.
Now you
One display that groups the marks yet keeps every exact value. Which?
One chart to compare their sizes at a glance. Which?
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 shown by the area of a bar, not its height, and the bars touch because the data are continuous. With equal class widths, area and height are proportional, so it looks like a bar chart with the gaps closed. See Histograms. When the widths differ, plot frequency density, the frequency divided by the class width. A class from 0 to 10 minutes holding 30 people has density 30 ÷ 10 = 3, and a class from 10 to 30 minutes holding 40 people has density 40 ÷ 20 = 2. The second bar is lower but wider, and its area, 20 × 2 = 40, still equals its frequency. Plain heights would overstate the wide class, which collects more values just by being wider. The vertical axis is then labeled frequency density. Check by finding each bar's area, 10 × 3 = 30 for the first, and adding the areas to reach the total: 30 + 40 = 70 people.
height = frequency whatever the width: the class of width 20 is drawn as tall as it would be at width 10, so a wide class becomes a false peak
Switch to density and widen the last class to 40
Once the class widths differ, height alone stops being a fair comparison.
Frequency density is a rate in items per unit. 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.
Now you
A bar has frequency density 6 over a width of 2. What frequency does it hold?
A bar has frequency density 10 over a width of 5. What frequency does it hold?
How do charts mislead?
A chart can show every number correctly and still mislead, usually through its scale or its shapes. The most common trick is a vertical axis that does not start at 0. Suppose two shops sell 97 and 100 items. With the axis starting at 94, the bars are 3 and 6 units tall, since 97 − 94 = 3 and 100 − 94 = 6, so one looks twice the other although the real gap is 3 in 100. Readers compare bar lengths, and a length only matches its value when the bar starts from zero. Pictures mislead another way: a drawing twice as tall and twice as wide covers 2 × 2 = 4 times the area, so a doubling looks like a quadrupling. A line graph may start above zero if its axis shows a clear break. Read the axis numbers and work out the real ratio: 100 ÷ 97 is barely more than 1.
baseline 0: the bars share their anchor, so their heights are in the ratio of the values, 102 : 98 = 1.04
Raise the baseline to 95 and watch a 4% difference become a gap
- A truncated vertical axis. 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. A picture that grows in two directions grows twice over.
- 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.
Now you
A bar chart starts its scale at 30 instead of 0. What does that do?
A bar chart starts its scale at 40 instead of 0. What does that do?
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.
Your turn
Three to try — tap what you get.
A bar chart shows 8 red and 5 blue. How many more red?
In a pictogram one symbol means 4 books. What do 3½ symbols mean?
Best chart for change over time?
Questions and answers
- What is the difference between a bar chart and a histogram?
- A bar chart shows categories, so its bars have gaps between them and can be reordered without losing meaning. A histogram shows a continuous measurement split into intervals, so its bars touch and the order is fixed by the number line. The deeper difference is what carries the frequency: in a bar chart it is the height, but in a histogram it is the area, which is why unequal intervals need frequency density on the vertical axis.
- When should you use a pie chart?
- Use a pie chart only when the categories make up a whole and you want to show each one as a share of it, and only when there are few enough slices to tell apart, usually under six. A pie chart is poor at comparison, because people judge angles less accurately than lengths, and it cannot show a total at all. If the question is which category is largest, a bar chart answers it faster.
- What is frequency density and why do histograms need it?
- Frequency density is frequency divided by class width, and it is the height that makes a histogram's areas honest. If one class spans 10 units and the next spans 40, plotting raw frequency as height makes the wide class look four times more crowded than it is. Dividing by the width converts each bar into a rate per unit, so the area of every bar equals its frequency and the eye compares like with like.
- How do you tell if a graph is misleading?
- Check the vertical axis first: if it does not start at zero, differences are exaggerated, sometimes deliberately. Then check that the intervals along the horizontal axis are equal, that a pictogram's symbols all represent the same count, and that any area or volume used to represent a quantity has been scaled in one dimension rather than two. A chart can be entirely accurate and still be built to be read wrongly.
- What is a two-way table for?
- A two-way table cross-classifies the same people by two things at once, such as year group against method of travel, so you can see whether the two are related. The row and column totals give each variable on its own, and the interior cells give the combinations. It is the layout that conditional probability questions are usually asked from, because reading down one column instead of the whole table is exactly what conditioning means.