Frequency Tables, Bar Graphs and Pie Charts

A survey starts with data: facts or values that answer a question. Before counting, decide what kind of answers you have.

Name the kind of data

Categorical data name groups, such as favorite fruit. Numerical data give numbers from counting or measuring. Counts are discrete: a student can have 0, 1 or 2 pets. Measurements are continuous: a time can fall anywhere in a range, including between whole seconds.

A category can have a number as its name. A bus route number is a label, not a measurement.

Count once with tallies

A frequency tells how often an answer occurs. A frequency table lists each answer beside its count. Add a tally column while sorting the raw answers.

1. Write every possible category, including any that receive no answers.
2. Read the answers in order. Make one tally for each answer, then cross that answer off.
3. Draw four upright tally strokes. Cross them with the fifth stroke. Start a new group after five.
4. Count the tallies and write a number in the frequency column.

Seven tally marks form one crossed group of five and two extra strokes. The frequency column contains the number 7.

Build and read a table

Twenty students choose a favorite fruit. Their answers are:

"A B G G O A B G A G O A G B A G O B A G"

Here A means apple, B banana, G grapes and O orange. Tallying each answer gives:

Add the frequencies: $6+4+7+3=20$. That matches the number of answers. A matching total can still hide two errors that cancel. Check the tallies against the answers too.

Grapes has the highest frequency, so grapes is the mode. Apple has two more votes than banana, because $6-4=2$. Banana and orange together have seven votes.

Find a share as a percentage

Divide a frequency by the total, then multiply by 100. For grapes:

So 35 percent of the students chose grapes. The exact percentages add to 100 percent. Rounding can make the displayed total slightly different, such as 99 or 101 percent. Keep the counts as well as the percentages when making a graph: the counts show how many answers the shares describe.

A table can become long when nearly every value differs. A grouped frequency table puts values into classes, or intervals. Each row gives the number of values in one range.

Make classes that fit

Whole-number reading times could use 0–14, 15–29, 30–44 and 45–59 minutes. Each class holds 15 possible whole numbers. This is its class width. Subtracting 14 minus 0 gives 14, but misses the included value 0.

Use equal widths to make the groups easy to compare. About four to eight classes often works for a small data set. Include a class even when its frequency is zero.

Every value must fit exactly once. Classes 0–10 and 10–20 overlap at 10. For whole numbers, 0–9 and 10–19 avoid that overlap. Check the smallest and largest values so the classes cover all the data.

Sort reading times

Twenty students recorded these whole-number minutes:

12, 35, 47, 5, 28, 30, 18, 52, 41, 22, 15, 9, 33, 44, 27, 58, 20, 36, 14, 31.

Read one number at a time. The first value, 12, goes in 0–14. The next, 35, goes in 30–44. Cross each value off after tallying it.

Check: $4+6+7+3=20$. The modal class is 30–44 minutes, because its frequency is greatest. At least 30 minutes includes the final two classes, giving $7+3=10$ students.

Place measurements at boundaries

For continuous measurements, whole-number labels can leave gaps. Use inequalities to show which endpoints belong:

This means heights from 140 cm up to, but excluding, 150 cm. The next class is $150\le h<160$. A height of exactly 150 cm goes there. A height of 149.8 cm stays in the first class.

Know what the table hides

The row 30–44 tells you that seven times lie in that range. It does not tell you their exact values. You cannot recover the number who read exactly 30 minutes, or the longest reading time, from the grouped table alone. Return to the raw list for those questions.

A double bar graph places two bars beside each category. One bar belongs to each data set. It is also called a dual, multiple or comparative bar graph. A simple bar graph may be called a bar chart or column graph.

Plan the graph

Start with a table. Students in two grades chose one favorite sport:

Put the sports along the category axis. Label the other axis "Number of students". The largest count is 12. A scale from 0 to 14 in steps of 2 gives enough space.

Equal distances on the axis must show equal increases. With labels 0, 2, 4 and 6, each step means two students. A bar halfway between 8 and 10 shows 9 students.

Draw the pairs

Choose a color or shading for each grade. A key, also called a legend, tells the reader which group each style means.

For soccer, draw the Grade 6 bar to 12 and its neighbor to 9. Repeat this for each sport. Make every bar the same width. Keep the pair close together and leave a gap before the next sport. Use the same group order in every pair.

Add the title "Favorite sports in Grades 6 and 7". Check all eight heights against the table. The key, both axis labels and the scale are part of the information the reader needs.

Compare across and within groups

To compare grades, read the two bars for one sport. Basketball has 11 Grade 7 votes and 8 Grade 6 votes:

Three more Grade 7 students chose basketball.

To compare sports within Grade 6, follow that grade's color. Soccer has 12 votes and tennis has 3:

Soccer was chosen four times as often. A difference uses subtraction; "times as many" uses division.

The most popular sport also differs between grades. Soccer leads in Grade 6, while basketball leads in Grade 7.

Check the size of each group

Both grades here have 28 students, so their counts compare equally sized groups. In groups of different sizes, a higher count can still be a smaller share. Find each count as a percentage of its own group's total when comparing how common a choice is.

A stacked bar graph shows one total as a bar split into parts. It is also called a composite, compound, component or segmented bar graph. A colored part is a segment.

Build a bar from running totals

An ice-cream stand records sales:

Keep the flavors in the same order in every bar. On Monday, vanilla starts at 0 and ends at 20. Add chocolate to reach $20+15=35$. Add strawberry to reach $35+10=45$.

These running totals mark the segment edges. They do not give the size of each segment. Use equal-width bars, a zero-starting scale, a title and a key.

Read a segment by subtraction

Wednesday's chocolate segment starts at 15 and ends at 45. Its size is:

So 30 chocolate cones were sold. Reading the top edge as 45 would count the vanilla cones too.

The whole bar ends at 60, showing Wednesday's total. Bottom segments are easy to compare because they share the zero line. Middle segments may start at different heights. Calculate their sizes before comparing them: Tuesday's chocolate has 20 cones, while Monday's has 15.

Compare shares with percentage bars

A percentage stacked bar graph makes every bar reach 100 percent. First divide each flavor's count by that day's total.

For chocolate, Monday gives about 33 percent, Tuesday 40 percent and Wednesday 50 percent. All bars have the same height, even though their sales totals differ. The bars compare the split within each day. Keep the totals labeled if readers also need amounts.

Use one divided bar

A divided bar shows one whole as a rectangle split into shares. It works like a pie chart stretched into a straight bar. If the bar is 10 cm long, 30 percent takes 3 cm:

Choose double bars when comparing each category across groups matters most. Choose stacked bars when readers need each total and its parts. Percentage bars help compare the shares within unequal totals.

A pie chart, also called a circle graph, shows how one whole is shared. Each slice is a sector. Its size matches its category's share of the total.

Connect a sector to a full turn

The full circle is 100 percent and a full turn is 360°. A quarter of the data takes a quarter of that turn:

A sector's center angle, area and curved edge each have the same fraction of the circle. Read the angle where the two straight edges meet at the center.

A diameter splits the circle into halves. A right angle marks a quarter. Use those familiar shapes to judge a flat chart. A slice must extend beyond half a turn to be more than half.

Read a sports chart

Twenty-four students choose one favorite sport. Soccer takes half the circle, basketball a quarter, and swimming and tennis an eighth each.

Half of 24 is 12 students. A quarter is 6. Each eighth is 3. Check the whole: $12+6+3+3=24$.

Without the total of 24, you could still compare shares. Soccer has twice basketball's share. You could not find the number of soccer fans.

Check that the parts form one whole

Pie-chart categories must share the same whole without overlap. A survey asking each student for one favorite sport fits. A survey allowing several sport choices cannot show those choices as nonoverlapping shares of the students.

Labels may give names, counts, percentages or angles. Exact percentages of all parts total 100 percent. Rounded labels can total slightly above or below 100.

Count squares in a waffle diagram

A waffle diagram uses a grid instead of a circle. On a 10-by-10 grid, 100 squares show the whole. Each square represents 1 percent, so 35 colored squares mean 35 percent.

Use flat charts when judging sizes. Tilting a pie chart changes how its angles and areas look. In a tilted chart, read the numerical labels or find a flat view.

A sector gives a fraction of the whole. To turn it into an amount, apply that fraction to the chart's total.

Find an amount from an angle

A travel survey covers 60 people. The bus sector measures 72°:

So bus travelers make up one fifth of the group:

There are 12 bus travelers. The general rule is:

You can also find the angle for one person. Here $360÷60=6$ degrees per person. Then $72÷6=12$ people. The 72° angle itself is not a count of 72 people.

Find an amount from a percentage

When 25 percent of 80 students walk, calculate:

Twenty students walk. Divide percentages by 100, but angles by 360°. These are two ways to describe the same share.

Work backward to the total

A 45° sector represents 15 students. Since 45° is one eighth of 360°, the full circle contains eight such shares:

The total is 120 students. Check backward: one eighth of 120 is 15. Scaling 15 students across eight equal portions avoids treating one degree as one student.

Find a missing part

All sector angles total 360°. If three angles are 150°, 90° and 60°, their sum is 300°. The missing sector measures ${360}^{\circ }-{300}^{\circ }={60}^{\circ }$.

Within one chart, angle comparisons also compare amounts. A 120° sector contains twice the amount of a 60° sector, because both belong to the same total.

Compare two charts fairly

School A has 200 students, and 30 percent walk. School B has 500 students, and 20 percent walk.

School A has the larger walking share, but School B has more walkers. Equal-sized circles do not tell you equal totals. Calculate each amount from its own total before comparing numbers of people.

Each frequency needs the same amount of angle per item. First add the frequencies to find the total. Then divide 360° among all the items.

Turn counts into angles

Thirty students choose a favorite pet: dog 12, cat 9, fish 4 and rabbit 5. One student's share of the circle is:

Multiply every frequency by 12°:

The check gives $144+108+48+60=360$. Every vote has the same angle, so larger counts make larger sectors.

For any total:

Multiplying a frequency by 360 without dividing by the total would give each item a whole circle.

Start from percentages or fractions

One percent of the full turn is 3.6°. Multiply the percentage number by 3.6°. For bus 35 percent, walk 25 percent, car 15 percent and bike 25 percent, the angles are 126°, 90°, 54° and 90°. They total 360°.

For a fraction, multiply the fraction by 360°. A three-eighths share has angle:

Keep decimals until the final angle

Frequencies 3, 2 and 2 total 7. Calculate each angle using the full division:

• First: $3\times 360÷7\approx 154.29$ degrees.
• Second and third: $2\times 360÷7\approx 102.86$ degrees each.

To the nearest degree, use 154°, 103° and 103°. These total 360°. Rounding the angle per item to 51° first gives 153°, 102° and 102°, leaving a 3° gap.

Close a small rounding gap

Frequencies 5, 5 and 4 total 14. Their angles are about 128.57°, 128.57° and 102.86°. Rounded separately, 129°, 129° and 103° total 361°.

Reduce one 129° sector to 128° for the drawing. This one-degree adjustment closes the circle. Keep the original counts and the rounding work in the table so the small adjustment is clear.

Use a compass or a round object to draw the circle. Have a ruler, protractor and sharp pencil ready. Calculate all sector angles before drawing.

Place the first two radii

Mark the circle's center clearly. Draw a radius from the center to the edge as your starting line. Pointing it straight up is a useful choice.

For the pet chart, start with the dog angle of 144°:

1. Put the protractor's center mark on the circle's center.
2. Line up its zero line with the starting radius.
3. Follow the scale that begins at that zero.
4. Mark 144°, then draw a radius through the mark.

The space between the two radii is the dog sector. The circle's size does not change the required angle.

Move the starting line

The cat sector is 108°. Align the protractor with the new dog edge. Measure 108° from that edge in the same direction around the circle. Draw the next radius.

Now measure 48° from the cat edge for fish. The remaining space should be 60° for rabbit.

Measuring 144°, 108° and 48° from the original line would place edges in the wrong order. Each individual angle starts from the previous sector's edge.

Use running totals as an alternative

You can keep one starting radius if you use running totals:

• Dog ends at 144°.
• Cat ends at $144+108=252$ degrees.
• Fish ends at $252+48=300$ degrees.

A semicircular protractor stops at 180°. To mark 252° from the start, measure $360-252=108$ degrees in the opposite direction.

Check before labeling

A protractor has two scales. The wrong scale turns a 48° reading into 132°, since $180-48=132$. Check which zero sits on your starting radius before reading.

Measure the remaining rabbit sector. If it measures 64° instead of 60°, check the earlier edges and redraw the error. About one degree of difference is normal in a hand drawing.

Label every sector with its category, directly or through a key. Add counts or percentages where useful. Finish with a title such as "Favorite pets of 30 students". Small sectors can have labels outside the circle, joined by a clear guide line.

Choose a display by asking what the reader needs to find. A list of exact counts, a comparison and a share of a whole each need a different view.

Match the display to the question

A table is useful for looking up 7 or 12 exactly. A graph makes a pattern easier to see quickly. Choose the view that serves the question.

Check the kind of data

Named categories can go in a table, bar graph or pie chart. Discrete counts can form bar-graph categories too. For example, use 0, 1, 2 and 3 pets along the category axis. Each bar counts the students with that many pets.

Measurements with many different values can first go into equal-width classes in a grouped frequency table. Keep the units in the class labels.

Decide whether a pie fits

A pie chart needs categories that make one whole without overlap. One favorite lunch per student fits. Students checking every lunch they like may appear in several categories, so those counts cannot be shares of the students in one pie.

Pie charts work best with a few clear parts. Many tiny sectors become hard to label and compare. A bar graph can make those categories easier to read.

Small differences are hard to judge from slices. Sectors of 23 percent and 27 percent look similar. Bars beside a shared scale show the four-percentage-point difference more clearly.

Choose between bar displays

"Did two grades choose sports differently?" calls for double bars. Readers can compare the pair for each sport. "How did each day's ice-cream sales split into flavors?" calls for stacked bars, showing both each day's total and its parts.

Middle stacked segments lack a shared starting line. Readers may need subtraction to compare them. For a question focused on those categories, double bars can be easier.

When group sizes differ, choose percentage bars to compare shares. Pie charts can also show each group's shares, with totals labeled. To compare amounts, calculate the counts from those shares and totals.

A graph can support some statements and leave others unanswered. Read what was measured before deciding what the numbers mean.

Read the information around the picture

Check the title, axis labels, units, scale and key. Find the source and total too. A bar reaching 6 means six items only if the frequency scale says so. The same height could mark 30 on a scale counting in fives.

Reading a value is the first step. Comparing values adds a calculation. Predicting a new group's choices goes beyond the recorded answers and needs care.

Name the comparison

Grade 6 sports counts include 12 soccer votes and 3 tennis votes. The difference is:

Nine more students chose soccer. The multiple is:

Four times as many chose soccer. "Nine more" and "four times as many" answer different questions.

Separate counts from shares

Class A has 10 cyclists among 20 students. Class B has 12 among 30:

Class B has more cyclists. Class A has the higher cycling share. The phrase "Class B bikes more" leaves the comparison unclear. Say whether you mean people or percentages.

Decide what can be told

A Grade 7 favorites graph gives basketball 11 votes and soccer 9. "Basketball received more votes than soccer" is true. "Soccer received more votes" is false.

"Grade 7 is better at basketball" cannot be told. The survey measured favorites, not skill. The graph also does not tell you why the students chose basketball.

A true statement fits the recorded values. A false statement conflicts with them. "Cannot be told" means the needed information was not measured or shown.

Keep the conclusion with the surveyed group

A survey of 28 students describes those 28 students. Another class could answer differently. A few changed answers can strongly affect a small group's percentages.

A clear conclusion names the group, the comparison and the numbers: "Basketball received 11 of the 28 Grade 7 votes, more than any other sport." A claim about every child in the country would need evidence from beyond this class.

A misleading graph makes the numbers seem different from what they are. This can happen by accident or by design. Read the values before trusting the picture's message.

Check where the bars begin

Sales rise from 50 to 55. A bar axis starts at 45, so the visible lengths are:

The second bar looks twice as long. Actual sales rose by 5, which is 10 percent of 50. They did not double. This cut-off scale is a truncated axis. Start bar axes at zero so lengths match amounts.

Check equal steps and widths

Labels 0, 10, 20, 50 and 100 cannot sit at equal distances on a regular bar scale. Their increases differ. Use equal numerical steps at equal distances.

Bars must also have equal widths. A wide bar can look more important even when its height matches another bar. Keep all categories visible; dropping one can change the story.

Watch picture sizes

One bag icon represents 1,000 bags sold. A second result is 2,000 bags. Making its icon twice as tall and twice as wide gives four times the area:

The picture exaggerates a doubling. Use two copies of the same-size icon to show 2,000 bags instead.

Check pie-chart design and labels

A tilted, 3D-style pie changes the apparent angles and areas. A sector at the front can look larger than an equal sector at the back. Pulling one slice out also draws extra attention. Use a flat chart and check numerical labels.

A sector labeled 40 percent must not be drawn as half a circle. Half means 50 percent. Exact percentages for one whole total 100 percent; rounded ones can differ slightly. For a pie with four categories rounded to whole percentages, a total of 110 percent is too far away to be rounding.

Repair the message

Look for a title, units, key, source, date and total. Missing information can stop you checking what a display means.

Redraw bars from zero with equal widths and equal scale steps. Use a flat pie and include every category. Replace an exaggerated headline with the real comparison: "Sales rose from 50 to 55, an increase of 10 percent.

An infographic combines displays and short statements to tell a data story. It might use a table, bars, a pie chart and a large fact box. Read both the individual displays and the links between them.

Find the setting

Begin with the title and headline. Find the source, date and number of people or items covered. These tell you whose answers you are reading and when they were collected.

Read each display's labels and scale on its own. A number in a fact box may be a count, while a nearby sector label is a percentage. Identify the whole behind each percentage.

Check that the numbers agree

An infographic called "How we get to school" covers 150 students:

For walking, $\frac{40}{100}\times 150=60$. The count matches the table. The four counts total $60+45+30+15=150$. Check each remaining share in the same way.

If a fact box said "70 walkers", it would disagree with both the table and 40 percent of 150. A clear layout cannot repair that number error.

Test the headline

Last year, 45 percent of 120 students walked:

Now 60 students walk. "More of us walk than last year" is true for counts. Yet walking's share fell from 45 percent to 40 percent. A careful headline might say "More walkers, but a smaller share of students".

You needed both totals to see the difference. Comparing 45 percent directly with 60 students would mix two kinds of quantity.

Check each design choice

Bar axes should begin at zero and rise in equal steps. Pie labels should fit the sectors. Percentages for one whole should total 100, allowing small differences from rounding. Icons should keep the same size for the same amount.

Finally, compare the dates and groups named under the displays. A class survey and a whole-school survey do not share the same total. State which group each claim describes before combining their messages.

Start an infographic with a question and a reader. For a school newsletter, the question could be "How do students in our grade get to school?" This gives you a reason to select the numbers and displays.

Plan the findings

Organize the data in a frequency table. Check the total, then pick two to four useful messages. A message should tell the reader something about the data, such as the most common travel choice or a change from last year.

For a 150-student travel survey, walking has 60 votes, bus 45, car 30 and bike 15. Walking is the most common choice, at 40 percent. It is not a majority: 40 percent is less than half.

Give each display a job

A pie chart shows the travel shares. A table gives exact counts. A double bar graph can compare this year's and last year's travel counts when both sets are available. A fact box can highlight "60 of 150 students walk".

Do not draw several charts just to fill space. Each one should answer a clear question. A stacked bar would be useful if you wanted to show several totals and their parts.

Arrange the page

Use a title that identifies the data. Place an accurate headline below it, such as "Walking is the most common choice: 40 percent walk". Put the displays in a reading order from top to bottom and left to right.

Write a short caption under each chart. Give the source, date and number of students asked at the bottom. Keep labels large enough to read and use the same color for walking wherever it appears.

Make the drawings fair

Start bar axes at zero. Use the same scale on separate graphs meant for direct comparison. Choose flat charts, equal-width bars and same-size icons for equal amounts.

Check that percentages total 100 percent, apart from small rounding differences. Counts, sector labels and fact boxes must agree. A pretty page with conflicting numbers leaves readers unsure which value to trust.

Check it as a reader

Can someone find the group size and source? Does every caption match its display? Does the headline fit the numbers? Ask someone to explain the message using the charts alone.

You can work by hand or use spreadsheet or design software. Software draws the selected chart from your table. You still choose the display, scale and labels, and check what the headline claims.