Bar Graphs, Picture Graphs and Tables

How could you find your class's favorite fruit? You need data: the answers you collect. A guess about your friends is not a count of your class.

Plan your question

Ask, "Which fruit is your favorite? Choose one." Give everyone the same choices: apples, bananas, grapes, oranges, or other fruit. Each answer has one clear home. Choices such as "dog" and "pet" would overlap in a pet survey. A dog belongs to both groups, so the choices would be confusing.

The answer groups are called categories. Some questions give word answers, such as kinds of fruit. Others give counted numbers. For "How many brothers and sisters do you have?", use groups such as 0, 1, 2, and 3. Keep these number groups in order.

Choose how to collect

A survey means asking people questions. Ask everyone in the group your question is about. If you want the whole class's favorite sport, ask the whole class. Asking only the soccer team tells you about that team.

You can also watch and count. To find which vehicles pass the school, count cars, buses, and bikes for 10 minutes. Or use facts already recorded, such as a list of library books. Choose a way that answers your question.

Keep track with tallies

Make one tally mark for each answer. Draw four lines for the first four answers. For the fifth, draw a line across those four. Start a new group after five.

Suppose you have three groups of five and two single lines. Count 5, 10, 15, then 16, 17. The tally shows 17 answers. Each complete group holds five marks.

The frequency is how many times an answer appears. Write each tally as a number. If 24 children answer once each, all the frequencies must add to 24. A total of 25 tells you to check for an extra mark or repeated answer. A total of 23 tells you an answer may be missing.

Use your answers

Working with data follows a cycle. Ask a question, collect answers, organize them in a table, and show them in a graph. Then use the numbers to answer your question. You might next ask, "Would another class choose the same fruit?" That starts a new round of collecting.

A table puts information into rows and columns. Rows go across. Columns go down. The title and headings tell you what the numbers mean.

Make a frequency table

Here is "Our favorite fruit". Each child chose one fruit.

Each fruit is a category. Its number is its frequency. To make this table, write a title and headings, list each category once, and turn its tally into a number. A tally column can help while you collect. The frequency column holds numbers. Add the frequencies for the total: $7+5+9+3=24$ children.

Read two headings together

Some tables show two things at once. "How we get to school" shows the class and the way children travel.

How many children in Class 3B come by bus? Start at the row "3B". Follow it across to the column "Bus". The number where they meet is 10. Your finger or a ruler can help you stay on the correct row.

Add across or down

A row total adds across one row. Class 3A has $12+6+8=26$ children. Class 3B has $9+10+5=24$.

A column total adds down one column. For walking, add 12 and 9 to get 21. For bus travel, add 6 and 10 to get 16. For car travel, add 8 and 5 to get 13.

The grand total counts everyone in the table. Add the row totals: $26+24=50$. Check by adding the column totals: $21+16+13=50$. Both ways count the same children.

Do not add the grand total to the other numbers. It already includes them. Adding 26, 24, and 50 would count everyone twice.

Answer with the correct group

"How many come by bus altogether?" asks for the whole bus column: 16 children. "How many more in 3B than 3A come by bus?" asks for two bus entries: $10-6=4$ more children.

Word categories, such as fruit names, can appear in any order. Putting the biggest count first can help you compare. Counted number categories, such as 0, 1, 2, and 3 siblings, stay in number order.

A picture graph shows data with pictures or symbols. Each row or column is a category. The title says what was counted, and labels name the groups. You may also hear "pictograph" or "pictogram".

Read the key first

The key says how much one whole picture stands for. In a scaled picture graph, one picture means more than one thing. A picture might stand for 2, 5, or 10.

"Our favorite sport" uses one smiley picture for 5 children. Here are its rows, described in words.

For soccer, count four groups of five: 5, 10, 15, 20. Or multiply: $4\times 5=20$ children. The answer is 20 children, not 4. Four counts the pictures.

Swimming has $2\times 5=10$ children. Basketball has $3\times 5=15$, and tennis has 5. Altogether, $20+10+15+5=50$ children answered.

Read a half picture

A half picture is worth half the key. If a whole picture means 10 books, half means 5 books. If a whole picture means 2 birds, half means 1 bird.

In "Library books borrowed", the key is one book picture for 10 books. Tuesday has three whole pictures and half a picture. First read the whole pictures: $3\times 10=30$. Add the half: $30+5=35$ books. Counting the half as 10 would give 40; leaving it out would give 30.

Compare two rows

Rows in one graph use the same key and picture size. More pictures therefore show a bigger amount. Soccer has two more pictures than swimming. Those two pictures stand for $2\times 5=10$ more children. You can also subtract the amounts: $20-10=10$.

Notice what a different key changes

The same 20 children need four pictures with a key of 5. With a key of 10, they need two pictures. With a key of 2, they need ten pictures. The count of children stays 20. The number of pictures changes because each picture now stands for a different amount.

To draw a picture graph, start with the counts you collected. You will turn each count into a row of pictures. Choose a simple symbol you can draw several times.

Choose a helpful key

Look at all the numbers before choosing the key. For counts of 20, 35, 15, and 40, a key of 5 works well. Each amount is a whole number of fives.

Dividing tells you how many pictures to draw. For 35, ask, "How many fives make 35?" Seven fives do, so draw seven pictures.

Also think about space. With a key of 1, a count of 40 needs 40 pictures. With a key of 10, it needs just four. Choose a key that keeps the rows easy to draw and read.

Use half pictures when you need them

For "Birds we saw", use one bird picture for 2 birds. The counts are robins 8, sparrows 13, crows 6, and blue jays 3.

Robins need four whole pictures. Crows need three. For 13 sparrows, six whole pictures show 12 birds. Half a picture shows the last bird, so draw six and a half pictures. Three blue jays need one and a half pictures.

With a key of 10, a count of 25 needs two whole pictures and a half. The two whole pictures show 20. The half adds 5.

A key of 5 is awkward for a count of 12. Two pictures show 10, but half a picture would add two and a half. A key of 2 makes 12 easy: draw six whole pictures.

Lay out your graph

Write the title, list the categories, and write the key beside the graph. Work out each row before drawing it. Make every whole picture the same size. Start the rows at the same edge and leave equal spaces between pictures.

A row can look longer just because its pictures are bigger or farther apart. Matching sizes and spaces makes the comparison fair.

Check the finished rows by reading them back. For six and a half pictures with a key of 2, six pictures show 12 and the half shows 1. The count is 13, matching the sparrow count. A missing key would leave the reader unable to work out these amounts.

In a block graph, each block stands for one thing. Stack eight blocks to show eight children. A bar graph uses solid bars instead. Read each bar's length using the numbers beside it.

Find the title and labels

A bar graph has two axes: the lines along the bottom and side. One axis names the categories. The other shows the number scale. Its label says what you counted, such as "Number of children".

Here is a description of a vertical bar graph.

The source names where the data came from. It tells you whose answers the graph shows.

Match length to amount

The scale tells you the value of each equal step. On this graph, each step is 2. Grid lines across the page help you follow a bar's end to the scale.

Make every bar the same width and leave equal gaps. Length shows the amount. A wider bar must not mean more children. Gaps show that the colors are separate groups.

Bars can go up the page, called vertical, or across it, called horizontal. Vertical bar graphs are sometimes called column graphs. "Bar chart" is another name for a bar graph. On a horizontal graph, follow the end down or up to its number axis.

Put categories in a useful order

Colors and fruits can go in any order. Ordering them from biggest to smallest helps you compare. Number categories, such as 0, 1, 2, and 3 siblings, stay in number order.

Begin at zero

A bar graph's scale starts at 0. This lets bar lengths show the sizes fairly. Imagine a graph for 12 and 20 that began at 10. The visible lengths would be 2 and 10 steps of 1. The bar for 20 would look five times as long, although 20 is not five times 12.

You may meet a bar-line graph. It draws each bar as a thin line. Read the end of each separate line using the scale. A line graph is different: it joins points together, often to show change over time.

Before reading a bar, check the number scale. Equal steps might count by 2, 5, or 10. Counting the squares as ones can give the wrong amount.

Find the step size

Look at two numbers next to each other. A scale marked 0, 5, 10, 15, 20 rises by 5 each time. Subtracting two neighbors checks it: $10-5=5$.

For a bar six squares tall on that scale, count six fives. You get $6\times 5=30$. The bar shows 30 things, not 6.

Follow the bar's end

For vertical bars, find the top and follow straight across to the scale. For horizontal bars, follow straight up or down to the number axis. A ruler helps you stay level.

Suppose "Books on shelves" has a scale starting at 0 and counting by 10 to 50. Its categories are shelf A, shelf B, and shelf C. Their bar ends are 40, halfway between 30 and 40, and 20.

Half of the ten-book gap is five, so shelf B has 35 books. Reading only the line below the end would lose five books.

Read lines without numbers

Sometimes only a few lines have numbers beside them. Count the equal spaces between numbered lines. If five spaces lie between 0 and 10, each space is worth $10÷5=2$.

Start at the labeled line for 10. A bar ending three small spaces higher shows $10+2+2+2=16$. Do not count the line for 10 as the first extra space.

Read between the main lines

On a scale of 2, halfway between 6 and 8 is 7.

A scale of 5 needs more care. Between 10 and 15 are 11, 12, 13, and 14. Imagine five equal small spaces, each worth 1. A bar for 12 ends two small spaces above 10. A bar for 13 ends three small spaces above 10. The middle is twelve and a half, so neither 12 nor 13 ends exactly halfway.

A clear bar graph starts with your table. Find the largest count first. That tells you how high the number scale must reach.

Choose a scale that fits

Imagine you have ten squares up the side. A step of 2 reaches 20, a step of 5 reaches 50, and a step of 10 reaches 100.

A step of 2 cannot fit a count of 45 in ten squares. Its scale only reaches 20. Larger steps let bigger amounts fit. Smaller steps can make nearby amounts easier to read.

Draw the axes and numbers

Draw the bottom and side axes with a ruler. Label the category axis and the number axis. Write 0 where they meet. Number the lines in equal steps.

For a step of 5, write 0, 5, 10, 15, 20. A scale marked 0, 5, 10, 20, 30 has unequal jumps. Equal spaces on the page must mean equal amounts.

Work through a color graph

The table "Favorite colors" has red 8, blue 14, green 5, and yellow 11. The largest count is 14. Use a scale of 2 from 0 to 16. Label the bottom "Color" and the side "Number of children".

Draw four equal-width bars with equal gaps. End red on 8 and blue on 14. Green ends halfway between 4 and 6, at 5. Yellow ends halfway between 10 and 12, at 11. Write each color under its bar and put the title above the graph.

A bar need not end on a numbered line. On a scale of 10, a count of 25 ends halfway between 20 and 30. Do not stretch it to 30 just to reach a line. If many counts are awkward to place, consider a smaller step that still fits your largest number.

Read your drawing back

Check each bar against the table. Does the blue bar show 14? Does the green bar show 5? Check the zero, equal steps, labels, widths, and gaps too. A source such as "Class color survey" tells readers where the answers came from.

A bar graph's scale and a picture graph's key do similar jobs. Both tell you what one step or picture is worth. You can change from one graph to the other while keeping every count the same.

Start by finding what each amount means. In one bar graph, the longest bar shows the biggest count. The shortest shows the smallest. Matching bar lengths show equal amounts.

Find the most and the fewest

For a picture graph, read the key first. Then compare rows that use the same key and matching pictures. For a table, compare the numbers.

"Our favorite fruit" has apples 7, bananas 5, grapes 9, and oranges 3. Grapes have the most votes. Oranges have the fewest. From most to fewest, the order is grapes, apples, bananas, oranges.

In a favorites survey, you can call the biggest group most popular. For a count of vehicles or books, say which group has the most. Those counts do not tell you what anyone likes.

Find how many more

Read both counts. Subtract the smaller from the bigger.

The picture graph "Our favorite sport" has a key of one smiley for 5 children. Soccer has four whole pictures, basketball three, swimming two, and tennis one. Soccer therefore has 20 children and basketball 15.

"How many more chose soccer than basketball?" asks for the gap:

Five more children chose soccer. Adding would give the number in both groups, which is a different question.

Find how many fewer

"How many fewer chose basketball than soccer?" looks at the same gap. Basketball has 5 fewer children than soccer. You still subtract 15 from 20. "How many less" is another way you may hear this question.

Check what form your answer needs. "Which sport?" needs a sport name. "How many more children?" needs a number of children.

Use steps carefully

A bar graph can show the gap as a number of steps. If one bar ends two equal steps higher, and each step is 5, the difference is $2\times 5=10$. Two steps do not mean two children on that scale.

Compare by doubling

Soccer has 20 votes and swimming has 10. Since $2\times 10=20$, twice as many children chose soccer. On a bar graph starting at 0, the soccer bar would be twice as long. This is a different comparison from asking for the gap of 10.

A question may ask for a total, a missing count, or a comparison after joining groups. Read the whole question and name the groups you need.

Add for a total

"Our favorite fruit" records one choice per child: apples 7, bananas 5, grapes 9, and oranges 3. To find everyone who answered, add every category once:

The total is 24 children. Words such as "altogether" and "in all" can signal adding. Still check which groups the question names. "Apples and bananas altogether" needs only those two groups.

Solve two steps

Ask, "How many more children chose apples or bananas than grapes?"

1. Join the apple and banana groups: $7+5=12$.
2. Compare that group with grapes: $12-9=3$.

The answer is 3 more children. Twelve is only the first step. It counts apples and bananas together but does not give the difference.

Find who is left

"How many did not choose grapes?" means everyone outside the grape group. Subtract grapes from the total: $24-9=15$.

You can check by adding the other fruits: $7+5+3=15$. Both methods count the same children. The subtraction check also works backward: $15+9=24$.

Find a missing category

Suppose 30 children vote for four choices. Three choices get 8, 10, and 7 votes. First add the known counts: $8+10+7=25$. Then take them from the total: $30-25=5$. The missing choice has 5 votes. Taking just 8 from 30 would leave the other known votes in your answer.

Change pictures into amounts

The graph "Library books borrowed" uses one book picture for 10 books. Its Monday row has three whole pictures and its Wednesday row has five. Read the counts first: Monday 30 and Wednesday 50. Then add: $30+50=80$ books on those two days.

Counting eight pictures is a useful start, but the answer asks for books. Eight pictures mean eight tens, or 80 books.

For a longer question, write the first answer with a label, such as "apples and bananas: 12". Then look back at the question to find what you still need to do.

The same answers can appear in a tally chart, table, picture graph, or bar graph. Changing the display must not change the counts.

Pick a useful display

Picture graphs can be quick to read. Half pictures and large keys can make exact counts take more work. Choose a display that helps answer your question.

To turn a picture graph into a table, multiply each row's pictures by the key. To turn a table into a bar graph, choose a scale and draw bars for those same counts. For the fruit survey, apples stay 7, bananas 5, grapes 9, and oranges 3 in every display.

Check a statement with numbers

"More children chose apples than oranges" is true: 7 is greater than 3.

"Half the class chose grapes" is false. There were 24 children, and half of 24 is 12. Only 9 chose grapes. Having the biggest group does not mean having half of everyone.

Check scales before comparing graphs

Two bars on different graphs may look equally tall. That does not mean their counts match.

Imagine both bars reach five steps above 0. The first graph counts by 2. Its bar shows $5\times 2=10$. The second counts by 10. Its bar shows $5\times 10=50$. Read each scale before comparing the amounts.

A picture key can change too. Four pictures with a key of 5 show 20. Two pictures with a key of 10 also show 20. Compare the amounts the pictures stand for.

Write a conclusion

A conclusion answers your question using the data. You could write, "Grapes were our class's favorite fruit: 9 of the 24 children chose them." The sentence names the group and gives a number to support the answer.

Your class survey tells you about your class. It cannot tell you every child's favorite fruit across the country. It also does not explain why children made their choices. To learn their reasons, ask another question, such as "Why did you choose that fruit?" To compare classes, collect the same kind of answers from another class.

A Venn diagram sorts items with circles. Each circle has a label giving a rule. Items that fit the rule go inside. Items that do not fit go outside. A box around the circles shows everything you are sorting.

Start with one rule

Sort 3, 8, 10, 15, 20, 21, and 25 using "Multiples of 5". Numbers you can count in fives belong inside: 10, 15, 20, and 25. The numbers 3, 8, and 21 stay outside the circle, inside the box.

Use two overlapping circles

Two circles can share a middle part called the overlap. It holds items that fit both rules. There are four possible places: only the first circle, the overlap, only the second circle, and outside both.

Sort 2, 5, 6, 7, 10, 12, 15, 18, and 20. The circles are "Even numbers" and "Multiples of 5".

Ten is even and a multiple of 5. Write it once in the overlap. Writing it separately in both circles would count the same number twice. Seven is neither even nor a multiple of 5, but it still belongs in the box.

Count the whole circle

The overlap is inside both circles. To count all even numbers, include the four in "Even only" and the two in the overlap: $4+2=6$. Counting only the four would miss 10 and 20.

For all multiples of 5, include 5 and 15 plus 10 and 20: four numbers. To count every item, add the four separate parts once: $4+2+2+1=9$.

Read a diagram with counts

A diagram can show counts instead of names. In a class of 24, 6 children have only a dog, 4 only a cat, 3 both, and 11 neither.

All dog owners include the overlap: $6+3=9$. All cat owners include it too: $4+3=7$. For the class total, count the overlap just once: $6+4+3+11=24$.

Some pairs of rules cannot overlap. No whole number is both even and odd. Their overlap is empty, and you may see those circles drawn apart.

A Carroll diagram sorts items into boxes. Each rule has a yes side and a no side, such as "Even" and "Not even". One rule needs two boxes. Two rules make four boxes.

Test both rules

Put one rule along the columns and the other along the rows. Use "Even" and "Not even" across the top. Use "Multiple of 5" and "Not a multiple of 5" down the side.

Sort 2, 5, 6, 7, 10, 12, 15, 18, and 20.

For 10, answer both questions: "Is it even?" Yes. "Is it a multiple of 5?" Yes. Follow those two headings to the same box. For 7, both answers are no. It goes where "Not even" meets "Not a multiple of 5".

Every item has one box. There is no outside area to use. The "not" boxes already hold items that fail one rule or both rules.

Match boxes to circles

These are the same nine numbers sorted by the same rules as a Venn diagram. The "Even" and "Multiple of 5" box matches the circles' overlap. The box for neither rule matches the space outside both circles. Each display keeps the same items together.

Read counts from the boxes

Here is a class of 24 children sorted by pets.

To count all children with a dog, add down that column: $3+6=9$. To count children with a cat, add across that row: $3+4=7$. Add the four boxes once to count everyone: $3+4+6+11=24$.

Make the rules clear

A rule must give a clear yes or no. "Big" can mean different things to different people. "Longer than 10 centimeters" gives a clear test. An object exactly 10 centimeters long goes in "Not longer than 10 centimeters".

"Not red" includes every color except red: blue, green, yellow, and more. It does not mean just blue.

After sorting, count every item in the boxes. The total must match your starting list. This helps you spot a missing item or an item written twice.