Making bar graphs
You have counted how many of each kind there are. How do you draw those counts so that anyone can see them at a glance, and read the exact numbers back?
Where you are 4 in Bar graphs and pictographs
When you reach for it
Feruza’s class has a sidewalk chalk art day. Each child picks one color of chalk to start with, and Feruza keeps the count in a table (Data tables, tallies and frequency tables): Blue 7, Pink 5, Yellow 3, Green 6. The table holds every number, but a reader has to compare four numbers in their head to see which color won. A bar graph shows the same counts as bars, one for each color, and the longest bar is the winner before anyone reads a number.
The child has already read bar graphs (Reading a bar graph): find the scale, follow the end of a bar across to it, read the number. Making one runs that the other way. Start from a count, and draw a bar whose end sits level with that count on the scale. Every part a reader relies on (a title, labels, a scale from 0 in equal steps) is now something the maker has to put there.
You reach for this whenever counts sorted into a few kinds need to be shown: a class vote, a tally from the playground, things brought in for a project. Every graph here has at most four categories, and the counts come from a table, a tally, or a plain list.
The lesson takes three days. Day 1 builds a bar graph from a table or a tally on a scale that counts by 1, with bars up the page and across it. Day 2 checks a finished graph against its table, and starts from a list in which one category was chosen by nobody. Day 3 is short: a scale that counts by 2 or 5 when the counts are big, and one look at a double bar graph.
The procedure
Day 1: from a table to a bar graph
The graph is built in two parts: first the frame, then the bars. The frame is everything a reader needs before reading a bar, so it goes down first.
- A title says what was counted: “Chalk Colors We Picked”.
- A label for each category goes under the bottom line, spaced evenly, one for each row of the table. The order can be the table’s order; any order gives a correct graph.
- A scale runs up the left side. It starts at 0 at the bottom line and goes up in equal steps. On Day 1 each step is 1. It has to reach at least the biggest count, here 7; going one step past it, to 8, leaves a little room at the top.
- A scale label says what the numbers count: “Number of children”.
Grid paper makes the frame easy, since each square is one step. Without it, a ruler keeps the steps equal: mark the scale every centimeter, or every half inch.
Then the bars. For each category, go up from its label to the line level with its count on the scale, and draw a bar from the bottom line up to there. Blue’s count is 7, so Blue’s bar stops level with the 7.
Each part of the frame has a reason.
The scale starts at 0, and every bar starts on the bottom line. The count is the length of the bar measured from 0, the way a ruler measures from its 0. A bar that starts above the line, or a scale whose bottom number is not 0, would make a bar look shorter or longer than its count.
The steps are equal. Each space on the scale stands for the same number. That is what lets a reader carry a bar’s end over to the scale and trust the number there, and what makes a bar twice as long mean twice as many.
The bars are the same width, with a gap between each two. Only the length carries the count. A wider bar has more shading and looks like more even when it is the same height, so every bar gets the same width. The gaps show that the categories are separate: no child picked “between blue and pink”.
A good way to see why this works is with linking cubes. Give each child one cube in the color they picked, and snap the cubes of each color into a tower. Stand the towers side by side on the table. That is a bar graph made of cubes: one cube for each child, each tower as tall as its count, all standing on the same table. Drawing the graph is drawing those towers, with the scale doing the counting that the cubes did.
Bars can also go across the page. Then the labels go down the left side, the scale runs along the bottom, and each bar grows to the right from the left line. Everything else is the same.
When the counts arrive as a tally, read each row first. Each bundle of five marks is 5, and the loose marks are counted on from there, so a bundle and 3 marks is 5 + 3 = 8. Writing the counts as a table first keeps the drawing simple.
Day 2: checking a graph against its table
A finished graph is checked part by part against the table it came from. The checks are the reading moves of Reading a bar graph, made on purpose.
- Is there a title, and does it say what was counted?
- Is there a label for every row of the table, and no extra one?
- Does the scale start at 0 and go up in equal steps? Is there a scale label?
- Follow each bar across to the scale. Does it read the number in the table?
- Are the bars the same width, with gaps?
One more check covers all the bars at once. Each thing counted is in exactly one row of the table and exactly one bar, so the bars added together must make the table’s total. If the totals differ, some bar is off.
A category with a count of 0. Sometimes a choice was offered and nobody took it. The table still has its row, with 0. On the graph it keeps its label, with no bar above it, because a bar of length 0 is no bar at all. Leaving the label off would hide the fact that the choice was there.
Starting from a list. When the data is a list of answers, one per child, make a tally or table first, crossing each answer off as it is counted so none is counted twice or missed. The number of answers in the list must equal the table’s total.
Day 3: bigger counts, and two bars side by side
Choosing what the scale counts by. On Day 1 every count fit on a scale that counts by 1. With counts like 16 and 18, a scale by 1 needs 18 steps or more: either a very tall graph or steps too small to draw neatly. A scale that counts by 2 (Counting by twos, fives, threes and fours) needs only half as many steps. With counts like 25 and 40, counting by 5 fits even better (Skip counting). The rule is the same as before: start at 0, take equal steps, and reach at least the biggest count. Choose a step that keeps the graph a sensible size and puts as many bar ends as possible on labeled lines.
A bar that ends between lines. On a scale by 2, the labeled lines are the even numbers. A bar for 11 goes up to the line for 10 and then half a space more, since half of a space worth 2 is 1. That is the only place 11 can be, because it is the one whole number between 10 and 12. On a scale by 5, several whole numbers sit between two lines, and a bar between them is hard to read exactly; it is best to pick the step so that does not happen. Scales like these are the main work of Scaled picture graphs and of reading graphs in Reading a graph: titles, labels and units.
Two bars side by side. Sometimes two groups answer the same question: two classes, or the morning and the afternoon. A double bar graph puts both on one frame. Each category gets a pair of bars, touching, one for each group, with a gap between pairs. The two groups’ bars look different (one shaded, one plain), and a key in the corner says which is which. Without a key a reader cannot tell whose bar is whose. The pairs make one comparison easy: inside a category, which group had more.
Worked examples
Examples 1 and 2 go with Day 1, Examples 3 and 4 with Day 2, and Examples 5 to 7 with Day 3.
Day 1: Feruza’s chalk colors
Feruza’s table says Blue 7, Pink 5, Yellow 3, Green 6. Make a bar graph of it.
SolutionStart with the frame. The title says what was counted: “Chalk Colors We Picked”. Under the bottom line go four labels, Blue, Pink, Yellow and Green, evenly spaced. The biggest count is 7, so the scale must reach at least 7; it starts at 0 and counts by 1 up to 8. The scale label is “Number of children”. That is the empty frame shown in the procedure.
Now one bar for each color, all the same width, with gaps. Blue’s bar goes from the bottom line up to the line for 7. Pink’s stops at 5, Yellow’s at 3, Green’s at 6.
Check two ways. The tallest bar is Blue, and 7 is the biggest count in the table, so they agree. And the bars together make the table’s total:
so the graph shows 21 children, one bar’s worth for each child in the table.
Day 1: from a tally, with bars going across
On the box fort building day, children bring empty boxes. Bahrom keeps a tally of each kind. Make a table from his tally, then a bar graph with the bars going across.
Read each row. Shoe box has one bundle of five and 3 loose marks, so 5 + 3 = 8. Cereal box has 4 loose marks, so 4. Pizza box has 2. Moving box has one bundle and 1 mark, so 5 + 1 = 6.
| Kind of box | Number of boxes |
|---|---|
| Shoe box | 8 |
| Cereal box | 4 |
| Pizza box | 2 |
| Moving box | 6 |
For bars going across, the four kinds go down the left side and the scale runs along the bottom, from 0 by 1s. The biggest count is 8, so the scale reaches 8. Each bar starts at the left line and grows to the right until it is above its count: Shoe box to 8, Cereal box to 4, Pizza box to 2, Moving box to 6.
Check: the table says 8 + 2 = 10 and 4 + 6 = 10, so 20 boxes. The bars add to the same 20, and the longest bar, Shoe box, is the biggest count.
Day 2: does the graph match the table?
At Gulnora’s playdough table, each child makes one animal. Her table says Snake 9, Bird 4, Cat 6, Turtle 5. Check this graph of it against the table.
Go part by part.
The title, “Playdough Animals”, says what was counted. There is a label for each of the four rows, and no extra. The scale starts at 0, counts by 1 in equal steps, and has a scale label. The frame is right.
Now each bar. Snake goes across to 9, as in the table. Bird reads 4, as in the table. Cat reads 7, but the table says 6: that bar is one step too tall. Turtle reads 5, which matches, but its bar is much wider than the others. A wide bar looks like more children even at the right height, so it should be redrawn the same width as the rest.
The totals show the Cat problem too. The table’s total and the total of the bars as drawn are
and they differ by 25 - 24 = 1. The fixed graph has the Cat bar at 6 and the Turtle bar the same width as the others.
Day 2: from a list, and a color nobody chose
Gulnora puts out four colors of playdough: red, blue, green and yellow. As each child comes to the table she writes down the color they take:
red, blue, red, green, blue, red, red, blue, green, red, blue, red, green, blue
Make a table and a bar graph.
SolutionGo along the list once, making a tally mark in the right row for each word and crossing the word off. Red gets 6 marks, blue 5, green 3. Yellow gets none, since it is not in the list at all.
| Color | Number of children |
|---|---|
| Red | 6 |
| Blue | 5 |
| Green | 3 |
| Yellow | 0 |
The list has 14 words, and the table adds to 6 + 5 + 3 + 0 = 14, so every answer was counted once.
The biggest count is 6, so a scale by 1 up to 7 fits. Red’s bar ends at 6, Blue’s at 5, Green’s at 3. Yellow keeps its label, with no bar above it: its count is 0, and a bar of length 0 does not rise off the bottom line.
Keeping the Yellow label tells the reader that yellow was on the table and nobody took it. Leaving it off would make the graph look as if only three colors were offered.
Day 3: a scale that counts by 2
On the box fort day, four rooms bring boxes: Room 3 brings 16, Room 5 brings 11, Room 6 brings 18, Room 8 brings 14. Choose a scale and make the graph.
SolutionThe biggest count is 18. A scale by 1 needs at least 18 steps, which makes a very tall graph. Counting by 2 reaches 18 in 9 steps: 2, 4, 6, \ldots, 18. Going one step more, the scale runs 0, 2, 4 and so on to 20.
Three counts are even, so their bars end on labeled lines: Room 3 at 16, Room 6 at 18, Room 8 at 14. Room 5 has 11, which is not a labeled line. It is 1 more than 10, and one space on this scale is worth 2, so the bar goes up to the line for 10 and then half a space more:
A scale by 5 would be shorter still, but then 16, 11, 18 and 14 would all end between lines, with four whole numbers sitting in each space, and a reader could not tell 16 from 17. Counting by 2 is the better choice here.
Day 3: a scale that counts by 5
At the end of the chalk day, Feruza counts the pieces of chalk the school used up: Blue 25, Pink 40, Yellow 15, Green 30. What should the scale count by?
SolutionEvery count is in the skip-counting-by-fives list 5, 10, 15, 20, 25, 30, 35, 40. So a scale that counts by 5 from 0 to 40 puts every bar end on a labeled line, and it needs only 8 steps. Counting by 1 would need 40 steps, and by 2 would need 20.
Blue’s bar goes up 5 steps, to the line for 25; Pink’s goes 8 steps, to 40; Yellow’s 3 steps, to 15; Green’s 6 steps, to 30.
A scale by 10 would work too, with 4 steps, but then Blue and Yellow would end halfway between lines, and the eye would have to judge the middle of a space that holds nine whole numbers.
Day 3: one look at a double bar graph
On the chalk day, each child in Feruza’s class and in Bahrom’s class draws one picture: a sun, a flower or a rocket. Feruza’s class: Sun 5, Flower 7, Rocket 3. Bahrom’s class: Sun 6, Flower 4, Rocket 8. Show both classes on one graph. Then use it: in which picture did Feruza’s class draw more than Bahrom’s, and by how many? How many flowers were drawn in all?
SolutionThe frame is as usual: a title, a label for each picture, and a scale from 0 by 1s up to 8, since the biggest count in either class is 8. Above each picture go two bars, touching: a shaded one for Feruza’s class and a plain one for Bahrom’s. A key in the corner shows a shaded square beside “Feruza’s class” and a plain square beside “Bahrom’s class”.
Now compare inside each pair. Sun: shaded 5, plain 6. Flower: shaded 7, plain 4. Rocket: shaded 3, plain 8. Only for the flower is the shaded bar taller, so Feruza’s class drew more flowers, by
Flowers in all is both bars of the Flower pair: 7 + 4 = 11.
Two separate graphs would show the same numbers, but the reader would have to look back and forth. Side by side, the taller bar in each pair stands out at once.
Boundaries
- The scale starts at 0 and every bar starts on the bottom line. The count is the bar’s length from 0. Start anywhere else and a longer bar no longer means more in the right amount.
- The steps of the scale are equal. If one space were worth 1 and the next 2, a bar’s end could not be carried to the scale and read.
- The scale reaches the biggest count. A bar taller than the scale has no number to end on.
- Only length carries the count. Bars of different widths, colors that mean nothing, or pictures on the bars must not change how much a bar seems to show.
- Each thing counted is in exactly one bar. That is why the bars add to the total. If a child could pick two colors, the bars would add to more than the number of children.
- A bar between two lines is exact only when one whole number fits there. On a scale by 2 it is the number halfway. On a scale by 5 or more, pick the step so bars end on lines, or the reader can only say “about”.
- A double bar graph needs its key. Without it the two bars in a pair cannot be told apart.
Where this goes next
Making a graph and reading one are the same skill from two ends, and every check here is a question a reader asks (Reading a bar graph). A picture graph shows the same kind of count with one picture for each thing; a bar graph is that row of pictures pressed into one bar, with the counting moved to the scale. Later, a picture can stand for several things (Scaled picture graphs), and scales count by 5, 10 or more, so the choice of step made on Day 3 becomes the main work (Reading a graph: titles, labels and units).
A dot plot shows counts too, but its categories are numbers on a number line and it has one mark for each thing (Reading dot plots: clusters, gaps and peaks, Making a dot plot). Much later, double bar graphs grow into the graphs used to compare two groups (Comparing two distributions) and to see whether two answers go together (Association in a two way table and segmented bar graphs).
Exercises
Day 1: from a table to a bar graph
Exercises 1 to 4 use this table of what children made at the playdough table: Pizza 4, Cookie 7, Cupcake 5, Pretzel 2.
- Write a title and a scale label for a bar graph of this table.
- The scale starts at 0 and counts by 1. What is the smallest top number that fits every bar?
- Draw the bar graph. Where does each bar end, and which bar is the tallest?
- How many children do the bars stand for in all?
Exercises 5 and 6 use Bahrom’s tally of fort supplies. Tape rolls: one bundle of five and 3 marks. Blankets: two bundles of five. Pillows: one bundle of five and 1 mark. Tubes: 4 marks.
- Write the count for each row.
- Bahrom draws the bars going across. Which bar is the longest, and what number must the scale reach at least?
- Why must every bar in a bar graph be the same width?
- A table has the row “Purple 0”. What does the bar graph show for purple?
Day 2: checking a graph against its table
Exercises 9 to 11 use this graph of a table of games played on the chalk day: Hopscotch 8, Four square 5, Tag 6, Jump rope 3.
- Which bar does not match the table? What should it be?
- What part of the graph is missing?
- Add the table’s counts, and add the bars as drawn. How do the two totals show something is wrong?
Exercises 12 and 13 use this list of the tool each child took at the playdough table, where rollers, cutters, stamps and scissors were set out:
roller, cutter, roller, stamp, roller, cutter, roller, roller, stamp, cutter, roller
- Make a table of the four tools.
- In the bar graph, which tool gets a label but no bar? How do you know every answer in the list was counted?
- One graph of a table puts the bars in the order Blue, Pink, Yellow, Green. Another graph of the same table puts them in the order Green, Yellow, Pink, Blue. Can both be right? Explain.
- A table has four rows that add to 20. Its graph has bars for 6, 5 and 4, and the bar for Purple is missing. How tall should the Purple bar be?
Day 3: bigger counts, and two bars side by side
Exercises 16 and 17 use a scale by 2 from 0 to 20 for the number of boxes in four forts: Castle 13, Rocket 17, Cave 9, Tower 20.
- Which bar ends exactly on a labeled line?
- Between which two labeled lines does the Castle bar stop, and where in that space?
Exercises 18 to 20 use the paper cups used at the playdough table: Monday 20, Tuesday 35, Wednesday 10, Thursday 45.
- Should the scale count by 1, by 2 or by 5? Explain.
- On a scale by 5, how many steps tall is the Tuesday bar?
- Will a scale by 5 that stops at 40 fit every bar? Explain.
Exercises 21 to 23 use this double bar graph.
- Which colors were picked more in the afternoon than in the morning?
- How many children picked blue in all, morning and afternoon together?
- What would a reader lose if the key were left off this graph?
Explain
- Why must the scale of a bar graph start at 0?
- Why must the steps on the scale be equal?
Answers
- For example, title “Playdough Snacks We Made” and scale label “Number of children”. The title says what was counted; the scale label says what the numbers count.
- 7. The biggest count is 7 (Cookie), so the scale must reach at least 7; going to 8 is fine too.
- Pizza ends at 4, Cookie at 7, Cupcake at 5, Pretzel at 2. The tallest is Cookie, because 7 is the biggest count.
- 18. Make two nines: 7 + 2 = 9 and 4 + 5 = 9, then 9 + 9 = 18.
- Tape rolls 5 + 3 = 8. Blankets 5 + 5 = 10. Pillows 5 + 1 = 6. Tubes 4.
- Blankets, with 10. The scale must reach at least 10, the biggest count.
- Only the length of a bar carries its count. A wider bar has more shading and looks like more even at the same length, so different widths would mislead the eye.
- The label Purple, with no bar above it. A count of 0 is a bar of length 0, and keeping the label shows that purple was a choice nobody picked.
- Four square. Its bar ends at 4, but the table says 5, so it should go up one more step.
- The title. Without it a reader does not know the graph counts games played on the chalk day.
- The table adds to 8 + 5 + 6 + 3 = 22 and the bars to 8 + 4 + 6 + 3 = 21. Each child is in one row and one bar, so the totals should match; they differ by 22 - 21 = 1, so a bar is off by 1.
- Roller 6, cutter 3, stamp 2, scissors 0. Go along the list once and tally each word.
- Scissors gets a label but no bar, since its count is 0. The list has 11 words and the table adds to 6 + 3 + 2 + 0 = 11, so every answer was counted once.
- Yes. Each bar still ends at its own count, and the labels say which bar is which. The order of the categories does not change any count.
- 5. The four bars must add to 20, and 6 + 5 + 4 = 15, so Purple is 20 - 15 = 5.
- Tower, at 20. The labeled lines are the even numbers, and 13, 17 and 9 are odd.
- Between 12 and 14, exactly halfway, since 12 + 1 = 13 and half a space on this scale is 1.
- By 5. Every count is in the list 5, 10, 15, \ldots, so every bar ends on a labeled line, and reaching 45 takes 9 steps. By 1 would take 45 steps and by 2 more than 20, with 35 and 45 ending between lines.
- 7 steps. Counting by fives, 5, 10, 15, 20, 25, 30, 35 is 7 counts.
- No. Thursday’s count is 45, and 45 > 40, so its bar would run past the top. The scale must reach at least 45.
- Red (7 > 4) and green (5 > 3). Blue went the other way, 6 in the morning and 2 in the afternoon.
- 8. Add the two Blue bars: 6 + 2 = 8.
- The reader could not tell which bar in each pair is the morning and which the afternoon, so no comparison between the two times could be made. The bars would show numbers with no meaning.
- The count is the length of the bar measured from 0, like a ruler. If the scale started at, say, 2, every bar would look 2 shorter than its count, and a bar for 4 would look twice as long as a bar for 3 instead of just a little longer.
- A reader carries the end of a bar over to the scale and reads the number there. That only works if every space is worth the same amount. With unequal steps a bar’s length would no longer match its count, and a bar twice as long would not mean twice as many.