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Dot Diagrams

A number line with one dot for each value: how to draw a dot diagram, read it, describe what it shows and compare two of them.

What You Should Already Know

  • Categorical data are names or labels, such as favourite fruit or mode of transport. Numerical data are numbers got by counting or measuring.
  • Numerical data are discrete if they can take only certain values, such as whole numbers (the number of siblings: 0, 1, 2, …), and continuous if they can take any value in a range (a height of 152.4 cm, a time of 21.7 s).

What Is a Dot Diagram?

Ten pupils wrote down how long they took to travel to school, in minutes: 20, 22, 21, 21, 18, 18, 22, 20, 30, 23. A list like this is hard to take in at a glance. A dot diagram shows it as a picture: a number line, with one dot above it for each pupil.

Dot diagram of the travel times of 10 pupils on a line from 18 to 30 minutes: 2 dots at 18, 2 at 20, 2 at 21, 2 at 22, 1 at 23 and 1 at 3018192021222324252627282930Time (minutes)

Two pupils took 18 minutes, so there are two dots above 18, one on top of the other. No pupil took 19 minutes, but 19 is still marked on the line, and so are 24 to 29. Because every minute has its place, you can see at once that most pupils took 18 to 23 minutes, and that the pupil who took 30 minutes is far from the rest.

Dot diagram

A dot diagram is a horizontal number line with a dot above it for each value in the data.

Each dot stands for one value (one pupil, one match, one day). Dots for the same value are stacked one above another, all the same size and the same distance apart.

The number line has equal steps and covers every value from the smallest to the largest (it may start a little before the smallest and end a little after the largest). Every value in between is marked, even one with no dots. Number every value, or every second value with a tick at each.

Drawing a Dot Diagram

Steps

1. Find the smallest and the largest value.

2. Draw a number line that covers the smallest to the largest value, with equal steps, and mark every value the data can take in between; number each one, or every second one.

3. Count how many times each value occurs (cross the values off the list as you go, so that none is missed or counted twice).

4. Draw that many dots above each value, stacked at equal steps. Give the line a name with its unit.

5. Check: the number of dots is the number of values in the list.

Example 1: Drawing a dot diagram

A group of pupils sent art pieces to a school exhibition. The numbers of pieces sent by the 22 pupils were: 6, 13, 12, 5, 6, 12, 12, 6, 8, 8, 10, 9, 10, 4, 5, 14, 6, 8, 10, 9, 13, 12. (a) Draw a dot diagram of the data. (b) Which numbers of art pieces were the most common?

Solution

(a) The smallest value is 44 and the largest is 1414, so the line goes from 44 to 1414 in steps of 11.

Count each value: 44: 1, 55: 2, 66: 4, 77: 0, 88: 3, 99: 2, 1010: 3, 1111: 0, 1212: 4, 1313: 2, 1414: 1.

Check: 1+2+4+0+3+2+3+0+4+2+1=22,1 + 2 + 4 + 0 + 3 + 2 + 3 + 0 + 4 + 2 + 1 = 22\text{,} the number of pupils.

No pupil sent 7 or 11 pieces, but 7 and 11 are still marked on the line, with no dots above them.

Dot diagram of the number of art pieces sent by 22 pupils, on a line from 4 to 14: 1 dot at 4, 2 at 5, 4 at 6, none at 7, 3 at 8, 2 at 9, 3 at 10, none at 11, 4 at 12, 2 at 13 and 1 at 144567891011121314Number of art pieces

(b) The tallest columns are at 66 and at 1212, with 4 dots each. Both are the most common numbers of pieces: a set of data can have two most common values.

(a) The dot diagram is drawn above, with 22 dots. (b) 6 pieces and 12 pieces were the most common (4 pupils each).

Reading a Dot Diagram

Each dot is one value, so a question about "how many" is answered by counting dots. Read the words carefully: "at least 25" includes 25, "more than 22" does not include 22, and "fewer than 18" does not include 18.

Example 2: Counting and percentages

The dot diagram shows the time, in minutes, that each pupil of Class 2E takes to travel to school. (a) How many pupils are in the class? (b) How many pupils take exactly 23 minutes? (c) How many take at least 25 minutes? (d) How many take fewer than 18 minutes? (e) What percentage of the class takes more than 22 minutes? (f) What is the most common time?

Dot diagram of the travel times of Class 2E on a line from 16 to 28 minutes: 1 dot at 16, 2 at 17, 3 at 18, 2 at 19, 4 at 20, 3 at 21, 2 at 22, 3 at 23, 1 at 24, 2 at 25, none at 26, 1 at 27 and 1 at 2816171819202122232425262728Time (minutes)

Solution

(a) Count every dot: 1+2+3+2+4+3+2+3+1+2+0+1+1=251 + 2 + 3 + 2 + 4 + 3 + 2 + 3 + 1 + 2 + 0 + 1 + 1 = 25 pupils.

(b) There are 3 dots above 23.

(c) "At least 25" means 25 or more: 25, 26, 27 and 28. That is 2+0+1+1=42 + 0 + 1 + 1 = 4 pupils.

(d) "Fewer than 18" means 16 or 17: 1+2=31 + 2 = 3 pupils.

(e) "More than 22" means 23 or more: 3+1+2+0+1+1=83 + 1 + 2 + 0 + 1 + 1 = 8 pupils.

Percentage =825×100%=32%.= \frac{8}{25} \times 100\% = 32\%\text{.}

(f) The tallest column is above 20, with 4 dots.

(a) 25 pupils (b) 3 pupils (c) 4 pupils (d) 3 pupils (e) 32% (f) 20 minutes.

Describing What a Dot Diagram Shows

Describing data from a dot diagram

Where the values lie: say where they go from and to ("the ages go from 14 to 25 years"). The range is one number: the largest value minus the smallest.

Where most values are: a group of values close together (a cluster), and the most common value.

A value far from the rest, and any gaps where there are no values.

Example 3: Describing a distribution

The dot diagram shows the ages, in years, of the 12 contestants in a karaoke contest at a community centre. Describe the ages.

Dot diagram of the ages of 12 karaoke contestants on a line from 13 to 26 years: 1 dot at 14, 1 at 20, 2 at 22, 4 at 23, 3 at 24 and 1 at 251314151617181920212223242526Age (years)

Solution

Where the values lie: the youngest is 14 and the oldest is 25. Range =25−14=11= 25 - 14 = 11 years.

Where most values are: 10 of the 12 contestants are aged 22 to 25, and the most common age is 23 (4 contestants).

Values far from the rest: the contestant aged 14 is far below the others. There is a gap from 15 to 19, where there are no contestants, and no contestant is 21.

The ages go from 14 to 25 years (range 11 years). Most contestants (10 of 12) are 22 to 25, the most common age being 23. One contestant, aged 14, is far younger than the rest, and no contestant is aged 15 to 19.

Completing and Comparing Dot Diagrams

Example 4: Completing a dot diagram

Mr Lim recorded how many library books each of the 24 pupils of his class borrowed in a month. He drew the dots for the first 18 pupils, shown below. The other 6 pupils borrowed 3, 5, 2, 5, 7 and 3 books. (a) Complete the dot diagram. (b) What is now the most common number of books borrowed?

Dot diagram of the books borrowed by the first 18 pupils, on a line from 0 to 8: 1 dot at 0, 2 at 1, 3 at 2, 4 at 3, 3 at 4, 2 at 5, 2 at 6, none at 7 and 1 at 8012345678Number of books borrowed

Solution

(a) Add one dot for each of the 6 values: one more at 2, two more at 3, two more at 5 and one at 7.

The completed dot diagram of the books borrowed by 24 pupils: 1 dot at 0, 2 at 1, 4 at 2, 6 at 3, 3 at 4, 4 at 5, 2 at 6, 1 at 7 and 1 at 8012345678Number of books borrowed

Check: 1+2+4+6+3+4+2+1+1=241 + 2 + 4 + 6 + 3 + 4 + 2 + 1 + 1 = 24 dots, one for each pupil.

(b) The tallest column is now at 3, with 6 dots.

(a) The completed diagram is drawn above. (b) 3 books (borrowed by 6 pupils).

To compare two sets of data, draw their dot diagrams on the same scale, one above the other. Then you can compare where the values lie and how spread out they are.

Example 5: Comparing two dot diagrams

The dot diagrams show the travel times to school, in minutes, of 12 pupils from Class 2A and 12 pupils from Class 2B. (a) Find the range of each class's times. Which class's times are more spread out? (b) How many pupils in each class take 20 minutes or less? Which class generally takes longer to travel to school?

Two dot diagrams on one scale from 10 to 30 minutes. Class 2A: 15, 16, 16, 17, 18, 18, 18, 19, 20, 21, 22, 23. Class 2B: 11, 13, 17, 20, 22, 24, 24, 25, 25, 26, 27, 30Class 2A101112131415161718192021222324252627282930Class 2B101112131415161718192021222324252627282930Time (minutes)

Solution

(a) Class 2A: range =23−15=8= 23 - 15 = 8 minutes. Class 2B: range =30−11=19= 30 - 11 = 19 minutes.

Class 2B's range is larger, and its dots are spread along most of the line, while Class 2A's dots are bunched between 15 and 23.

(b) Class 2A: 15, 16, 16, 17, 18, 18, 18, 19, 20, so 9 pupils take 20 minutes or less. Class 2B: 11, 13, 17, 20, so 4 pupils.

Also, 7 pupils of Class 2B take 24 minutes or more; no pupil of Class 2A does.

(a) Class 2A: 8 minutes; Class 2B: 19 minutes. Class 2B's times are more spread out. (b) 9 pupils in Class 2A and 4 in Class 2B. Class 2B generally takes longer.

When to Use a Dot Diagram

A dot diagram shows every value, so you can read each one, count them and see where they cluster. It works well when the data are numbers that take only a few different values. It does not suit data with many different values over a wide range (the line becomes very long and the dots thin out), or a very large set of data (too many dots to draw and count), or categorical data (there is no number line for names).

Example 6: Is a dot diagram a good choice?

For each set of data, say whether a dot diagram is a good way to show it, and why. (a) The favourite fruit of 30 pupils. (b) The number of siblings of each of 30 pupils. (c) The number of books read in a year by each of 50 pupils, from 0 to 45, with many different values. (d) The heights of 30 pupils, measured to the nearest 0.1 cm.

Solution

(a) Favourite fruit is categorical data: there are no numbers to put on a line. A bar graph or a pie chart suits it.

(b) The number of siblings is discrete numerical data with only a few values (0, 1, 2, 3, …). A dot diagram is a good choice.

(c) The line would need 46 values from 0 to 45, and the 50 dots would be spread thinly along it, so no pattern would show. Grouping the values into intervals is better (a histogram, lesson 12.4), or a stem-and-leaf diagram (lesson 12.2).

(d) Heights are continuous data. Measured to 0.1 cm, almost every height is different, so nearly every dot would stand alone on a very long line.

A dot diagram is a good choice only for (b).

Example 7: Spot the error

A football team scored these numbers of goals in 12 matches: 0, 1, 1, 2, 2, 2, 3, 3, 4, 6, 1, 2. Two pupils drew dot diagrams of the data. What is wrong with each? Draw it correctly.

Diagram A:

Diagram A: a number line marked 0, 1, 2, 3, 4, 6 with equal gaps, 5 left out; 1 dot at 0, 3 at 1, 4 at 2, 2 at 3, 1 at 4 and 1 at 6012346Number of goals

Diagram B:

Diagram B: a number line from 0 to 6; 1 dot at 0, 3 dots at 1 drawn far apart, 4 dots at 2, 2 at 3, 1 at 4 and 1 at 60123456Number of goals

Solution

Diagram A leaves 5 off the line, because no match had 5 goals. Then 6 sits right next to 4, and the match with 6 goals does not look far from the rest, as it is.

Diagram B stacks the dots at 1 further apart than the others, so the column of 3 dots looks taller than the column of 4 dots at 2.

Correct diagram: every value from 0 to 6 marked, and every column stacked at the same step.

The correct dot diagram of the goals in 12 matches, on a line from 0 to 6: 1 dot at 0, 3 at 1, 4 at 2, 2 at 3, 1 at 4, none at 5 and 1 at 60123456Number of goals

Diagram A leaves out the value 5; Diagram B stacks the dots at 1 at a different step. The correct diagram marks 0 to 6 and stacks every column at the same step.

Common Mistakes

  • Leaving a value with no dots off the number line. Every value from the smallest to the largest must be marked, or values far apart look close together.
  • Numbering the line only every 5 when the dots stand at single values: no one can then tell whether a dot is at 22 or 23. Number every value (or every second value, with a tick at each).
  • Drawing dots of different sizes, or stacking some columns more widely than others. A column's height must show only how many dots it has.
  • Misreading the words: "at least 25" includes 25, but "more than 22" and "fewer than 18" do not include 22 and 18.
  • Giving the range as two numbers ("18 to 23"). The values go from 18 to 23; the range is 23−18=523 - 18 = 5.
  • Counting the columns instead of the dots when finding how many values there are.

Practice Questions

Try each question first, then open the solution to check your working.

Question 1

Say whether each set of data is categorical or numerical. If it is numerical, say whether it is discrete or continuous. (a) The number of pets each pupil has. (b) The time taken to run 2.4 km. (c) The CCA each pupil takes part in. (d) The shoe sizes of the pupils in a class. (e) The mass of a school bag.

ShowHide solution

(a) Numerical, discrete: it is a count (0, 1, 2, …).

(b) Numerical, continuous: a time can take any value in a range (11 min 23.4 s).

(c) Categorical: a CCA is a name, not a number.

(d) Numerical, discrete: shoes come only in certain sizes (6, 6.5, 7, …), with none in between.

(e) Numerical, continuous: a mass can take any value in a range.

(a) numerical, discrete (b) numerical, continuous (c) categorical (d) numerical, discrete (e) numerical, continuous.

Question 2

The numbers of text messages sent by 15 pupils on a Saturday were: 12, 15, 18, 12, 20, 18, 15, 10, 18, 20, 15, 12, 18, 15, 20. (a) Draw a dot diagram of the data. (b) What is the most common number of messages? (c) How many pupils sent at least 18 messages?

ShowHide solution

(a) The smallest value is 10 and the largest 20. Count: 1010: 1, 1212: 3, 1515: 4, 1818: 4, 2020: 3. Check: 1+3+4+4+3=15.1 + 3 + 4 + 4 + 3 = 15\text{.}

Dot diagram of the messages sent by 15 pupils on a line from 10 to 20: 1 dot at 10, 3 at 12, 4 at 15, 4 at 18 and 3 at 201011121314151617181920Number of messages

(b) The columns at 15 and 18 are the tallest, with 4 dots each, so both are the most common.

(c) "At least 18" means 18 or 20: 4+3=74 + 3 = 7 pupils.

(a) Drawn above. (b) 15 and 18 messages (4 pupils each). (c) 7 pupils.

Question 3

The dot diagram shows how many times each pupil of Class 2C visited the school library last month. (a) How many pupils are in the class? (b) How many pupils visited at least 4 times? (c) What fraction of the class visited fewer than 2 times? Give it in its lowest terms. (d) What percentage of the class visited exactly 3 times? Give your answer to 1 decimal place.

Dot diagram of library visits by Class 2C on a line from 0 to 7: 3 dots at 0, 5 at 1, 7 at 2, 6 at 3, 4 at 4, 2 at 5, none at 6 and 1 at 701234567Number of visits
ShowHide solution

(a) 3+5+7+6+4+2+0+1=283 + 5 + 7 + 6 + 4 + 2 + 0 + 1 = 28 pupils.

(b) 4, 5, 6 or 7 visits: 4+2+0+1=74 + 2 + 0 + 1 = 7 pupils.

(c) 0 or 1 visit: 3+5=83 + 5 = 8 pupils. Fraction =828=27= \frac{8}{28} = \frac{2}{7}.

(d) 6 pupils visited exactly 3 times. Percentage =628×100%=21.42…%≈21.4%.= \frac{6}{28} \times 100\% = 21.42…\% \approx 21.4\%\text{.}

(a) 28 (b) 7 (c) 27\frac{2}{7} (d) 21.4%

Question 4

The dot diagram shows the scores, out of 10, of the pupils in a spelling quiz. A pupil passes with a score of at least 6. (a) How many pupils took the quiz? (b) What percentage of them passed? (c) How many more pupils scored 7 than scored 4?

Dot diagram of spelling quiz scores on a line from 0 to 10: 1 dot at 3, 2 at 4, 3 at 5, 4 at 6, 5 at 7, 3 at 8, 1 at 9 and 1 at 10012345678910Score
ShowHide solution

(a) 1+2+3+4+5+3+1+1=201 + 2 + 3 + 4 + 5 + 3 + 1 + 1 = 20 pupils.

(b) A score of 6, 7, 8, 9 or 10 passes: 4+5+3+1+1=144 + 5 + 3 + 1 + 1 = 14 pupils. Percentage =1420×100%=70%.= \frac{14}{20} \times 100\% = 70\%\text{.}

(c) 5 pupils scored 7 and 2 scored 4: 5−2=35 - 2 = 3 more.

(a) 20 (b) 70% (c) 3 more.

Question 5

The dot diagram shows the masses, to the nearest kilogram, of 15 school bags. Describe the masses: where they lie and the range, where most of them are, and any value far from the rest.

Dot diagram of the masses of 15 school bags on a line from 3 to 11 kg: 1 dot at 3, 3 at 4, 5 at 5, 3 at 6, 2 at 7, none at 8, 9 and 10, and 1 at 1134567891011Mass (kg)
ShowHide solution

The masses go from 3 kg to 11 kg. Range =11−3=8= 11 - 3 = 8 kg.

Most bags are 4 kg to 7 kg: 3+5+3+2=133 + 5 + 3 + 2 = 13 of the 15. The most common mass is 5 kg (5 bags).

One bag, of 11 kg, is far heavier than the others: there are no bags of 8, 9 or 10 kg.

The masses go from 3 kg to 11 kg (range 8 kg). Most bags (13 of 15) are 4 kg to 7 kg, and 5 kg is the most common mass. One bag of 11 kg is far heavier than the rest, with a gap from 8 kg to 10 kg.

Question 6

Siti asked 20 classmates how many siblings they have. She drew the dots for the first 15 answers, shown below. The last 5 answers were 2, 0, 3, 1 and 2. (a) Complete the dot diagram. (b) How many of the 20 classmates have at least 2 siblings?

Dot diagram of the first 15 answers on a line from 0 to 5: 2 dots at 0, 5 at 1, 5 at 2, 2 at 3, none at 4 and 1 at 5012345Number of siblings
ShowHide solution

(a) Add one dot at 0, one at 1, two at 2 and one at 3.

The completed dot diagram of 20 answers: 3 dots at 0, 6 at 1, 7 at 2, 3 at 3, none at 4 and 1 at 5012345Number of siblings

Check: 3+6+7+3+0+1=20.3 + 6 + 7 + 3 + 0 + 1 = 20\text{.}

(b) 2, 3, 4 or 5 siblings: 7+3+0+1=117 + 3 + 0 + 1 = 11 classmates.

(a) Drawn above. (b) 11 classmates.

Question 7

Two teams of 10 pupils each took a quiz marked out of 10. The dot diagrams show their scores. (a) Find the range of each team's scores. (b) How many pupils in each team scored 8 or more? (c) Which team's scores are more consistent (closer together)? Give a reason.

Two dot diagrams on one scale from 0 to 10. Team Red: 4, 5, 5, 6, 6, 6, 7, 7, 8, 9. Team Blue: 2, 3, 5, 6, 7, 8, 8, 9, 10, 10Team Red012345678910Team Blue012345678910Score
ShowHide solution

(a) Team Red: 9−4=59 - 4 = 5. Team Blue: 10−2=810 - 2 = 8.

(b) Team Red: 8 and 9, so 2 pupils. Team Blue: 8, 8, 9, 10, 10, so 5 pupils.

(c) Team Red's range is smaller, and 7 of its 10 scores are between 5 and 7. Team Blue's scores are spread from 2 to 10.

(a) Team Red 5, Team Blue 8. (b) Team Red 2, Team Blue 5. (c) Team Red, because its range is smaller and its scores are bunched together.

Question 8

Say whether a dot diagram would be a good way to show each set of data, and explain. (a) The number of people living in each of the 40 flats of an HDB block, from 1 to 7. (b) The marks, out of 100, of 200 pupils in a Mathematics examination.

ShowHide solution

(a) Yes. The data are whole numbers with only 7 possible values, so the line has 7 marks and the 40 dots stack into columns that show the pattern clearly.

(b) No. The marks can take any of 101 values from 0 to 100, and there are 200 of them: the line would be very long and the dots too many to draw and count. Grouping the marks into intervals (0 to 9, 10 to 19, …) and drawing a histogram suits them better.

(a) Yes: few different values. (b) No: too many values over too wide a range; a grouped histogram is better.

Question 9

Ten pupils did as many push-ups as they could in one minute: 21, 22, 22, 23, 25, 25, 25, 26, 26, 28. Ahmad drew this dot diagram of the data. What has he done wrong? Draw the diagram correctly.

Ahmad's dot diagram: a number line marked 21, 22, 23, 25, 26, 28 with equal gaps; 1 dot at 21, 2 at 22, 1 at 23, 3 at 25, 2 at 26 and 1 at 28212223252628Number of push-ups
ShowHide solution

He has left 24 and 27 off the line, because no pupil did 24 or 27 push-ups. The gap from 23 to 25 is drawn the same as the gap from 22 to 23, so the line's steps are not equal.

The correct dot diagram on a line from 21 to 28: 1 dot at 21, 2 at 22, 1 at 23, none at 24, 3 at 25, 2 at 26, none at 27 and 1 at 282122232425262728Number of push-ups

He left out 24 and 27, so the steps of his line are not equal. The correct line marks every value from 21 to 28.

Question 10

A school floorball team played 16 matches. The dot diagram shows how many goals it scored in each match, but the column for 2 goals has been rubbed out. (a) In how many matches did the team score 2 goals? (b) In what percentage of the matches did it score at least 3 goals?

Dot diagram of goals in 16 matches on a line from 0 to 5, the column at 2 rubbed out: 2 dots at 0, 4 at 1, 3 at 3, 1 at 4 and 1 at 5012345Number of goals
ShowHide solution

(a) The dots shown: 2+4+3+1+1=11.2 + 4 + 3 + 1 + 1 = 11\text{.} There were 16 matches, so 16−11=516 - 11 = 5 matches had 2 goals.

(b) 3, 4 or 5 goals: 3+1+1=53 + 1 + 1 = 5 matches. Percentage =516×100%=31.25%.= \frac{5}{16} \times 100\% = 31.25\%\text{.}

(a) 5 matches (b) 31.25%

Key Points

  • A dot diagram is a number line with one dot above it for each value; dots for the same value are stacked at equal steps.
  • The line has equal steps and covers every value from the smallest to the largest; every value in between is marked, even a value with no dots.
  • Count dots to answer "how many"; "at least" includes the value, "more than" and "fewer than" do not.
  • A percentage of the data is number of dotstotal number of dots×100%.\frac{\text{number of dots}}{\text{total number of dots}} \times 100\%\text{.}
  • The value with the tallest column is the most common value; there can be two.
  • Describe a dot diagram by where the values lie (and the range, largest minus smallest), where they cluster, and any value far from the rest or gaps.
  • Compare two sets of data with two dot diagrams on the same scale.
  • A dot diagram suits numerical data with a few different values, not categorical data or data with many different values.

‹ 11.8 Frustums of a Cone 12.2 Stem-and-Leaf Diagrams ›

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