One dot for each value
Eight children take a short quiz, and their scores are 3, 4, 4, 5, 5, 5, 6 and 7. To show them all at once, draw a number line that covers every score from 2 to 8. Then put one dot above the line for each child, at the score that child got.
This drawing is called a dot diagram, or a dot plot. Each dot is one child, and its place along the line is that child’s score.
Eight dots, one for each child, each above the score that child got.
A stack is a count
When two children get the same score, their dots are stacked one above the other. The stack above 5 has 3 dots, so 3 children scored 5. The number under a stack is the score, and the height of the stack is how many children got that score.
To find how many children took the quiz, count the dots, not the stacks. There are 5 stacks, but they hold 1 + 2 + 3 + 1 + 1 = 8 dots, so 8 children took the quiz.
The stack above 5 is 3 dots tall: 3 children scored 5.
The shape of the data
The number line keeps the scores in order and equally spaced, so the diagram shows how the scores are spread out. The lowest score is 3 and the highest is 7. The scores pile up in the middle, at 5, and thin out on either side. Nobody scored 2 or 8; an empty place on the line is information too.
The diagram keeps every value. Each dot can be read back as one child’s exact score, so nothing is lost by drawing it.
Why dots suit numerical data
A dot diagram is for numerical data, such as scores, counts or measurements, when there are not too many values. A bar chart of categories has no number scale along the bottom: its bars could be put in any order. On a dot diagram the values sit on a number line, in order, and a score that nobody got still has its own place.
When there are many values, the stacks become too tall to count, and a vertical line chart, which reads each count off a scale, does the same job.
The mean and the median
To total the quiz scores, multiply each score by the number of dots above it, because each dot is one child, and add: 3 × 1 + 4 × 2 + 5 × 3 + 6 × 1 + 7 × 1 = 3 + 8 + 15 + 6 + 7 = 39. The mean is the total shared out equally among the 8 children: 39 ÷ 8 = 4.875.
A second kind of average, the median, is the middle value once the values are in order. On a dot diagram they are already in order, so count the dots in from the left. With an odd number of values there is one middle value. With an even number there are two, and the median is halfway between them. The 8 quiz scores have the 4th and the 5th in the middle, and both are 5, so the median is 5.
A value far from the rest
A dot diagram shows at once a value that lies far away from all the others. Such a value is called an outlier. It may be a mistake in the data, or a real but unusual result.
The mean uses every value, so an outlier drags it toward itself. The median depends only on which value is in the middle, so an outlier leaves it where it is.
the median is the middle value once the data is in order, so it depends on rank and not on size — moving x₅ past the others never moves it
Drag the last value away from the rest to make it an outlier
The values 2, 3, 4, 5 and 6 have a mean of 20 ÷ 5 = 4 and a median of 4. Drag the gold value away from the rest: the total grows, so the mean follows it, but the middle value is still 4, so the median does not move.
Worked example: A Dot Diagram of Goals with One Outlier, and What It Does to the Mean
Question A dot diagram shows the goals scored by a hockey team in each of its 10 matches. There is 1 dot at 0, there are 2 dots at 1, 3 dots at 2, 2 dots at 3, 1 dot at 4 and 1 dot at 12. (a) Find the mean number of goals in a match. (b) The match with 12 goals was against a team that had only 7 players. Leave that match out, find the new mean, and say whether the mean or the median describes a usual match better.
1.Find the total number of goals by multiplying each value by its number of dots: 0 × 1 + 1 × 2 + 2 × 3 + 3 × 2 + 4 × 1 + 12 × 1 = 0 + 2 + 6 + 6 + 4 + 12 = 30 goals.
Each dot is one match. Multiply each value by its number of dots: 0 + 2 + 6 + 6 + 4 + 12 = 30 goals. 2.(a) The mean is 30 ÷ 10 = 3 goals in a match. Only 2 of the 10 matches had more than 3 goals, so the mean is higher than most of the dots.
(a) The mean is 30 ÷ 10 = 3 goals. Only 2 matches had more than 3 goals. 3.The dot at 12 is an outlier: it is far to the right of all the other dots. Without it there are 30 − 12 = 18 goals in 9 matches, and the mean is 18 ÷ 9 = 2 goals.
The dot at 12 is an outlier. Without it the mean is 18 ÷ 9 = 2 goals. 4.Find the median both ways. With all 10 matches, the 5th and 6th values are both 2, so the median is 2. Without the outlier there are 9 values and the 5th is 2, so the median is still 2.
The 5th and 6th values are both 2, and with 9 values the 5th is 2, so the median is 2 both ways. 5.(b) Without the outlier the mean falls from 3 goals to 2 goals, while the median stays at 2 goals. The median describes a usual match better, because one unusual match does not change it.
(b) The mean falls from 3 to 2 and the median stays at 2, so the median describes a usual match better.
Answer: (a) the mean is 3 goals; (b) the mean falls to 2 goals and the median stays at 2 goals, so the median describes a usual match better
Common mistakes
- Adding the six values on the axis that have dots, 0 + 1 + 2 + 3 + 4 + 12 = 22, and dividing by 6. Each dot is a match, so a value with 3 dots must be counted 3 times, and the division is by the 10 matches.
- Dividing the 18 goals by 10 after the outlier is removed. Leaving out the match removes one value as well as its 12 goals, so there are only 9 matches to divide by.