One line for each value
A class takes a quiz with four questions, and each child scores 0, 1, 2, 3 or 4. Two children score 0, five score 1, seven score 2, four score 3 and one scores 4. The number of times a value occurs is its frequency, so the frequency of the score 2 is 7.
A vertical line chart draws a thin line straight up from each score, as tall as that score’s frequency. The scores are written along the bottom, in order, and the vertical axis carries a scale for the frequency.
One line for each score. The line above 1 stops halfway between 4 and 6 on the scale, so 5 children scored 1.
Read the height off the scale
This scale goes up in twos: 0, 2, 4, 6. The line above 0 reaches the mark for 2, so 2 children scored 0. The line above 1 stops halfway between 4 and 6, so its height is 5, and the line above 3 reaches the mark for 4. The line above 2 is the tallest; it stops half a step above 6, at 7.
Keep the two numbers apart. The number under a line is the score. The height of the line is how many children got that score.
Why a line, and not a bar
A score on this quiz is a whole number. Nobody can score 1.5 or 2.3, so the data are discrete. A score of 2 is one exact point on the number line, not a range of values, and so it is drawn as a line with no width, standing exactly on 2.
A bar with a width would cover the values on either side of 2 as well, which is what the bars of a histogram mean for measurements such as heights. For categories such as colors, a bar chart is right, because there is no number scale along the bottom at all.
A dot diagram shows the same data as stacks of dots. Each line stands where a stack would, and its height is the number of dots in the stack. A tall stack has to be counted dot by dot; a line is read straight off the scale, so a vertical line chart can show frequencies in the hundreds.
The same 19 scores as a dot diagram: the stack above 2 has 7 dots, and the line above 2 is 7 tall.
Add the heights for the total
Every child is counted in exactly one line, the line for that child’s score. So adding the heights counts every child once, and nobody twice: 2 + 5 + 7 + 4 + 1 = 19 children took the quiz.
The total is not the number of lines. There are 5 lines, one for each possible score, but a line of height 7 stands for 7 children, not one.
The mode, and the total of the scores
The value that occurs most often is called the mode. On a vertical line chart it is the value under the tallest line. Here the tallest line stands above 2, so the mode is a score of 2. The height of that line, 7, is its frequency, not the mode.
To add up all the scores, notice that the 7 children who scored 2 bring 2 × 7 = 14 to the total. Multiply every score by its frequency and add: 0 × 2 + 1 × 5 + 2 × 7 + 3 × 4 + 4 × 1 = 0 + 5 + 14 + 12 + 4 = 35. The mean score is the total shared among the 19 children: .
Worked example: A Vertical Line Chart of People in Each Car: The Mode and the Total
Question A student counted the people in each car that entered a car park and drew a vertical line chart. The line at 1 person has height 15, the line at 2 people has height 12, the line at 3 has height 7, the line at 4 has height 4 and the line at 5 has height 2. (a) State the mode, and find how many cars were counted. (b) Find the total number of people in the cars, and the mean number of people in a car.
1.Read the heights of the lines: 15, 12, 7, 4 and 2 cars. The tallest line stands at 1 person, so the mode is 1 person. The mode is the value under the tallest line, not the height 15.
The height of each line is a number of cars. The tallest line stands at 1 person, so the mode is 1. 2.(a) The mode is 1 person in a car, and 15 + 12 + 7 + 4 + 2 = 40 cars were counted.
(a) The mode is 1 person, and 15 + 12 + 7 + 4 + 2 = 40 cars were counted. 3.Multiply each value by its frequency to count the people: 1 × 15 + 2 × 12 + 3 × 7 + 4 × 4 + 5 × 2 = 15 + 24 + 21 + 16 + 10 = 86.
Each value times its frequency is a number of people: 15 + 24 + 21 + 16 + 10 = 86. 4.(b) There were 86 people, and the mean is 86 ÷ 40 = 2.15 people in a car. The mean need not be a whole number, although every single car holds a whole number of people.
(b) 86 people, and a mean of 86 ÷ 40 = 2.15 people in a car.
Answer: (a) the mode is 1 person, and 40 cars were counted; (b) 86 people, a mean of 2.15 people in a car
Common mistakes
- Giving the mode as 15. The height 15 says how many cars had 1 person. The mode is the value that occurs most often, which is 1 person.
- Adding the heights to find the number of people. 15 + 12 + 7 + 4 + 2 = 40 counts the cars. A car at the line for 3 carries 3 people, so each height must be multiplied by its value before adding.