Chapter 13 · Statistics and data displays

Missing Values and Measures of Center from Displays

Use the mean to recover a total and find a missing value. Then read the mean, median and mode from a line plot by counting every mark.

15 min read

In this lesson, a data set is a group of numbers, such as a list of quiz scores. You already know how to find its mean: add all the values, then divide by the number of values. The mean is the equal share. It is what every value would be if the total were spread out evenly.

That idea also works backward. Suppose 3 friends have a mean of 5 stickers. If their stickers were spread out evenly, each friend would have 5. Spreading them out does not change how many stickers there are, so together the friends have 5+5+5=155 + 5 + 5 = 15 stickers. The mean and the number of values tell you the total.

You will use that total in two ways. First, you will find a missing value when you know the mean. Then you will find the mean, and two other measures you already know, from a data display, which is a picture or table that organizes a data set.

Key concept

The mean times the number of values gives the sum

Every data set has three linked numbers: the sum of the values (the total), the number of values, and the mean. The mean is the sum divided by the number of values. Multiplying by the number of values undoes dividing by it, so the mean times the number of values gives the sum back.

This holds for any data set, as long as you count every value. A value that is missing still counts as one value.

Know the mean and the number of values? Multiply to find the sum. Know the sum and the number of values? Divide to find the mean.

sum of values=mean×number of values\text{sum of values} = \text{mean} \times \text{number of values}

Worked example

Find a missing quiz score

Four quiz scores have a mean of 88 points. Three of the scores are 77, 99 and 66 points. What is the fourth score?

  1. Let x stand for the fourth score, in points. Write what the mean tells you: the sum of all four scores, divided by 4, equals 8. You divide by 4, not 3, because the missing score is one of the values.

    7+9+6+x4=8\frac{7 + 9 + 6 + x}{4} = 8

  2. Multiply both sides by 4. Multiplying by 4 undoes dividing by 4, and doing the same thing to both sides keeps them equal. The left side is now the sum of all four scores.

    7+9+6+x=8×4=327 + 9 + 6 + x = 8 \times 4 = 32

  3. Add the three scores you know: 7 + 9 = 16, and 16 + 6 = 22. Writing 22 in place of 7 + 9 + 6 does not change the equation, because 7 + 9 + 6 and 22 have the same value.

    22+x=3222 + x = 32

  4. Subtract 22 from both sides. Subtracting 22 undoes adding 22, so x is left by itself, and both sides stay equal.

    x=32−22=10x = 32 - 22 = 10

  5. Check the answer. Put 10 in place of x and find the mean of all four scores. It should be 8.

    7+9+6+104=324=8\frac{7 + 9 + 6 + 10}{4} = \frac{32}{4} = 8

Answer

The fourth score is 1010 points.

A line plot is a data display made from a number line with marks above it. Each mark stands for exactly one data value and sits above that value on the number line. Marks for the same value pile up in a stack. A value with no marks has an empty stack. Values are not grouped into ranges, so a mark above 2 means the value is exactly 2.

The line plot below shows how many books each student in a reading club read last month. The number line counts books and starts at 0. Each mark is one student. Three marks sit above 1, so three students read 1 book each. No marks sit above 0 or 4, so no student read 0 books or 4 books.

To count the data values, add the heights of all the stacks: 0+3+2+1+0+1+2=90 + 3 + 2 + 1 + 0 + 1 + 2 = 9. The plot shows 9 students, even though only 5 stacks have marks.

Line plotLine plot. Value axis: each step is 1, up to 3. Measure axis: each step is 1 book, up to 6 books. 0 books, 0. 1 book, 3. 2 books, 2. 3 books, 1. 4 books, 0. 5 books, 1. 6 books, 2.01×××2××3×45×6××books
Line plot
Books read last month. Each mark is one student, so the 99 marks show 99 students.

You can find three measures of center from a line plot. A measure of center is one number that describes the middle or typical value of a data set. You know all three:

  • The mean is the sum of the values divided by the number of values.
  • The median is the middle value when the values are in order from least to greatest. With an even number of values, it is the mean of the two middle values.
  • The mode is the value that appears most often.

A table can help you find the sum. The table below is a second data display of the same data set. Each row matches one number on the number line. The first column is that number of books. The second column counts the marks in its stack, which is the number of students who read that many books. The third column is how many books those students read altogether: the value times its number of marks. The last row adds up the second and third columns.

Books read last month, one row for each number on the line plot
Books readMarks in the stack (students)Books for the whole stack
00
00
0×0=00 \times 0 = 0
11
33
1×3=31 \times 3 = 3
22
22
2×2=42 \times 2 = 4
33
11
3×1=33 \times 1 = 3
44
00
4×0=04 \times 0 = 0
55
11
5×1=55 \times 1 = 5
66
22
6×2=126 \times 2 = 12
Total
99
2727

Worked example

Mean, median and mode from a line plot

Use the line plot and the table of books read by 99 students. Find the mean, the median and the mode of the number of books read.

  1. Find the number of values. Each mark is one student, so add the marks in every stack. The result matches the total of the second column.

    0+3+2+1+0+1+2=90 + 3 + 2 + 1 + 0 + 1 + 2 = 9

  2. Find the sum of the values. The three students above 1 read 1 + 1 + 1 books, which is the same as 1 × 3. So each stack adds its value times its number of marks. Add those products. The result matches the total of the third column.

    0+3+4+3+0+5+12=270 + 3 + 4 + 3 + 0 + 5 + 12 = 27

  3. Find the mean. Divide the sum by the number of values, because the mean shares the 27 books equally among all 9 students.

    27÷9=327 \div 9 = 3

  4. Find the median. Reading the stacks from left to right puts the values in order, because the number line goes from least to greatest. With 9 values, the middle one is the 5th: 4 values come before it and 4 come after. The stack above 1 holds the 1st, 2nd and 3rd values. The stack above 2 holds the 4th and 5th.

    median=2\text{median} = 2

  5. Find the mode. The stack above 1 has 3 marks, and every other stack has fewer. So 1 is the value that appears most often.

    mode=1\text{mode} = 1

  6. Check the mean with the key idea: the mean times the number of values should give the sum back.

    3×9=273 \times 9 = 27

Answer

The mean is 33 books, the median is 22 books, and the mode is 11 book.

Not every number line counts by ones. The next line plot shows how many minutes each of 8 students spent on homework one evening. Its number line counts by tens, from 0 to 50. A mark's value is the number printed under its stack, so each mark above 20 means 20 minutes. One mark sits above 10, three above 20, one above 30, one above 40 and two above 50. The stack above 0 is empty.

You do not need a table this time. You can multiply and add in one line. The plot shows 8 students, an even number, so there are two middle values to find.

Line plotLine plot. Value axis: each step is 1, up to 3. Measure axis: each step is 10 minutes, up to 50 minutes. 0 minutes, 0. 10 minutes, 1. 20 minutes, 3. 30 minutes, 1. 40 minutes, 1. 50 minutes, 2.010×20×××30×40×50××minutes
Line plot
Minutes of homework. Each mark is one student. The number line counts by tens from 00 to 5050.

Worked example

A line plot that counts by tens

Use the line plot of homework minutes for 88 students. Find the mean, the median and the mode of the minutes.

  1. Count the values by adding the marks in every stack, from 0 to 50. Each mark is one student.

    0+1+3+1+1+2=80 + 1 + 3 + 1 + 1 + 2 = 8

  2. Find the sum. Multiply each value on the number line by its number of marks, then add the products. The empty stack above 0 adds nothing.

    (10×1)+(20×3)+(30×1)+(40×1)+(50×2)=10+60+30+40+100=240(10 \times 1) + (20 \times 3) + (30 \times 1) + (40 \times 1) + (50 \times 2) = 10 + 60 + 30 + 40 + 100 = 240

  3. Divide the sum by the number of values to find the mean. You can check it by multiplying: 30 × 8 = 240.

    240÷8=30240 \div 8 = 30

  4. Find the median. With 8 values, the two middle ones are the 4th and 5th, with 3 values on each side of them. Counting from the left, the 1st value is 10, the 2nd to 4th are 20, and the 5th is 30. The median is the mean of 20 and 30.

    20+302=502=25\frac{20 + 30}{2} = \frac{50}{2} = 25

  5. Find the mode. The stack above 20 has 3 marks, more than any other stack.

    mode=20\text{mode} = 20

Answer

The mean is 3030 minutes, the median is 2525 minutes, and the mode is 2020 minutes.

The sum connects everything you did. When you know the mean and the number of values, multiply to get the sum. Then subtract the values you know to find the missing one. On a line plot, every mark is one value. Add the stack heights to count the values. Multiply each value by its number of marks and add to build the sum. Then divide to get the mean. Count marks from the left to find the middle value, or the mean of the two middle values, which is the median. When one stack is taller than all the others, its value is the mode.

Next, you will see how changing one data value affects the mean and the median, and you will learn to spot mistakes in finding measures of center.

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Changing Data Values and Checking Measures of Center

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