Mean, Median, Mode and Range

Worked example

1. 1

   $8,5,10,2,5,6$ — The list, as the question gives it. It has 6 numbers.

2. 2

   $2,5,5,6,8,10$ — Write it again from smallest to largest. Tick each number off so none is lost. 5 appears twice.

3. 3

   $\text{mode}=5$ — The mode is the value that appears most often. No other number repeats, so it is 5.

4. 4

   $\text{position}=\frac{6+1}{2}=3.5$ — Find where the middle is. 3.5 means halfway between the 3rd and 4th numbers.

5. 5

   $\text{median}=\frac{5+6}{2}=5.5$ — The 3rd number is 5 and the 4th is 6. Add them and halve to get the point halfway between.

6. 6

   $2+5+5+6+8+10=36$ — For the mean, first add every number to get the total.

7. 7

   $\text{mean}=36÷6=6$ — Divide the total by how many numbers there are, 6. The mean does not have to be in the list.

8. 8

   $\text{range}=10-2=8$ — Biggest take away smallest. It tells you how spread out the numbers are.

Worked example

1. 1

   $8\times 5=40$ — Mean × how many gives the total of all five. The mean was the total shared out equally, so multiply to undo it.

2. 2

   $6+11+3+9=29$ — Add the four numbers you know.

3. 3

   $40-29=11$ — What is left of the total must be the missing number.

4. 4

   $\frac{6+11+3+9+11}{5}=\frac{40}{5}=8$ — Check: put 11 back and work out the mean. It gives 8, so the answer is right.

Worked example

1. 1

   $4\times 70=280$ — Total of the first 4 tests: mean × how many.

2. 2

   $5\times 73=365$ — Total she needs after 5 tests. Use the new mean and the new count, 5.

3. 3

   $365-280=85$ — The 5th test must make up the difference. So the least she needs is 85.

Worked example

1. 1

   $12\times 15=180$ — Mean × how many gives the boys' total points.

2. 2

   $8\times 20=160$ — Same for the girls' total: 8 girls × mean 20.

3. 3

   $180+160=340$ — Add the two totals. This is every point scored by all the children.

4. 4

   $340÷20=17$ — Divide by everyone: 12 + 8 = 20 children. The mean for all is 17.

Worked example

1. 1

   $30\times 64=1920$ — Mean × how many gives the total for all 30 students.

2. 2

   $12\times 70=840$ — Mean × how many again, for the 12 girls.

3. 3

   $1920-840=1080$ — Take the girls away from everyone. What is left is the other students' total.

4. 4

   $30-12=18$ — Count the other students too: 18 of them.

5. 5

   $1080÷18=60$ — Divide their total by how many they are. Their mean is 60.

6. 6

   $\frac{840+1080}{30}=\frac{1920}{30}=64$ — Check: put both groups back together. The mean of all 30 is 64, as given.

Worked example

1. 1

   $\text{mode }10:c=10$ — Start with the mode. c appears twice and every other number at most once, so c must be the mode, 10.

2. 2

   $\text{range }11:14-a=11$ — The list is in order, so a is the smallest and 14 the biggest. Range = biggest − smallest.

3. 3

   $a=14-11=3$ — Take 11 from 14 to find the smallest number.

4. 4

   $\frac{b+10}{2}=8$ — Six numbers, so the median is halfway between the 3rd (b) and the 4th (c = 10).

5. 5

   $b+10=16$ — Multiply both sides by 2 to clear the fraction.

6. 6

   $b=6$ — Take 10 from both sides.

7. 7

   $3,5,6,10,10,14$ — Write the full list and check every clue: mode 10, range 14 − 3 = 11, median (6 + 10) ÷ 2 = 8.

Worked example

1. 1

   $2+3+3+4+5+25=42$ — Add them all for the mean.

2. 2

   $\text{mean}=42÷6=7$ — The mean is 7, but 5 of the 6 friends did fewer than 7 hours.

3. 3

   $\text{median}=\frac{3+4}{2}=3.5$ — The list is already in order. The median sits between the 3rd and 4th values, close to most of the data.

4. 4

   $25\text{is an extreme value}$ — The 25 is far bigger than the rest and pulls the mean up. So the median is the better average here.

Worked example

1. 1

   $3,4,6,7,9,12$ — Put the new value in its place. It goes first, below the median.

2. 2

   $\text{median}=\frac{6+7}{2}=6.5$ — Now there are 6 numbers, so the median is halfway between the 3rd and 4th. It went down from 7, because the new value is below the median.

3. 3

   $\text{mean}=41÷6=6.83(3\text{ s.f.})$ — The total is 38 + 3 = 41. The mean went down from 7.6, because 3 is below the mean. A value above the mean would push it up.

4. 4

   $\text{range}=12-3=9$ — 3 is the new smallest value, so the range grows from 8 to 9.

Worked example

1. 1

   $\text{median }7+2=9,\text{mean }7.6+2=9.6$ — Every value moves up by 2, so every average moves up by 2. The mode would too.

2. 2

   $\text{range}=14-6=8$ — The biggest and smallest both move up by 2, so the gap between them stays 8.