These questions ask you to calculate or interpret the arithmetic average of a short list. You may need to handle a zero, a negative number, repeated values or a result that is not a whole number.
Use total, count, divide
Think of this average as the value each item would have if the whole total were shared equally. First add every recorded value, keeping its sign. Next count every observation, including zeros and repeats. Finally divide the total by that count. For example, the values 1, 4 and 10 total 15, and 15 shared across three values gives an arithmetic average of 5.
| Signal in the data | What to do |
|---|---|
| A recorded zero | Include it in the count even though it adds nothing to the total. |
| A negative value | Add it with its minus sign, so it lowers the total. |
| A repeated value | Count every occurrence, not just each different number. |
| A decimal result | Keep the exact decimal requested; the result need not be one of the recorded values. |
| A new value joins the set | Rebuild the total and the count. A value equal to the old average leaves the result unchanged. |
When a new value is added, do not give the old average and the new value equal weight unless the old result represented just one value. Recover the old total from the original data, add the newcomer, and divide by the new count. This preserves the weight of every earlier observation.
The traps
| Tempting route | How to reject it |
|---|---|
| Choose the middle or most frequent value | Those routes use position or frequency. The requested arithmetic average must use the total and the number of observations. |
| Stop after adding | The sum describes everything together. Divide it by the count to find the equal share. |
| Ignore zero or count only different values | A recorded zero and every repeated occurrence occupy places in the data, so all belong in the divisor. |
| Turn negative values positive | Changing a sign changes the total. Keep each value on its correct side of zero before dividing. |
| Average an old result directly with one new value | The old result can represent several observations. Recover their total so that one newcomer receives only one share of the weight. |
Try it
Four books have 3, 4, 6 and 8 chapters. What is the mean number of chapters per book? Enter the number of chapters. Give your answer as a decimal.
The values 4, 6, 8 and 10 have a mean of 7, and a fifth value of 7 is then added to the set. Which statements about the new set of five values are true? Select all that apply.
- Adding a fifth value must change the mean.
- The total of the five values is 35.
- The mean of the five values rises to 7.5.
- The mean of the five values is 8.75, because 35 is divided by 4.
- The mean of the five values is still 7.
On four winter mornings the temperature was -2, 0, 3 and 7 degrees Celsius. What is the mean of these four temperatures? Enter the number of degrees Celsius.