These questions ask you to choose a useful average or explain what happens when an extreme value is added or removed. Your job is to match the purpose to the mean, median or mode, then check whether an outlier makes the result misleading.
Choose by purpose, not habit
Start by identifying what must be represented. Use the mean when every numerical value should contribute: add all values and divide by their count. Use the median when the centre by position matters, especially when one end value is far from the rest. Put the values in order and take the middle one, or average the two middle values when the count is even. Use the mode when you need the value or category that occurs most often. It is the only one of these averages that works for unordered categories such as pet types.
| Signal in the task | Method to choose |
|---|---|
| Every numerical value must affect the summary | Use the mean: total divided by count. |
| One extreme value should not dominate the centre | Use the median: order first, then find the middle position or positions. |
| The most common category or usable size is wanted | Use the mode: compare frequencies. |
For a quick outlier check, compare the calculation before and after the extreme value. With 6, 7, 8 and 39, the total-based average is 15, but removing 39 leaves an average of 7. The ordered centre changes far less. If a value is removed, recalculate both the total and the count. Remember that an unusually small value can pull the mean downward just as an unusually large value can pull it upward.
Where mistakes come from
| Tempting route | How to reject it |
|---|---|
| Choose the mean whenever all values are visible | Using every value is useful only when that matches the purpose. It also makes an extreme value influential, so test whether the result still represents the main cluster. |
| Try to calculate a mean or median for named categories | Categories without numerical size cannot be totalled or ordered by magnitude. Count repeats and use the mode. |
| Use the mean when the most frequently needed item is wanted | A mean can be a value that never occurs. The mode identifies the actual item appearing most often. |
| Remove an outlier from the total but keep the old divisor | Removing one observation changes two things: subtract its value and reduce the count by one before dividing again. |
| Look only for unusually large outliers | Extreme values work in both directions. A very small observation can drag the mean below the main cluster. |
Try it
Five employees earn 18, 19, 20, 21 and 90 thousand per year. The manager wants the average that best represents a typical salary in this group rather than one that the single very high salary inflates. Enter that average in thousands per year.
The values 2, 2, 30, 31 and 32 are recorded. Which average of this data set is fixed by the only repeated value and therefore sits far below the bulk of the data, and what is its value?
- The median, 30
- The mode, 2
- The mean, 19.4
- The mode, 30
The values 10, 12 and 14 have a mean of 12 and a median of 12. The value 100 is now added to the data set. Select all statements that are true about the four values that result. Select all that apply.
- The new median is still 12, because 12 was the middle value before
- The new mean is 34
- The new mean is still 12, because one added value cannot change a mean
- The new median is 13