Mean, median, mode and range
Use the original recorded values and their units. Repeated values still count as separate observations
- Mean Add every observation and divide by the number of observations \(\displaystyle \bar{x}=\frac{\text{total}}{n}\)
- Median Order the values first. For an odd count take the middle value; for an even count average the two middle values
- Mode Report every value tied for greatest frequency. If several distinct values all have the same frequency there is no unique mode. A constant list has one mode
- Range Largest minus smallest describes the full span. It is sensitive to extreme values \(\displaystyle R=\max-\min\)
Worked example
The recorded lengths are 1.2, 0.8, 1.4, 0.8 and 1.8 metres. Find all four summaries
- \(\displaystyle 0.8,\ 0.8,\ 1.2,\ 1.4,\ 1.8\)Order the observations without dropping the repeated 0.8
- \(\displaystyle \bar{x}=\frac{6.0}{5}=1.2\)The total is 6.0 m, so the mean is 1.2 m.
- \(\displaystyle \text{median}=1.2\)The third ordered value is 1.2 m.
- \(\displaystyle \text{mode}=0.8\)The most frequent value is 0.8 m.
- \(\displaystyle R=1.8-0.8=1.0\)The full span is 1.0 m.
Common mistakes
Difficulty levels
Questions about this topic
Can the median be absent from the data?+
Yes. With an even count the average of the middle pair may not be observed
Do zero observations count in the mean?+
Yes. They add zero to the total but still contribute to the count
Can a dataset have several modes?+
Yes. Several values may tie for greatest frequency
Is a constant dataset without a mode?+
No. Its one distinct value is the unique mode; its range is zero
Can a range be negative?+
No. Subtract the minimum from the maximum, including when the minimum is negative
Does adding a constant change range?+
No. Both endpoints move equally. Mean, median and modes shift by that constant
Does an unusual observation have to be removed?+
No. Check its accuracy and relevance. Comparing summaries with and without it does not by itself justify deletion
Must summaries have the same decimal precision as individual observations?+
No. An average can lie between the recorded values. Follow any stated rounding instruction and do not round intermediate totals
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 5: Statistics