Mode and outliers
The mode is identified from frequencies, while the mean and median describe the centre in different ways. A distant outlier can change the mean substantially while leaving the median close to the middle of the data
- Tally each value Count how many times every value occurs before deciding which value is most common
- Name the modal value The mode is the data value with the greatest frequency, not the frequency itself \(\displaystyle f(5)=3\;\Rightarrow\;\operatorname{mode}=5\)
- Recognise ties All values tied for greatest frequency are modes. Two modal values make a bimodal data set; three or more ties are also possible. Use the stated no-mode convention for all-distinct data \(\displaystyle f(4)=f(7)=2>f(9)=1\)
- Check the effect of an outlier A distant value usually pulls the mean more than the median because the mean uses every value \(\displaystyle \bar{x}=\frac{\sum x}{n}\)
- Investigate before excluding Check unusual values against source records and context. Being extreme does not make an observation invalid
Worked example
The numbers of pages read by seven students are 8, 10, 10, 11, 12, 13 and 35. Find the mode and compare the mean and median before and after removing the distant value 35.
- \(\displaystyle f(10)=2\)Every other value occurs once, so the mode is 10.
- \(\displaystyle \bar{x}=99/7\approx14.1\)Seven observations total 99 pages
- \(\displaystyle \bar{x}_{\mathrm{without}}=64/6\approx10.7\)Removing 35 leaves total 64 and count 6.
- \(\displaystyle \operatorname{median}_{\mathrm{all}}=11\)The fourth of seven values is 11.
- \(\displaystyle \operatorname{median}_{\mathrm{without}}=10.5\)The central values 10 and 11 average 10.5. The mean moves further; keep full precision before comparing
Common mistakes
Difficulty levels
Questions about this topic
Can a numerical data set have two modes?+
Yes. If two values share the greatest frequency, the data set is bimodal. If all values occur once, it has no mode
Why does an outlier usually affect the mean more than the median?+
The mean includes the numerical size of every value, while the median depends mainly on the middle position or positions
Is the largest value always an outlier?+
No. A value is an outlier only when it is sufficiently separated from the main group according to the rule being used
Can there be more than two modes?+
Yes. Report every value tied for greatest frequency. Three tied modal values are not described as bimodal
Should an outlier always be deleted?+
No. Check its validity and the purpose of the analysis. A genuine extreme observation may be important; disclose exclusions
Does removing a large value always change the median?+
No. The central positions and their values determine whether the median changes. It can stay the same or change
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics