Averages, Measures of Spread and Outliers
Three averages, and the marks are for choosing the right one. Plus quartiles, the interquartile range, and how to prove a value is an outlier rather than just calling it one.
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Three averages, one decision 📊
Most people can calculate a mean. Far fewer can say **why the mean was the wrong thing to calculate**, and that is where the interpretation marks live. An average is a **choice**. One dataset can honestly produce three different "averages", and picking the one that suits the data is a skill this specification tests directly. So this module does the arithmetic and the judgement together, on one small set of numbers you will get to know well.
What each average is good for 🗂️
Know the definition and the reason you would reach for it:
Match each situation to the average you would use
- The favourite colour of 200 pupils
- Salaries in a company where one director earns far more than anyone else
- The heights of 30 pupils, with no unusual values
- Yearly growth rates that multiply on each other [Higher]
- the mode, since the data are categories
- the median, since one extreme value would distort the mean
- the mean, since it uses every value and none is extreme
- the geometric mean
Find the median 🧮
An interactive activity.
Two ways to measure spread ↔️
Both describe how spread out the data are. They disagree about what to do with the extremes, and that disagreement is usually the point of the question.
Find the interquartile range 🔢
An interactive activity.
Which tier needs what 🎓
This sub-section splits sharply between the tiers, so it is worth knowing which side of the line you are on. **Both tiers:** mode, median, mean, range, quartiles, percentiles, the interquartile range, and spotting an outlier by looking at the data. **Higher tier only:** weighted mean, geometric mean, interpercentile and interdecile range, standard deviation, identifying an outlier by **calculation**, and the **standardised score**, (x - mu) / sigma. **One warning about that last one.** The standardised score formula is **not given** in the assessment, so it has to be memorised. Most of the formulae you meet in this subject are provided; this is one that is not.
Why the median? ❓
One pupil in that group actually read 60 books, not 12. What happens to the averages?
- The mean rises noticeably while the median stays where it is
- Both the mean and the median rise by the same amount
- The median rises and the mean is unaffected
- Neither changes, because it is only one value out of seven
The outlier boundary [Higher tier only] 🚩
An interactive activity.
Which are true? ✅
Select the TWO correct statements about outliers and averages.
- An outlier should be identified by a test, not just because a value looks unusual
- The interquartile range is unaffected by an outlier, while the range is not
- An outlier should always be deleted from the data
- An outlier changes the mode more than the mean
Choose the measure 🧭
An interactive activity.
Order the method 🪜
An interactive activity.
The summary 📝
The _____ is the only average that works for categories such as favourite colour. The _____ is the middle value once the data are in order, and one extreme value does not move it. The range is the largest minus the smallest, while the _____ range describes only the middle half, so an outlier cannot inflate it. At Higher tier an outlier can be identified by calculation, using _____ times the interquartile range beyond a quartile. And whatever the question, the first thing you do with a list is put it in _____.
Your turn 📄
An interactive activity.