Shape of Data, Outliers and Best Measure of Centre
Theory
Look at the shape before calculating. An outlier, a value far from the rest, drags the mean towards itself but hardly moves the median. For stays of \(2, 2, 3, 3, 4, 4, 5, 5, 6\) and \(46\) nights, the mean is \(8\) but the median is \(4\). This Year 8 maths page shows how to choose.
Shape. Data is symmetrical when the two halves are close to mirror images, positively skewed when the tail stretches to the right (high values), and negatively skewed when the tail stretches to the left (low values). The tail names the skew.
Peaks. No mode is uniform, one mode is unimodal, two modes is bimodal. Also describe clusters (values bunched together), gaps (stretches with no values) and outliers, with a reason in context.
What an outlier does. The mean is pulled towards it, because every value is added in. The median barely moves, because it depends only on the middle values. The mode usually does not change. The range grows a lot.
Skew and order. Positively skewed usually gives mean \(>\) median; negatively skewed gives mean \(<\) median; symmetrical gives mean \(\approx\) median. It is a guide to check, not a law.
Working with a total: total \(=\) mean \(\times\) number of values. To remove a value, subtract it from the total and divide by one fewer.
Comparing two sets: compare their centres, their spreads and their shapes, in context.
Which measure of centre?
| The data | Best measure | Why |
|---|---|---|
| close together, no outliers | mean | it uses every value |
| has an outlier, or is strongly skewed | median | an outlier barely moves it |
| sizes or categories; the most common is wanted | mode | the mean may not be a real size |
Describing and summarising data
- Describe the shape: symmetrical or skewed (name the tail), the number of peaks, clusters, gaps and outliers.
- Calculate the mean and the median (with and without any outlier).
- Choose: mean for close data, median with an outlier or strong skew, mode for the most common size or category.
- Explain the choice in context.
(i) Describe the shape.
(ii) Unimodal or bimodal?
(iii) Name the gap and the outlier.
The stacks hold \(5, 6, 4, 2, 1\) and \(1\) dots.
(i) Positively skewed: the dots bunch at the left and the tail is on the right.
(ii) Unimodal (one tallest stack, at \(1\)).
(iii) A gap at \(5\) and \(6\); the lone \(7\) is an outlier, perhaps a flock passing through.
(i) the mean
(ii) the median
(iii) the mean without the outlier
(iv) the better measure for a typical booking.
| \(\text{Total}\) | \(=\) | \(80\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{80}{10} = 8\) |
| \(\text{Median}\) | \(=\) | \(\dfrac{4 + 4}{2} = 4\) |
| \(\dfrac{80 - 46}{9}\) | \(=\) | \(\dfrac{34}{9} \approx 3.8\) |
(i) \(8\) nights
(ii) \(4\) nights
(iii) about \(3.8\) nights
(iv) The median: removing the outlier drops the mean from \(8\) to \(3.8\), but the median does not move.
(i) the total of the first eight
(ii) the mean of all nine
(iii) the rise in the mean.
| \(460 \times 8\) | \(=\) | \(3680\) |
| \(\dfrac{3680 + 640}{9}\) | \(=\) | \(\dfrac{4320}{9} = 480\) |
| \(480 - 460\) | \(=\) | \(20\) |
(i) \(3680\) g
(ii) \(480\) g
(iii) \(20\) g. The median hardly changes: the heaviest jar moves the middle of the ordered list along only half a place.
(i) \(22\) measurements of the same steel rod, all between \(1.82\) m and \(1.86\) m.
(ii) \(15\) second-hand bikes: one at \(\$4200\), the rest between \(\$80\) and \(\$350\).
(iii) A shop choosing which shoe size to order most.
(i) Mean: the readings are close with no outlier, and the mean uses all \(22\).
(ii) Median: the \(\$4200\) bike would drag the mean above almost every price.
(iii) Mode: a mean shoe size such as \(8.3\) is not a size the shop can order.
Common pitfalls
Frequently asked questions
What is an outlier?
A value that is far away from the rest of the data.
How does an outlier affect the mean and the median?
It pulls the mean towards itself, sometimes a long way, but changes the median very little.
What does positively skewed mean?
Most of the data is at the low end and the tail stretches out to the high values.
When should I use the median instead of the mean?
When the data has an outlier or is strongly skewed, such as house prices or wages.
When is the mode the best measure?
When you want the most common size or category, such as the shoe size a shop sells most.