Before the method: the idea behind it
Where the middle is, and how far things spread
The previous lesson gave you probabilities. This one gives you summaries: a single number for the centre and a single number for the spread. Every formula is inherited from the discrete lesson, with integrals in place of sums, but continuous variables also bring medians and percentiles into play, which discrete variables handled only awkwardly.
Part one
Sums become integrals
Every formula from the discrete lesson survives intact. Only the machinery changes.
For a discrete variable you multiplied each value by its probability and added. For a continuous one you multiply x by the density and integrate. The structure is identical, because an integral is what a sum becomes when the values form a continuum.
| Quantity | Discrete | Continuous |
|---|---|---|
| Mean | \(\sum xP(X=x)\) | \(\int x\,f(x)\,dx\) |
| E(X²) | \(\sum x^2P(X=x)\) | \(\int x^2 f(x)\,dx\) |
| Variance | \(E(X^2)-[E(X)]^2\) | \(E(X^2)-[E(X)]^2\) |
| Linear change | \(E(aX+b)=aE(X)+b\) | identical |
The variance formula is unchanged, because once you have E(X) and E(X²) the rest is the same arithmetic. The linear rules survive too, and for the same reason: adding a constant slides the distribution without changing its spread, while multiplying stretches every deviation, and variance squares those deviations.
Part two
Three different centres
The mean, the median and the mode all describe the middle, and for a skewed distribution all three differ.
The mean is the balance point: where the shape would balance on a pivot. The median splits the area in half, with 50% either side. The mode is simply where the density is tallest.
For a symmetric density all three coincide at the centre, which is worth exploiting: if you can show a density is symmetric, you have the mean and median instantly with no integration.
When a density is skewed they separate in a predictable order. A long tail on the left drags the mean furthest towards it, because the mean is sensitive to how far values sit from the centre. The median moves less, since it only counts how much area lies either side, and the mode does not move at all.
If you already have the cumulative distribution function from the previous lesson, these become one-line equations: solve F(m) = 0.5 for the median, or F(x) = 0.3 for the 30th percentile. Building F once and reusing it is almost always quicker than integrating repeatedly.
Part three
Why spread needs its own measure
Two distributions can share a mean and behave completely differently.
A uniform distribution on [0, 4] and a triangular one peaking at 2 both have mean 2. But the uniform spreads probability evenly right out to the ends, while the triangular concentrates it near the middle. The uniform has twice the variance, and that difference is what the mean alone cannot capture.
Variance measures the average squared distance from the mean. Squaring stops positive and negative deviations cancelling, and it also weights distant values heavily, which is why a long tail inflates the variance so much. Taking the square root returns the standard deviation to the original units, which is why it is the figure usually quoted.
Where this shows up in real life
Why house prices use the median
Property prices have a long right tail: a handful of very expensive sales pull the mean well above what a typical buyer pays. Official statistics quote the median precisely because it resists that distortion.
Service level agreements
Companies promise response times as percentiles, such as “95% of calls answered within 30 seconds”, rather than as averages. An average can hide a long tail of very slow responses, while a percentile guarantees performance for a stated share of customers.
Next
Try it yourself
The working method, twenty practice questions, ten harder ones, and the ClassPad guide.