Mean, Median & Mode

At a glance 9 formulas

1. 1Median of n values$n \text{ odd: the } \left(\tfrac{n + 1}{2}\right)^{\text{th}} \text{ sorted value}$$n \text{ even: mean of the } \left(\tfrac{n}{2}\right)^{\text{th}} \text{ and } \left(\tfrac{n}{2} + 1\right)^{\text{th}}$
2. 2ModeThe mode is the value that occurs most often. A set can have two modes, or none when every value occurs equally often.
3. 3Median ignores the endsRaising the largest values or lowering the smallest leaves the median where it is; the mean moves with every change.
4. 4Frequency table medianKeep a running total. Odd $N$: the median is the $\tfrac{N+1}{2}$th value. Even $N$: average the $\tfrac{N}{2}$th and $(\tfrac{N}{2}+1)$th values, even if they differ.
5. 5Grouped data medianMedian $= l + \tfrac{N/2 - c}{f} \times h$: $l, f, h$ are the median class's start, frequency and width, $c$ the total before it, $N$ the total.
6. 6Empirical relation$\text{mode} \approx 3\,\text{median} - 2\,\text{mean}$$\text{mean} - \text{mode} = 3(\text{mean} - \text{median})$
7. 7Which is largestA tail of large values pulls the mean up: mean > median > mode. A tail of small values reverses the order; symmetric data has all three equal.
8. 8Median with an unknownWith an unknown $x$ in the data, the median depends on where $x$ sorts: take the cases below, between and above the middle values.
9. 9A single modeA unique mode $m$ must occur more often than every other value: if $m$ occurs twice, every other value occurs at most once.