z-scores and the 68-95-99.7 rule
Minutes and pounds cannot be compared, but their z-scores can. Give each value with the mean and the standard deviation of the distribution it comes from; the comparator standardises both, places them on the standard normal and says which is the more unusual.
Goes with chapter 8. The standard normal and z-scores
Compare two values
Value A
Value B
| Value | z = (x − μ)/σ | Share at or below, Φ(z) | Share at least as far out |
|---|---|---|---|
| Value A | 2 | 0.9772 | 0.0228 |
| Value B | 2.4 | 0.9918 | 0.0082 |
Value B is the more unusual: z = 2.4, against z = 2 for Value A.
The share at least as far out is the tail on the value's own side of the mean: the chance of a value at least as extreme in that direction.
The 68-95-99.7 rule
On value A's distribution, N(35, 8):
| Band | From – to | Exact share | The rule |
|---|---|---|---|
| μ ± 1σ | 27 – 43 | 0.6827 | 68 % |
| μ ± 2σ | 19 – 51 | 0.9545 | 95 % |
| μ ± 3σ | 11 – 59 | 0.9973 | 99.7 % |
What to look for
A z-score of 2 means two standard deviations above the mean, whatever the units. The further a value's z-score is from 0, on either side, the rarer it is; the share beyond it is the chance of a value at least as extreme on that side.
Use the mean and the standard deviation of the distribution the value comes from, the process, not of a few values at hand: with a small sample's own figures the same value gets a different, misleading z-score.