Library / Foundations: Maths / FMA-32
Foundations: Maths
Normal distribution: z-score
How many standard deviations a value lies from the mean.
- Ref
- FMA-32
- Rev
- v1
- Field
- Foundations: Maths
- Result
- Z
For normally distributed X subtract the mean μ and divide by the positive standard deviation σ to obtain a standard normal variable. The probability below X is Φ(z).
A z-score puts any normal quantity on a standard scale: how far the value is from average, measured in standard deviations.
Calculator
- Probability below X93.32 %
- Probability above X6.681 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- About 68%, 95% and 99.7% of a normal population lie within 1, 2 and 3 standard deviations of the mean.
- Z-scores allow comparison of values from different scales.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| σ Standard deviation σ | 2 to 20 | 0.75 | 7.5 | 6.75 | +10% | −9.09 % |
| X Value X | 60 to 100 | −1 | 3 | 4 | +10% | +56.7 % |
| μ Mean μ | 60 to 80 | 0.5 | 2.5 | 2 | +10% | −46.7 % |
Sweep of Value X
| Value X | z-score |
|---|---|
| 60 | −1 |
| 65 | −0.5 |
| 70 | 0 |
| 75 | 0.5 |
| 80 | 1 |
| 85 | 1.5 |
| 90 | 2 |
| 95 | 2.5 |
| 100 | 3 |
Sweep one variable
Variables and how each one contributes
z-score
What the equation gives you. Everything else feeds it.
Value X
The measured value. Raises z in proportion.
Mean μ
Mean of the distribution. Lowers z as it rises.
Standard deviation σ
Spread of the distribution. Inversely proportional: more spread means a smaller z.
Worked example with the values above
- Distance from the mean: X − μ = 85 − 70 = 15.
- Divide by σ: 15 / 10.
- Z = 1.5, so 93.32% of values lie below X.
Assumptions and limits
- The quantity is normally distributed.
- σ is positive.
Common mistakes
- Using the variance instead of the standard deviation.
- Reversing the subtraction.
Statistics and uncertainty
Estimated σ carries its own error from the sample size; z-scores beyond about 3 depend heavily on whether the true distribution has heavier tails than normal.
Where it is used
- Dispersion and accuracy of impact points.
- Quality limits and tolerances.
Maths you need
- Mean and standard deviation: centre and spread
- Normal curve: bell-shaped distribution