Library / Statistics / STA-10
Statistics
Normal probability density
Height of the bell curve at x.
- Ref
- STA-10
- Rev
- v1
- Field
- Statistics
- Result
- f(x)
μ is the mean and σ the standard deviation of the distribution; f(x) is the height of the bell curve at x, not a probability. Probabilities are areas under the curve: about 68 % lies within μ ± σ, 95 % within μ ± 2σ and 99.7 % within μ ± 3σ.
Height of the bell curve at x. About 68 % of values fall within one σ of the mean μ and 95 % within two.
Calculator
- Standard score z = (x − μ)/σ1
- Peak height at x = μ0.3989
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.
- The curve is symmetric about μ and is highest there; the peak of the default standard normal curve is 0.399.
- The density can exceed 1 when σ is small; that is fine, because only areas are probabilities.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Value | −100 to 100 | 0 | 0.3989 | 0.3989 | +4 | −100 % |
| μ Mean | −100 to 100 | 0 | 0.242 | 0.242 | +4 | −98.2 % |
| σ Standard deviation | 0.1 to 50 | 7.695e-22 | 0.242 | 0.242 | +10% | −0.85 % |
Sweep of Value
| Value | Density |
|---|---|
| −100 | 0 |
| −75 | 0 |
| −50 | 0 |
| −25 | 7.654e-137 |
| 0 | 0.3989 |
| 25 | 7.654e-137 |
| 50 | 0 |
| 75 | 0 |
| 100 | 0 |
Sweep one variable
Variables and how each one contributes
Density
What the equation gives you. Everything else feeds it.
Value
The value at which the height of the curve is wanted. Falls quickly as x moves away from μ: at one σ from the mean the height is 61 % of the peak.
Mean
The centre of the distribution. Shifts the whole curve; only the distance x − μ matters.
Standard deviation
The spread of the distribution. A larger σ lowers and widens the curve. At x = μ the height is inversely proportional to σ, but at the default x is exactly one σ from the mean, where the height hardly changes (a 10 % rise in σ lowers f by under 1 %).
Worked example with the values above
- Distance in standard deviations: z = (1 − 0) / 1 = 1.
- Exponent: −z²/2 = −0.5; ethat = 0.60653.
- f = 0.60653 / (1 × 2.5066) = 0.24197.
Assumptions and limits
- The quantity really is normally distributed (check with a plot of the data).
- μ and σ describe the whole population.
Common mistakes
- Reading f(x) as the probability of exactly x.
- Using the variance in place of σ in the formula.
Statistics and uncertainty
About 68 % of a normal population lies within ±1σ of the mean, 95 % within ±2σ and 99.7 % within ±3σ. The density value is not itself a probability; areas under the curve are.
Where it is used
- Modelling measurement scatter, such as miss distance or sensor noise.
- Setting tolerance limits from a standard deviation.
Maths you need
- Standard deviation: spread
- Exponentials: e^(−z²/2)