About the Normal Distribution Calculator
What is the Normal Distribution Calculator?
The normal distribution is the symmetric, bell-shaped curve that describes many natural measurements — heights, exam scores, measurement errors and IQ scores. It is fully described by its mean μ (the centre) and standard deviation σ (the spread).
Choose what to find: the probability below a value, above it, between two values, or the value at a given percentile (the inverse calculation). The calculator converts values to z-scores, uses the standard normal distribution and draws the bell curve for your mean and standard deviation.
How to use the Normal Distribution Calculator
- Mean (μ) (sample: 100)
- Standard Deviation (σ) (sample: 15)
- Find — choose the question; each option lists the fields it uses. (sample: P(X < x) – probability below x)
- Value (x) (sample: 115)
- Upper Value (x₂, for "between") — used only for “between two values”. (sample: 130)
- Percentile (for "value at percentile") — used only for “value at a given percentile”. (sample: 90)
- Press Calculate Probability (results also refresh as you type). The result card shows the Probability, along with z-score, P(X > x) and Within μ ± 1σ (85 to 115).
Formula used
P(X < x) = Φ((x − μ) ÷ σ)
- Φ = the cumulative standard normal distribution
- μ, σ = the mean and standard deviation
- x at percentile p = μ + z_p × σ, where Φ(z_p) = p
P(X > x) = 1 − Φ(z) and P(a < X < b) = Φ(z_b) − Φ(z_a). The density curve is f(x) = e^(−(x − μ)² / 2σ²) ÷ (σ√(2π)).
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Mean (μ) | 100 |
| Standard Deviation (σ) | 15 |
| Find | P(X < x) – probability below x |
| Value (x) | 115 |
| Upper Value (x₂, for "between") | 130 |
| Percentile (for "value at percentile") | 90 |
| Result | Value |
|---|---|
| P(X < 115) | 84.1345% |
| z-score | 1 |
| P(X > x) | 15.8655% |
| Within μ ± 1σ (85 to 115) | 68.27% |
Tips for using the Normal Distribution Calculator
- The 68–95–99.7 rule: about 68% of values lie within μ ± σ, 95% within μ ± 2σ and 99.7% within μ ± 3σ.
- For IQ scores (μ = 100, σ = 15), 115 is about the 84th percentile and 130 about the 98th.
- For a continuous distribution, P(X < x) and P(X ≤ x) are the same.