Answer first
What this calculator tells you
Calculate the probability that a normally distributed value falls between two limits. Turn a mean and a standard deviation into the share of results that land inside a range you care about. Formula: P(a < X < b) = Φ((b − μ) ÷ σ) − Φ((a − μ) ÷ σ), Φ the standard normal cumulative distribution. At the worked-example inputs, the probability between a and b is 68.3%. Holding every other input steady, moving mean (μ) from 80 to 120 moves the result from 36.0% to 68.3%.
Transparent method
The formula
Turn a mean and a standard deviation into the share of results that land inside a range you care about.
Worked example
Example inputs
How to interpret the result
A bell curve turns a mean and a spread into the share of results in any band. About 68 percent of values sit within one standard deviation of the mean, which for a mean of 100 and a spread of 15 is the range from 85 to 115. The two tail figures show what is left outside, split evenly here because the band is centered.
At the worked-example inputs the probability between a and b is 68.3%. It rises with upper value (b) and falls as lower value (a) and standard deviation (σ) increase; mean (μ) does not move it.
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability.
Before you rely on it
What to check
Confirm the data is roughly bell-shaped. Income, wait times and many counts are skewed, and the curve misstates their tails badly.
The common error
Where people go wrong with normal distribution probability calculator
Entering the variance as the standard deviation. Variance is the square, so a spread of 15 entered as 225 stretches the band far wider than the data.
Sensitivity evidence
How mean (μ) changes the probability between a and b
Holding every other input at the worked-example value, moving mean (μ) from 80 to 120 moves the probability between a and b from 36.0% to 68.3%: a spread of 32.3%, or 47% of the worked-example result.
| Mean (μ) | Probability between a and b | Probability below a | Probability above b |
|---|---|---|---|
| 80 | 36.0% | 63.1% | 0.982% |
| 90 | 58.3% | 36.9% | 4.8% |
| 100worked example | 68.3% | 15.9% | 15.9% |
| 110 | 58.3% | 4.8% | 36.9% |
| 120 | 36.0% | 0.982% | 63.1% |
Every input, tested
Which input moves the probability between a and b most
Of the 4 inputs, upper value (b) moves the probability between a and b most (35.9% across the range tested) and standard deviation (σ) moves it least (9.4%). Mean (μ) does not change it at all.
| Input | Tested from | To | Probability between a and b at each end | Swing |
|---|---|---|---|---|
| Upper value (b) | 104 | 127 | 44.6% to 80.5% | 35.9% (53%) |
| Lower value (a) | 77 | 94 | 77.9% to 49.7% | 28.2% (41%) |
| Standard deviation (σ) | 14 | 17 | 71.6% to 62.2% | 9.4% (14%) |
| Mean (μ) | 90 | 110 | 58.3% to 58.3% | none |
Two variables at once
Probability between a and b by mean (μ) and standard deviation (σ)
Across the grid the probability between a and b runs from 33.7% to 78.9%. Moving mean (μ) from 80 to 120 shifts it by 0.000% at the middle column, and moving standard deviation (σ) from 12 to 18 shifts it by 19.3% at the middle row, so standard deviation (σ) is the bigger lever here.
| Mean (μ) \ Standard deviation (σ) | 12 | 15 | 18 |
|---|---|---|---|
| 80 | 33.7% | 36.0% | 36.5% |
| 90 | 64.3% | 58.3% | 52.7% |
| 100 | 78.9% | 68.3% | 59.5% |
| 110 | 64.3% | 58.3% | 52.7% |
| 120 | 33.7% | 36.0% | 36.5% |
The highlighted cell is the worked example: 68.3%.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Mean (μ) | 100 | The center of the distribution. |
| Standard deviation (σ) | 15 | The spread of the distribution. It must be above zero. |
| Lower value (a) | 85 | The bottom of the range you are asking about. |
| Upper value (b) | 115 | The top of the range you are asking about. |
| Probability between a and b | 68.3% | |
| Probability below a | 15.9% | |
| Probability above b | 15.9% | |
Inputs, definitions and assumptions
Mean (μ)
The center of the distribution. The prefilled worked-example value is 100.
Standard deviation (σ)
The spread of the distribution. It must be above zero. The prefilled worked-example value is 15.
Lower value (a)
The bottom of the range you are asking about. The prefilled worked-example value is 85.
Upper value (b)
The top of the range you are asking about. The prefilled worked-example value is 115.
How to use this calculator
- 1Verify the inputs. Gather mean (μ), standard deviation (σ), lower value (a) and upper value (b) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the probability between a and b at 68.3%. Store your own version of it as Scenario A.
- 3Test one change. Start with upper value (b), the input with the biggest effect here: moving upper value (b) from 104 to 127 takes the probability between a and b from 44.6% to 80.5%, a swing of 53% of the worked-example figure.
- 4Check the extremes. At half the example upper value (b) (57.5) the probability between a and b is 0.000%; at double (230) it is 84.1%.
People also ask
Frequently asked questions
How do you calculate normal distribution probability?
P(a < X < b) = Φ((b − μ) ÷ σ) − Φ((a − μ) ÷ σ), Φ the standard normal cumulative distribution. At the worked-example inputs the probability between a and b is 68.3%.
What does the normal distribution probability result mean?
Turn a mean and a standard deviation into the share of results that land inside a range you care about. At the worked-example inputs the probability between a and b is 68.3%. It rises with upper value (b) and falls as lower value (a) and standard deviation (σ) increase; mean (μ) does not move it.
How much does mean (μ) change the probability between a and b?
Holding every other input at the worked-example value, moving mean (μ) from 80 to 120 moves the probability between a and b from 36.0% to 68.3%, a spread of 32.3%.
What are the limits of this normal distribution probability calculator?
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability. The tables on this page test mean (μ) only from 80 to 120; a value outside that range is not tabulated here.
Which input moves the probability between a and b most in the normal distribution probability calculator?
Ranked by how far each moves the probability between a and b across the range tested: upper value (b) (35.9%, 53%), lower value (a) (28.2%, 41%) and standard deviation (σ) (9.4%, 14%). Mean (μ) does not change it.
If I double upper value (b) in the normal distribution probability calculator, does the probability between a and b double?
Doubling it from 115 to 230 takes the probability between a and b from 68.3% to 84.1%, which is 1.23 times the worked-example figure. So it grows, but by less than double. Halving it to 57.5 gives 0.000%.
How much does standard deviation (σ) matter in the normal distribution probability calculator?
The worked example uses 15. With the other inputs left at the worked example, moving standard deviation (σ) from 14 to 17 takes the probability between a and b from 71.6% to 62.2%, a swing of 14% of the worked-example figure.
How much does lower value (a) matter in the normal distribution probability calculator?
The worked example uses 85. With the other inputs left at the worked example, moving lower value (a) from 77 to 94 takes the probability between a and b from 77.9% to 49.7%, a swing of 41% of the worked-example figure.
How much does upper value (b) matter in the normal distribution probability calculator?
The worked example uses 115. Holding every other input at its worked-example value, moving upper value (b) from 104 to 127 takes the probability between a and b from 44.6% to 80.5%, a swing of 53% of the worked-example figure.
Which inputs change the probability below a in the normal distribution probability calculator?
At the worked-example inputs it is 15.9%. Mean (μ) takes it from 36.9% to 4.8%, standard deviation (σ) takes it from 14.2% to 18.9% and lower value (a) takes it from 6.3% to 34.5%.
Which inputs change the probability above b in the normal distribution probability calculator?
At the worked-example inputs it is 15.9%. Mean (μ) takes it from 4.8% to 36.9%, standard deviation (σ) takes it from 14.2% to 18.9% and upper value (b) takes it from 39.5% to 3.6%.
Does a streak mean a coin is due to change?
No. Each independent flip has the same chance whatever came before. A long streak is unlikely, but after one the next flip is still even. Believing otherwise is called the gambler's fallacy.
What is the difference between exactly k and k or fewer?
Exactly k is the chance of one specific count. K or fewer adds up every count from zero to k. Ask which one your question needs, because the second is always the larger number.
Sources and evidence
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