What the normal distribution calculator gives you
The calculator answers the four questions a normal model is used for: the probability below a value (the normal CDF, normalcdf on a TI-84), the probability above it, the probability between two values, and the inverse question, which value sits at a given percentile (invNorm).[2] Pick the question, type the mean and the standard deviation of your distribution, and the shaded curve shows the area it computed. Areas come from the standard normal CDF Φ evaluated to about 15 digits, so there is no table rounding.[4]
Every normal distribution is the standard normal stretched and shifted: subtract the mean and divide by the standard deviation, and any value becomes a z-score with mean 0 and SD 1.[1] That is why one calculator handles test scores, measurement errors and fill weights alike. For the z-score itself and z-table lookups, use the z-score calculator.
Worked example: filling 500 mL bottles
Suppose a machine fills bottles with a mean of 500 mL and a standard deviation of 4 mL, and fill volumes are roughly normal (a made-up example to show the steps).
Three questions, one distribution (μ = 500, σ = 4)
Below 494 mL? z = (494 − 500) ÷ 4 = −1.5, and Φ(−1.5) = 0.0668: about 6.7% of bottles.
Between 495 and 505 mL? z runs from −1.25 to 1.25, so the area is Φ(1.25) − Φ(−1.25) = 0.7887.
Where does the fullest 5% start? The 95th percentile of the standard normal is 1.6449, so x = 500 + 1.6449 × 4 = 506.58 mL.
The empirical rule (68-95-99.7)
The empirical rule says that about 68% of a normal distribution lies within one standard deviation of the mean, about 95% within two and about 99.7% within three.[1][3] It is a quick check on any answer from the calculator: a value 2 SD above the mean should land near the 97.5th percentile, never near the 60th.
| Range | Inside | Above μ + kσ | Below μ − kσ |
|---|---|---|---|
| μ ± 1σ | 68.269% | 15.866% | 15.866% |
| μ ± 1.5σ | 86.639% | 6.681% | 6.681% |
| μ ± 1.96σ | 95% | 2.5% | 2.5% |
| μ ± 2σ | 95.45% | 2.275% | 2.275% |
| μ ± 2.576σ | 99% | 0.5% | 0.5% |
| μ ± 3σ | 99.73% | 0.135% | 0.135% |
Normal distribution probability: which area?
Match the wording of the question to the area; this is where most mistakes happen.
| Question says | Area | TI-84 |
|---|---|---|
| less than x, at most x, below | P(X < x) = Φ(z) | normalcdf(−1E99, x, μ, σ) |
| more than x, at least x, above | P(X > x) = 1 − Φ(z) | normalcdf(x, 1E99, μ, σ) |
| between a and b | Φ(z_b) − Φ(z_a) | normalcdf(a, b, μ, σ) |
| the value at the pth percentile | x = μ + z_p σ | invNorm(p, μ, σ) |
For a continuous distribution “less than” and “at most” give the same area, because a single point has probability 0. That is not true for counts: for a number of successes use the binomial distribution calculator, which handles the discrete steps exactly.
Is my data normal?The calculator assumes the model is normal. Check a histogram or the quartiles of your data with the statistics calculator first: strongly skewed data (incomes, waiting times) give wrong tail areas under a normal model.
Common mistakes
- Using the variance as σ. If you are given σ² = 16, the standard deviation is 4.
- Reading a left area as a right area. “Top 10%” is the 90th percentile.
- Using a sample SD for a sample mean. The spread of an average of n values is σ ÷ √n; use the confidence interval calculator for questions about means.
How we check this calculator
Tests compare the engine with the NIST standard normal table: Φ(1.96) = 0.9750021, Φ(−3) = 0.0013499, and the 97.5th percentile z = 1.959964.[4] The empirical-rule shares above are computed live from the same function, not typed in.
- 14 November 2026: page published.
More statistics calculators
- Statistics calculatorMean, median, mode, SD, quartiles and outliers
- Standard deviation calculatorSample and population SD, step by step
- Z-score calculatorz, percentile and area under the normal curve
- P-value calculatorFrom z, t, chi-square or F
- Confidence interval calculatorMeans and proportions, t or z
- t-test calculatorOne-sample, Welch, pooled and paired
- Binomial distribution calculatorExactly, at most, at least x successes
- Probability calculatorAnd, or, neither, given; at least once in n tries
- Sample size calculatorSurvey size for a margin of error, and the reverse
Frequently asked questions
How do you calculate a normal distribution probability?
Standardize with z = (x − μ) ÷ σ, then find the area under the standard normal curve: Φ(z) for below, 1 − Φ(z) for above, and the difference of two Φ values for a range.
What is the empirical rule?
About 68% of normal values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3 (exactly 68.27%, 95.45% and 99.73%).
What is an inverse normal calculator?
It goes from a probability to a value: for the 95th percentile, x = μ + 1.6449σ. Choose “z-score from a percentile” in the calculator and add the mean and SD.
What is normalcdf?
The TI-84 function for the area under a normal curve between two values; normalcdf(a, b, μ, σ) equals the “between” result above.