About Normal Distribution Calculator
This normal distribution calculator works for any mean μ and standard deviation σ, not only the standard normal. It opens on exam scores with a mean of 70 and a standard deviation of 10 and finds the share of students at 85 or more. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The normal curve is symmetric about the mean, and about 68%, 95% and 99.7% of the values lie within 1, 2 and 3 standard deviations. The calculator shades the exact region you ask about, so you can see the probability as an area.
Use the inverse mode to turn a percentile into a score: the value that cuts off the top 10% of a normal with mean 70 and standard deviation 10 is about 82.8.
How to use Normal Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
Normal density and the z-score
Any normal value can be converted to a z-score, the number of standard deviations from the mean, and then looked up in the standard normal.
f(x) = 1 / (σ × sqrt(2π)) × exp(−(x − μ)² / (2σ²))
z = (x − μ) / σ
mean = μ, variance = σ²Worked examples
Scores with mean 70, standard deviation 10: P(X ≥ 85) gives 0.066807
85 is 1.5 standard deviations above the mean, so about 6.7% of students score that high.
Standard normal: P(-1.96 ≤ Z ≤ 1.96) gives 0.95
The 95% central interval used in confidence intervals.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find a probability from a z-score?
Set the mean to 0 and the standard deviation to 1, choose P(X ≤ x) and enter the z-score. That is the same as a standard normal table lookup.
How do I find a percentile?
Choose Inverse and enter the percentile as a probability, for example 0.9 for the 90th percentile. The result is the score at that percentile.
What does the 68-95-99.7 rule say?
In a normal distribution about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.