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Free Online Normal Distribution Calculator With the Curve Drawn

Find the probability below, above, between or outside any values on a normal distribution, with the area drawn and shaded so you can see what the number means. Accurate far into the tails, where the approximations most calculators use are worse than the answer they are reporting.

Free Forever Nothing Uploaded Curve Drawn Accurate in the Tails

Free Online Normal Distribution Calculator With the Curve Drawn

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🔔 Normal Distribution Calculator The shaded area drawn · accurate in the tails

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The mean

The standard deviation

🔔 What do you want to know?

Pick the shape of the question. The shaded part of the curve is what you are asking for, which makes it hard to ask for the wrong thing by accident.

The question

The value

I am typing z-scores, not values With this on, the mean and deviation are ignored and your numbers are read as standard deviations from the mean.

Try one of these

The sixth is where the usual approximations stop being any use.

The probability

—

As a percentage—

As odds—

The z-scores—

Out of a million—

Step	What happens

The bands people quote from memory, with the exact figures. Two of them are not the round numbers everyone remembers.

Within	That range	Inside	Outside

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How to Use a Normal Distribution Table

Three steps, and the picture is doing most of the work.

Give it the mean and the standard deviation Or switch on the z-score option and type standard deviations directly, which is what an exam question usually gives you.

Choose the shape of the question Below, above, between or outside. Mixing these up is the most common mistake in this kind of problem, and the picture makes it visible before the answer matters.

Check the shaded area matches what you meant The curve shows exactly which part of the distribution has been counted. If the shading is not what you pictured, the number would have been wrong and you would not have known.

What to Know About the Normal Curve

Where the tails matter, and when the curve is the wrong model.

The picture is there to catch the mistake the number cannot. Between and outside, below and above: these are easy to pick wrongly and the resulting number looks entirely plausible either way. Shading the area means you check your question rather than just reading an answer. This is the main reason the curve is drawn rather than described.

The tail answers here are accurate, which is unusual. The approximation that circulates for the normal curve is good to about seven decimal places. At six standard deviations out the true answer is about one in a billion, so that approximation’s error is a thousand times larger than the answer. This page uses an incomplete gamma method accurate to around fourteen digits, checked against scipy, so the six sigma preset gives a real number.

A great many real quantities are not normally distributed. Heights, measurement errors, exam marks and the averages of repeated samples fit the curve well. Incomes, waiting times, insurance claims, file sizes, city populations and anything that cannot go below zero but has no ceiling do not. For those, the curve gives confident answers that are wrong, usually badly wrong in the tail, which is exactly where people look.

The 68, 95, 99.7 rule is a rounding of 68.27, 95.45 and 99.73. And the 95 per cent that statistics actually uses is 1.96 standard deviations rather than 2. Two deviations give 95.45 per cent. That gap is small and it is the reason confidence intervals are built on 1.96, so both appear in the bands table rather than only the memorable version.

The chance of exactly one value is zero, and that is not a trick. The normal distribution is continuous, so probability is area, and a single point has no width and therefore no area. Asking for the chance of being exactly 85 has the answer zero. What is usually meant is a narrow range around 85, which is what the between option is for. This is why there is no "equals" option here.

The standard deviation has to be above zero. A deviation of zero describes a distribution with no spread, where every value is the mean exactly. There is no curve to draw and no area to measure, so rather than returning zero or one it is reported as a question that does not apply.

Key Features & Capabilities

What it does, and what it deliberately does not.

The area drawn and shaded Scalable vector, so the text stays text, and the shading shows precisely which part of the curve was counted.

Four shapes of question Below, above, between and outside, which are the four that matter and the four people mix up.

Accurate deep in the tail Six and seven standard deviations give real numbers rather than the noise a seven-decimal approximation produces there.

Values or z-scores Type real units, or switch to standard deviations, which is the form exam questions arrive in.

The empirical rule, exactly One, two and three deviations with their real percentages, and 1.96 alongside two so the difference is clear.

Nothing is uploaded It runs in this tab and keeps working offline.

About the Normal Distribution

The normal distribution is the bell shaped curve, and its usefulness comes from a surprising fact: the average of a lot of independent things tends towards this shape whatever shape those things had individually. That is why it turns up in measurement error, in manufacturing tolerances, in sampling, and in anything built from many small additive effects.

Using it means answering one of four questions. What proportion is below a value, above a value, between two values, or outside two values. The arithmetic is identical in all four cases; what differs is which part of the area under the curve you want. And that is exactly where the mistakes happen, because between and outside are complements, below and above are complements, and all four produce numbers that look reasonable. A tool that prints 0.9332 without showing you what it measured has given you no way to notice. So the curve is drawn here and the part you asked for is shaded.

The second decision was about accuracy in the tails. There is a short approximation for the normal curve that appears in textbooks and gets copied into calculators everywhere, and it is accurate to about seven decimal places. Near the middle that is far more than enough. At five standard deviations out the answer itself is about three in ten million, and at six it is about one in a billion, so the approximation’s error dwarfs the answer it is reporting. Since quality control, reliability and anything that asks how rarely something happens all live out there, this page computes the curve through the incomplete gamma function instead, which holds around fourteen digits across the whole range. It was checked against scipy.

The caveat belongs in plain sight rather than in the small print. The curve is a model, and it fits some things beautifully and others not at all. Heights, measurement errors and sample averages, yes. Incomes, waiting times, insurance claims, file sizes and city populations, no, because those are skewed and bounded below. For a skewed quantity the normal curve will still hand you a precise looking probability, and it will be wrong, most badly in the tail. Knowing which kind of quantity you have is the part the calculator cannot do for you.

Frequently Asked Questions

Between, outside, tails and the empirical rule.

They are complements and together they make one. Between counts the area in the middle, outside counts the two tails. On a mean of 70 and a deviation of 10, between 50 and 90 is 95.45 per cent and outside them is 4.55 per cent. The shading shows which one you have selected.

Because the distribution is continuous, so probability is area under the curve, and a single point has no width. The answer is always zero. What people usually want is a narrow interval, which is what the between option provides.

That about 68 per cent of values lie within one standard deviation of the mean, 95 within two and 99.7 within three. The exact figures are 68.27, 95.45 and 99.73. The 95 per cent used in statistics is 1.96 deviations rather than 2, which is why both are in the bands table.

Yes, to around fourteen digits across the whole range. That matters because the common seven decimal approximation has an error larger than the answer itself past about five standard deviations, which is exactly where questions about rare events sit.

When the data is skewed or bounded. Incomes, waiting times, insurance claims and city sizes are not normal, and the curve will give a confident answer that is wrong, worst in the tail. Heights, measurement errors and the averages of samples are fine.

Yes, with the switch in the panel. The mean and standard deviation are then ignored and your numbers are read as standard deviations from the mean, which is the form most textbook questions use.

No. The curve and the probabilities are worked out in your own browser, and the page keeps working offline.