Normal Distribution Calculator
Already working with z scores? Leave the mean at 0 and the standard deviation at 1, then enter your z as the raw score.
On a continuous curve P(X = exactly x) is always 0, so probability questions are always about the area on one side of x, or between two values.
Enter a mean, a standard deviation and a value, and this normal distribution calculator returns the probability under the bell curve, the z-score, and every line of the working. It covers the five questions a stats course actually asks: below a value, above a value, between two values, outside two values, and the reverse question where you already know the probability and need the cutoff score.
No z-table lookup, no rounding to two decimals, no guessing which tail the textbook wants.
How to use it
- Pick what you are solving for. The first dropdown switches between the four probability types and the inverse mode.
- Enter the mean (μ) and the standard deviation (σ). For a standard normal distribution calculator, leave these at 0 and 1 and your x value is the z-score.
- Enter your value, or the two boundary values if you chose “between” or “outside”.
- Read the answer as a decimal probability and a percentage, with the shaded curve showing exactly which area was measured.
Everything below the answer is the working: the z conversion, the table areas used, and the subtraction or flip that produced the final number. Copy it straight into your homework or check it against your own steps line by line.
Probability below a value
A class has test scores that are roughly normal with a mean of 78 and a standard deviation of 9. What fraction of students scored under 90?
Convert 90 to a z-score first: z = (90 − 78) / 9 = 1.3333. That says 90 sits one and a third standard deviations above the class mean. The area to the left of z = 1.3333 is 0.908789, so about 90.88% of the class scored below 90.
This left-tail area is what a graphing calculator calls normalcdf with a lower bound of negative infinity. It is also the percentile: a score of 90 is roughly the 91st percentile in that class.
Probability above a value
Same class, same numbers, opposite question. What fraction scored above 90?
The whole curve holds an area of 1, so the right tail is whatever the left tail leaves behind: 1 − 0.908789 = 0.091211, or 9.12%. Roughly nine students in a hundred beat that score.
Students lose marks here more than anywhere else. A z-table only prints left-hand areas, so every right-tail question needs that subtraction step. Choosing P(X > x) in the mode dropdown does the flip for you and prints it, so you can see where the 1 − came from.
Probability between two values
Adult male heights in the US run close to normal with a mean near 69 inches and a standard deviation near 2.8 inches. What share falls between 66 and 72 inches?
Two z-scores this time: z = (66 − 69) / 2.8 = −1.0714 and z = (72 − 69) / 2.8 = 1.0714. Take the area left of the upper bound and cut away the area left of the lower bound. The result is 0.716023, so a little over 71% of men fall in that six-inch band.
Between-values problems are subtraction, never addition. If you find yourself adding two tail areas and getting a number above 1, you have added the wrong two pieces.
Probability outside two values
IQ scores are scaled to a mean of 100 and a standard deviation of 15. What proportion of people score below 85 or above 115?
Both tails count here. The calculator shades them together and returns 0.317311, about 31.7%. The complement is the familiar 68.27% who land inside one standard deviation of the mean.
Both tails together are what a two-tailed hypothesis test measures, which is why this mode earns its place. The classic example is z = ±1.96, where the outside area comes to 0.049996 and gives you the 5% significance level used throughout confidence interval work.
Working backwards from a probability
The harder direction: you know the percentage and need the score. A scholarship goes to the top 10% of applicants on a test with a mean of 75 and a standard deviation of 8. What is the cutoff?
Switch the mode to inverse and set the direction to “top”, because “top 10%” is a right-hand area and the underlying table is left-handed. The calculator converts P(X > x) = 0.1 into P(X < x) = 0.9, finds the z-score that carries 90% of the area below it (z = 1.2816), then rebuilds the raw score: x = 75 + 1.2816 × 8 = 85.2524.
Miss that direction step and you get 64.75, which is the bottom 10% cutoff. It looks like a perfectly reasonable answer and it is wrong by 20 points. An inverse normal distribution calculator that does not ask which tail you mean will hand you that mistake without a warning, so the direction selector here is deliberate.
The same problems on a TI-84
If your course requires calculator syntax on the exam paper, the two commands live under 2nd, VARS (DISTR).
- normalcdf(lower, upper, μ, σ) gives an area. For P(X < 90) in the test score example, that is normalcdf(-1E99, 90, 78, 9) = 0.9088. For P(X > 90), swap the bounds: normalcdf(90, 1E99, 78, 9) = 0.0912. For the height problem, normalcdf(66, 72, 69, 2.8) = 0.7160.
- invNorm(area, μ, σ) goes the other way and returns a score. The scholarship cutoff is invNorm(0.9, 75, 8) = 85.2524.
Two things trip people up. The 1E99 is just a stand-in for infinity, typed with the EE key, and older operating systems treat invNorm as left-tailed with no option to say otherwise, so a “top 10%” question still needs you to enter 0.9 rather than 0.1. The results above match this tool to four decimal places, so you can use one to check the other.
What the normal distribution actually describes
The normal distribution is a continuous probability curve defined by two numbers: where its center sits (μ) and how wide it spreads (σ). Change the mean and the whole bell slides along the axis. Change the standard deviation and it gets taller and narrower, or flatter and wider, while the area underneath stays pinned at exactly 1.
It shows up constantly because of the central limit theorem. Sample means from almost any population start looking normal once the samples get large enough, which is why sampling distributions, measurement error and standardized test scores all end up on the same curve. The NIST/SEMATECH e-Handbook of Statistical Methods gives the full density function and its properties if you need the formal definition.
Raw scores, z-scores and the standard normal
A z-score is just a raw value rewritten as a distance from the mean, measured in standard deviations:
z = (x − μ) / σ
A z of 2 means “two standard deviations above average” whether the original units were dollars, kilograms or exam marks. That is the whole trick behind the standard normal distribution: rescale everything to μ = 0 and σ = 1, and a single table of areas works for every problem.
If you are starting from a raw data set rather than a stated mean and spread, run it through the standard deviation calculator first to get σ, then bring those two numbers back here. The standard deviation graph tool plots the resulting bell curve if you want to see the shape your data produced.
The 68-95-99.7 rule
Statistics courses call this the empirical rule. For any normal curve, the area within a whole number of standard deviations from the mean is fixed:
| Range | Exact area | Rounded |
|---|---|---|
| μ ± 1σ | 0.682689 | 68.3% |
| μ ± 2σ | 0.954500 | 95.4% |
| μ ± 3σ | 0.997300 | 99.7% |
Use this as a sanity check on anything the calculator returns. If you asked for the probability inside two standard deviations of the mean and got 0.6 back, one of your inputs is wrong. The 95% figure people quote in practice comes from ±1.96σ, not ±2σ, which is why 1.96 turns up everywhere in confidence intervals.
Why P(X = exactly one number) is always zero
Ask this calculator for the probability that a value equals exactly 70 and the honest answer is 0. On a continuous curve, probability is area, and a single point has no width, so there is no area to measure.
That sounds like a technicality until an exam question asks for P(X = 70) and expects you to say zero. What people usually mean is a small interval around 70, such as P(69.5 < X < 70.5), which the “between” mode handles. This is also why the normal approximation to a binomial distribution needs a continuity correction: you are replacing a count that can only be a whole number with a curve that has no gaps.
When the bell curve is the wrong tool
The normal distribution assumes symmetry and thin tails. Plenty of real data breaks both assumptions.
- Income, house prices and website session lengths are right-skewed. A normal model will predict negative values that cannot exist.
- Counts of events, like defects per batch or arrivals per hour, follow Poisson or binomial patterns until the numbers get large.
- Small samples with an unknown population standard deviation belong to the t-distribution, not this one. Under roughly 30 observations, the t-curve has fatter tails and gives wider intervals.
- Bounded data, such as percentages trapped between 0 and 100, gets squeezed at the edges in a way the bell curve does not model.
Plot a histogram before you commit to a normal model. If it looks lopsided or has a hard floor at zero, the probability this calculator returns will be technically correct and practically useless. OpenStax covers the assumption checks in its chapter on the standard normal distribution if you want a second walkthrough.
Try these next
- Percentile Calculator: find percentiles directly from a list of raw values.
- Hypothesis Testing Calculator: run a one-sample or two-sample test with the test statistic shown.
- P Value Calculator: turn a z or t statistic into a p-value for either tail.
- Standard Error Calculator: measure how much a sample mean is likely to move.
Frequently Asked Questions (FAQs)
How do you find probability on a normal distribution calculator?
Enter the mean and standard deviation, pick which side of the curve you want, then enter your value. The calculator converts that value to a z-score and returns the matching area under the curve. A probability of 0.9088 means 90.88% of the distribution falls in the shaded region.
What is the difference between a normal distribution and a standard normal distribution?
They are the same curve at different scales. A normal distribution can have any mean and any positive standard deviation. The standard normal has a mean of 0 and a standard deviation of 1, which is what z-tables are printed for. Converting to z-scores moves any normal problem onto that single shared scale.
Can I use this as a z-score calculator?
Yes. Leave the mean at 0 and the standard deviation at 1, and whatever you type as x is read directly as a z-score. The calculator also prints the z conversion on every calculation, so you get the z-score even when you entered raw values with a different mean and spread.
What does normalcdf actually calculate?
It measures the area under the normal curve between two bounds, which is the probability of landing in that range. The four arguments are the lower bound, the upper bound, the mean and the standard deviation. Using negative infinity as the lower bound gives a left-tail probability, which is the same thing as a percentile.
What is invNorm used for?
invNorm runs the calculation backwards. Instead of giving it a score and getting a probability, you give it a probability and it returns the score that sits at that cutoff. Use it for questions phrased as “the top 5%”, “the bottom quartile” or “the 90th percentile”, where the percentage is known and the value is not.
How do you find a percentile from a normal distribution?
A percentile is a left-tail area. The 25th percentile of a distribution with a mean of 78 and a standard deviation of 9 is 71.93, because 25% of the curve sits below that point. Set the mode to inverse, choose the bottom direction, and enter 0.25.
Why does the calculator return 0 for an exact value?
Because probability on a continuous curve is area, and a single point has zero width. P(X = 70) is genuinely 0 for any normal distribution. If you need the chance of a measurement rounding to 70, ask for the probability between 69.5 and 70.5 instead.
Can I use this for a normal approximation to the binomial?
Yes, provided both np and n(1-p) are at least 10. Set the mean to np and the standard deviation to the square root of np(1-p), then apply a continuity correction by shifting each boundary half a unit outward. Below that sample size, use the exact binomial instead.
What do I enter if I only have raw data and no stated standard deviation?
Calculate the mean and standard deviation from the data set first, then enter those two numbers here. Use the population standard deviation when your data covers the entire group, and the sample standard deviation when it is a sample drawn from a larger population.