Probability
Normal Distribution Calculator
Turn one raw score, a mean, and a standard deviation into a z-score, a curve height, and the probability on each side of that score.
Worksheet
Press Calculate to see the result.
Turn one raw score, a mean, and a standard deviation into a z-score, a curve height, and the probability on each side of that score.
What this calculator answers
A normal model piles probability in a bell around a mean and lets the standard deviation set the width. This page starts from a measurement on that scale, such as a test score or a lab reading, and reports how many standard deviations it sits from the center. It also reports the height of the curve at that score and the probability on each side. Open it when the question is about a raw measurement rather than a z you already hold.
The page standardizes x to z, evaluates the standard normal density at that z, and divides by the standard deviation so the density is on the raw scale. It then reports the lower-tail probability and the survival probability above x. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Normal density and distribution function uses this relationship.
z = (x − mean) / sd, f(x) = φ(z) / sd, P(X ≤ x) = Φ(z), P(X > x) = 1 − Φ(z)
- x. The raw score on the user's scale.
- mean. The center of the normal curve.
- sd. The standard deviation, which must be positive.
- z. How many standard deviations x sits from the mean.
Subtracting the mean and dividing by the standard deviation maps any normal curve onto the standard normal. The standard density φ is a height on the z axis, so dividing by sd puts that height back onto the raw axis. The cdf Φ is already a probability, so it does not get divided by sd. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The score is 115, the mean is 100, and the standard deviation is 15. z = (115 − 100) / 15 = 1. The standard normal density at that z is 0.241971.
Divide by the standard deviation to put the height on the raw scale: 0.241971 / 15 = 0.016131. The lower-tail probability P(X ≤ x) is the standard normal cdf of 1, which equals 0.8413. The survival P(X > x) is 1 − 0.8413 = 0.1587.
The density was divided by 15. The cdf was not, because it is already a probability. These two probabilities add to 1. They locate one score on a normal curve, not a sample mean.
| z-score | 1 |
|---|---|
| Density f(x) | 0.016131 |
| P(X ≤ x) | 0.8413 |
| P(X > x) | 0.1587 |
A second situation
A score two standard deviations below the mean
The score is 70, the mean is 100, and the standard deviation is 15. z = (70 − 100) / 15 = −2. The standard normal density at that z is 0.053991.
Divide by the standard deviation to put the height on the raw scale: 0.053991 / 15 = 0.003599. The lower-tail probability P(X ≤ x) is the standard normal cdf of −2, which equals 0.0228. The survival P(X > x) is 1 − 0.0228 = 0.9772.
The density was divided by 15. The cdf was not, because it is already a probability. These two probabilities add to 1. They locate one score on a normal curve, not a sample mean.
| z-score | −2 |
|---|---|
| Density f(x) | 0.003599 |
| P(X ≤ x) | 0.0228 |
| P(X > x) | 0.9772 |
A score of 70 is two standard deviations below 100, so z changes sign and most of the probability moves to the right of the score. The density stays positive because the curve has height on both sides of the mean. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The score moved from 115 to 130, one further standard deviation above the mean. The headline result went from 1 to 2.
The nudged inputs
The score is 130, the mean is 100, and the standard deviation is 15. z = (130 − 100) / 15 = 2. The standard normal density at that z is 0.053991.
Divide by the standard deviation to put the height on the raw scale: 0.053991 / 15 = 0.003599. The lower-tail probability P(X ≤ x) is the standard normal cdf of 2, which equals 0.9772. The survival P(X > x) is 1 − 0.9772 = 0.0228.
The density was divided by 15. The cdf was not, because it is already a probability. These two probabilities add to 1. They locate one score on a normal curve, not a sample mean.
| z-score | 2 |
|---|---|
| Density f(x) | 0.003599 |
| P(X ≤ x) | 0.9772 |
| P(X > x) | 0.0228 |
How to read the result
Read z as a distance, the density as a height rather than a probability, and the cdf as the share of the curve at or below the score. The survival is what remains above it, and the two probabilities add to 1. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The measurement is modeled as continuous and normal.
- The mean and standard deviation describe the same population as the score.
- The standard deviation is a population scale, not a standard error.
- One score is being located on a curve, not being tested as a sample mean.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Dividing the cdf by the standard deviation. Only the density is rescaled.
- Using a standard error in the sd box. That answers a different question about a mean.
- Reading the density as the probability of the exact score. A continuous point has probability 0.
- Forgetting that P(X ≤ x) and P(X > x) share no overlap and add to 1.
Where a different calculator is the honest choice
Use the inverse normal calculator when you have a probability and want the raw score. Use the z-score probability calculator when you already have z and do not want a raw scale. Use the normal approximation to the binomial when the random variable is a count. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Inverse Normal Distribution Calculator, Z-Score Probability Calculator, Normal Approximation to Binomial Calculator.
Check yourself
A score of 115 comes from a normal model with mean 100 and standard deviation 15. What are z, the density, and the two tail probabilities?
z is 1. The density on the raw scale is 0.016131. P(X ≤ 115) is 0.8413 and P(X > 115) is 0.1587.
Using the result in a sentence
Standardized test scores, measurement error around a calibration mean, and biological traits that are roughly bell-shaped are the usual settings. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
A real histogram can be skewed or heavy-tailed. The probabilities on this page are exact only for the normal model you typed, not a certificate that the data are normal. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Normal Distribution Calculator commit to?
This page uses Normal density and distribution function: z = (x − mean) / sd, f(x) = φ(z) / sd, P(X ≤ x) = Φ(z), P(X > x) = 1 − Φ(z). Subtracting the mean and dividing by the standard deviation maps any normal curve onto the standard normal. The standard density φ is a height on the z axis, so dividing by sd puts that height back onto the raw axis. The cdf Φ is already a probability, so it does not get divided by sd. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Score x, Mean, Standard deviation, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the z-score is 1. Read z as a distance, the density as a height rather than a probability, and the cdf as the share of the curve at or below the score. The survival is what remains above it, and the two probabilities add to 1. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Dividing the cdf by the standard deviation. Only the density is rescaled. A real histogram can be skewed or heavy-tailed. The probabilities on this page are exact only for the normal model you typed, not a certificate that the data are normal. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the inverse normal calculator when you have a probability and want the raw score. Use the z-score probability calculator when you already have z and do not want a raw scale. Use the normal approximation to the binomial when the random variable is a count. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
z is 1. The density on the raw scale is 0.016131. P(X ≤ 115) is 0.8413 and P(X > 115) is 0.1587. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.