Normal Distribution Calculator

Calculate probabilities, z-scores, percentiles, critical values, and highlighted bell-curve regions.

Results are calculated automatically as you enter data.

Mean μ Standard deviation σ

Task

x value

Calculation steps

Result -

Bell curve

The highlighted area shows the selected probability region.

Disclaimer: All tools on this site are intended for educational and informational purposes only. Not professional advice or a decision tool. No guarantee of accuracy; verify data, assumptions, rules, model fit, and results before important use. See our Terms of Use.

Assumptions and limitations

Results are estimates for educational and exploratory use only and are not professional advice. No guarantee is made that outputs are accurate, complete, current, suitable, or error-free. Verify inputs, units, formulas, assumptions, rounding, data quality, model fit, and applicable rules independently. Do not rely on this calculator as the only basis for financial, legal, medical, safety, academic, business, official, eligibility, or regulated decisions; consult a qualified professional when stakes are material. Probability and inverse-percentile results use browser-based numerical approximations of the normal CDF and inverse CDF. No formal error bound is provided; verify high-stakes statistical work with validated software or tables. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Is a Normal Distribution?

A normal distribution is a bell-shaped probability model for a continuous variable. It is centered at its mean, spreads according to its standard deviation, and is symmetric: values the same distance above and below the mean have the same density.

Normal-distribution calculations answer questions about area under the bell curve. Because total area under a probability density curve is $1$, an area can also be read as a probability or percentage. For example, an area of $0.25$ means a probability of $25\%$.

The normal distribution is useful because many measurements are approximately bell-shaped, and many statistical methods use normal models or normal approximations. It is commonly used for test scores, measurement errors, biological measurements, quality-control data, standardized scores, confidence intervals, and hypothesis tests.

Why Normal Distribution Calculations Matter

A normal curve turns raw measurements into interpretable probabilities. Instead of only saying that a value is high or low, you can estimate how unusual it is under a chosen normal model.

Normal calculations help answer questions such as:

What percentage of values are expected to be below a cutoff?
What percentage are expected to be above a cutoff?
What is the probability of falling between two values?
What raw value marks the $90$th, $95$th, or another percentile?
How many standard deviations is a value from the mean?

These questions appear often in statistics classes, research summaries, quality assurance, analytics dashboards, and standardized comparisons.

Key Terms to Know

Random variable $X$: The quantity being modeled, such as a score, height, weight, time, or measurement error.
Mean $\mu$: The center of the normal distribution.
Standard deviation $\sigma$: The spread of the distribution. A larger standard deviation makes the curve wider and flatter.
Standard normal distribution: A normal distribution with mean $0$ and standard deviation $1$.
z-score: A standardized value that tells how many standard deviations a raw value is from the mean.
Probability density function: The curve itself. Its height is density, not probability.
Cumulative distribution function: The area to the left of a value.
Tail probability: The probability in one end, or tail, of the distribution.
Percentile: A cutoff value with a given percentage of the distribution at or below it.
Inverse normal CDF: The process of finding the z-score or raw value that corresponds to a chosen percentile.

How Normal Distribution Calculations Work

A normal distribution with mean $\mu$ and standard deviation $\sigma$ has density

$$ f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

This formula describes the height of the bell curve at $x$. Probabilities come from areas under the curve, not from the height at a single point.

The most important step is standardization. A raw value $x$ is converted to a z-score:

$$ z=\frac{x-\mu}{\sigma} $$

Where:

$x$ is the raw value.
$\mu$ is the mean.
$\sigma$ is the standard deviation.
$z$ is the number of standard deviations $x$ is above or below the mean.

A positive z-score means the value is above the mean. A negative z-score means it is below the mean. A z-score of $0$ means the value equals the mean.

Once a raw value has been standardized, the standard normal cumulative distribution function gives the left-tail area:

$$ \Phi(z)=P(Z\le z) $$

For the standard normal curve, this area can be written as

$$ \Phi(z)=\int_{-\infty}^{z}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}\,dt $$

This integral does not simplify into an elementary formula, so tables, calculators, and software use numerical methods.

The main probability relationships are:

$$ P(X\le x)=\Phi\left(\frac{x-\mu}{\sigma}\right) $$

$$ P(X\ge x)=1-\Phi\left(\frac{x-\mu}{\sigma}\right) $$

For an interval from $a$ to $b$:

$$ P(a\le X\le b)=\Phi\left(\frac{b-\mu}{\sigma}\right)-\Phi\left(\frac{a-\mu}{\sigma}\right) $$

For the outside area beyond an interval:

$$ P(X\le a\ \text{or}\ X\ge b)=1-P(a\le X\le b) $$

A percentile works in reverse. If $p$ is a proportion between $0$ and $1$, first find the standard normal value

$$ z=\Phi^{-1}(p) $$

Then convert it back to the original scale:

$$ x=\mu+z\sigma $$

Examples of Normal Distribution Calculations in Practice

Example 1: Left-tail probability

Suppose exam scores are normally distributed with mean $100$ and standard deviation $15$. To find the probability that a score is at or below $130$, first standardize:

$$ z=\frac{130-100}{15}=2 $$

A z-score of $2$ is two standard deviations above the mean. The standard normal left-tail area at $z=2$ is approximately $0.97725$, so

$$ P(X\le 130)\approx 0.97725=97.725\% $$

That means a score of $130$ is higher than about $97.7\%$ of values in this normal model.

Example 2: Probability between two values

Suppose heights are modeled with mean $170$ cm and standard deviation $8$ cm. To find the probability of a height between $160$ cm and $180$ cm, standardize both endpoints:

$$ z_1=\frac{160-170}{8}=-1.25 $$

$$ z_2=\frac{180-170}{8}=1.25 $$

The area between the two z-scores is

$$ P(160\le X\le 180)=\Phi(1.25)-\Phi(-1.25) $$

$$ P(160\le X\le 180)\approx 0.89435-0.10565=0.78870 $$

So the probability is approximately $78.870\%$.

Example 3: Finding a percentile value

Suppose the same height model has mean $170$ cm and standard deviation $8$ cm. To find the $95$th percentile, use the z-score whose left-tail area is $0.95$:

$$ z=\Phi^{-1}(0.95)\approx 1.64485 $$

Then convert the z-score back to the original unit:

$$ x=170+1.64485(8) $$

$$ x\approx 183.1588 $$

The $95$th percentile is about $183.16$ cm. In this model, about $95\%$ of values are at or below that height.

Example 4: Outside-area probability

For the standard normal distribution, the probability between $-1$ and $1$ is about $68.27\%$. The outside probability is the remaining area in both tails:

$$ P(Z\le -1\ \text{or}\ Z\ge 1)=1-P(-1\le Z\le 1) $$

$$ P(Z\le -1\ \text{or}\ Z\ge 1)\approx 1-0.68269=0.31731 $$

So about $31.731\%$ of the standard normal distribution lies more than one standard deviation away from the mean.

How to Interpret the Result

A normal-distribution result is an area, a z-score, or a raw percentile value.

A left-tail probability such as $P(X\le x)$ is the percentage of the distribution at or below $x$. If the result is $84\%$, then about $84\%$ of values are expected to be less than or equal to that cutoff under the model.

A right-tail probability such as $P(X\ge x)$ is the percentage of the distribution at or above $x$. Large right-tail probabilities mean the cutoff is relatively low; small right-tail probabilities mean the cutoff is relatively high.

A between probability measures area inside an interval. It is useful when the question is about being within a target range, tolerance range, or score band.

An outside probability measures the combined area in both tails beyond an interval. It is useful when the question is about being unusually low or unusually high.

A z-score is unitless. It is not measured in centimeters, dollars, seconds, points, or any other original unit. It tells position relative to the mean in standard-deviation units.

A percentile result is a raw cutoff value in the same unit as the original variable. A $95$th percentile value is not $95\%$ itself; it is the $x$ value with $95\%$ of the model at or below it.

For a continuous normal distribution, $P(X\le x)$ and $P(X<x)$ have the same value because the probability of landing on exactly one point is zero. The same idea applies to $P(X\ge x)$ and $P(X>x)$.

Common Mistakes and Misconceptions

Entering percentiles on the wrong scale. If a percentile input expects a percent, enter $95$ for the $95$th percentile, not $0.95$. The value $0.95$ would mean less than the $1$st percentile if the input is interpreted as a percent.

Using inconsistent units. The mean, standard deviation, and raw values must all use the same unit. Do not mix centimeters and inches, seconds and minutes, or dollars and cents unless you convert them first.

Confusing density with probability. The height of the curve at a point is not the probability of that exact value. Probability is area over an interval or tail.

Mixing up between-area and outside-area questions. The probability inside an interval and the probability outside an interval are complements. If one is $0.80$, the other is $0.20$.

Treating a z-score as a raw value. A z-score of $1.5$ means one and a half standard deviations above the mean. It does not mean the original value is $1.5$ in the original unit.

Assuming all data are normal. A normal model is useful only when it is reasonable for the situation. Strong skew, multiple peaks, outliers, bounded data, or categorical data may require a different method.

Rounding too early. Rounding intermediate z-scores can slightly change the final probability or percentile. Keep a few extra decimal places until the final answer.

When to Use Normal Distribution Calculations

Use normal-distribution calculations when:

A variable is reasonably modeled by a normal curve.
You know or are assuming a mean and positive standard deviation.
You need a probability below, above, between, or outside selected values.
You need to convert a raw value into a standardized z-score.
You need to convert a percentile into a raw cutoff value.
You are working with a statistical method that uses normal or approximately normal behavior.

Normal calculations are especially common in introductory statistics, standardized scoring, measurement analysis, process control, probability modeling, and normal-approximation methods.

Limitations and Things to Keep in Mind

Normal-distribution calculations depend on the assumption that the variable follows a normal distribution with the chosen mean and standard deviation. If the model is wrong, the probability may be misleading even when the arithmetic is correct.

The standard deviation must be greater than $0$. A standard deviation of zero would mean there is no spread, so the normal distribution formula is not valid.

Percentiles must be strictly between $0\%$ and $100\%$. The theoretical normal distribution extends forever in both directions, so the exact $0$th and $100$th percentiles are not finite values.

The normal curve extends from negative infinity to positive infinity. A graph may show only a practical window around the mean, but very small tail areas still exist beyond that window.

Probability and inverse-percentile values are numerical approximations. Results are usually rounded for display, so small differences can occur between tables, calculators, and software.

The normal model does not perform unit conversion. All raw values should be entered using one consistent unit, and the result should be interpreted in that same unit.

For high-stakes decisions involving money, health, safety, engineering, legal requirements, official records, or research conclusions, check the model assumptions and consult an appropriately qualified professional.

How to Use This Calculator

Enter the mean of the normal distribution.
Enter a standard deviation greater than $0$.
Choose the task: left-tail probability, right-tail probability, between probability, outside probability, z-score, or percentile.
Enter the value required by the selected task:

Use one $x$ value for a left-tail or right-tail probability.
Use one raw value for a z-score.
Use two endpoints for a between or outside probability.
Use a percentile as a percent, such as $95$ for the $95$th percentile.

Review the main result, summary values, formula steps, and shaded normal-curve graph.
If example buttons or a graph download option are available, use them to test typical inputs or save the visual.

For range calculations, the smaller endpoint is treated as the lower value and the larger endpoint as the upper value.

Frequently Asked Questions

What does a z-score tell me?

A z-score tells how many standard deviations a value is from the mean. A z-score of $2$ means the value is two standard deviations above the mean, while a z-score of $-2$ means it is two standard deviations below the mean.

What is the difference between a percentile and a probability?

A probability is an area under the curve, often shown as a percent. A percentile is a cutoff value: the $p$th percentile is the value with $p\%$ of the distribution at or below it.

Why are normal probabilities areas under a curve?

The normal curve is a probability density function. For continuous variables, probability comes from the area under the density curve across an interval, not from the height at one exact point.

Is $P(X\le x)$ different from $P(X<x)$?

For a continuous normal distribution, they are the same. The probability of being exactly equal to one specific value is $0$, so including or excluding the endpoint does not change the probability.

Why does the standard deviation have to be positive?

The standard deviation controls the spread of the normal curve. If it were $0$ or negative, the normal distribution formula would not describe a valid probability distribution.

Can I use this for data that are not normally distributed?

Only with caution. If the data are strongly skewed, have outliers, have multiple clusters, or are limited by hard boundaries, a normal model may give poor probability estimates.

Why are percentile inputs entered as numbers like $95$ instead of $0.95$?

A percentile is commonly written on a $0$ to $100$ scale. When a tool asks for a percentile percent, $95$ means the $95$th percentile; $0.95$ would mean $0.95\%$.

Sources and References

Books

Barbara Illowsky and Susan Dean. Introductory Statistics 2e. OpenStax, 2023. Chapter 6, Sections 6.1 “The Standard Normal Distribution” and 6.2 “Using the Normal Distribution.” Section 6.1 and Section 6.2.
David M. Diez, Mine Çetinkaya-Rundel, and Christopher D. Barr. OpenIntro Statistics. 4th ed., OpenIntro, 2019. Chapter 3.1 “Normal Distribution” and normal probability table material. OpenIntro book page and Chapter 3.1 via Statistics LibreTexts.

Online and Official Sources

National Institute of Standards and Technology. “Normal Distribution.” NIST/SEMATECH e-Handbook of Statistical Methods. Accessed July 4, 2026. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
Penn State Eberly College of Science. “Lesson 3: Probability Distributions.” STAT 500 Applied Statistics. Accessed July 4, 2026. https://online.stat.psu.edu/stat500/Lesson03
OpenIntro. “TI-83/84 Guide for Introductory Statistics.” OpenIntro, 2019. Distribution of random variables section on normal CDF and inverse normal calculations. Accessed July 4, 2026. https://openintro.org/go/?id=ti8384_instructions