Calculate the cumulative probability for a normal distribution quickly and accurately. Enter the mean, standard deviation, and X value to find the probability that a value falls below X.
Enter your values below to calculate the normal cumulative distribution function (CDF) and get the probability instantly.
What Is a Normal CDF?
A normal CDF (cumulative distribution function) gives the probability that a normally distributed variable is less than or equal to a specific value. It is written as:
In simple terms, the normal CDF tells you how much probability lies to the left of a given X value on a normal distribution curve.
To calculate a normal CDF, you need three key values:
Mean (μ\mu) — The center or average of the normal distribution.
Standard deviation (σ\sigma) — Measures how spread out the values are around the mean.
X value — The specific value for which you want to find the cumulative probability.
For example, if the CDF at X=70X=70 is 0.84, it means there is an 84% probability that a randomly selected value is 70 or less.
Normal CDF Calculator
Calculate normal probabilities, Z-scores, percentiles, and inverse CDF values.
Probability Visualization
Shaded area represents the probabilityShow Calculation
How to Use the Normal CDF Calculator
Using the Normal CDF Calculator takes just a few steps. Enter the distribution details, select the probability type, and calculate your result.
Step 1 — Enter the Mean
Enter the mean (μ\mu) of your normal distribution. The mean represents the center of the distribution.
Step 2 — Enter the Standard Deviation
Enter the standard deviation (σ\sigma) of the distribution. Make sure you use the standard deviation, not the variance.
Warning: Do not enter variance when the calculator asks for standard deviation. If you have variance, take its square root first to get the standard deviation.
Step 3 — Choose the Probability Type
Select the type of probability you want to calculate:
Below — Finds the probability of being less than or equal to X.
Above — Finds the probability of being greater than or equal to X.
Between — Finds the probability of falling between two X values.
Step 4 — Enter Your X Value
Enter the relevant X value for your selected probability type. For Between, enter both the lower and upper bounds.
Step 5 — Calculate
Click Calculate Normal CDF to get your result. The calculator shows the cumulative probability based on your mean, standard deviation, and selected X value or range.
The result is typically shown as a decimal probability and can also be interpreted as a percentage. For example, a result of 0.85 means an 85% probability.
Normal CDF Calculator Example
Suppose test scores are normally distributed with a mean of 75 and a standard deviation of 10. You want to find the probability that a randomly selected test score is 85 or lower.
We need to calculate:
Enter These Values
- Mean ((\mu)) = 75
- Standard deviation ((\sigma)) = 10
- X value = 85
- Mode = Below X
After entering these values, click Calculate Normal CDF.
Find the Z-Score
First, convert the X value to a z-score:
A z-score of 1 means the score of 85 is one standard deviation above the mean.
For z = 1, the standard normal cumulative probability is approximately:
Result
The Normal CDF Calculator gives a cumulative probability of approximately 0.8413, or 84.13%.
This means there is about an 84.13% probability that a randomly selected test score is 85 or lower.
Calculate Probability Above a Value
The Normal CDF Calculator can also find the probability that a value is greater than or equal to a specific X value. This is called the right-tail probability.
For example, to find the probability of a test score of 85 or higher, calculate:
The calculator first finds the cumulative probability below 85. It then determines the remaining probability to the right of 85.
The formula is:
So, if F(85) = 0.8413:
Calculate Probability Between Two Values
You can also use the calculator to find the probability that a normally distributed value falls between two X values.
For example:
The calculator finds the cumulative probability at both endpoints and subtracts the lower probability from the upper probability.
The formula is:
For a normal distribution with μ = 75 and σ = 10, you would enter 70 as the lower value and 90 as the upper value, then select the Between option.
The result represents the proportion of values that fall between 70 and 90. You can interpret the decimal result as a percentage by multiplying it by 100.
Inverse Normal CDF Calculator
A normal CDF works from an X value to a probability. An inverse normal CDF works in the opposite direction. It starts with a probability and finds the corresponding X value.
What Is Inverse Normal?
The basic relationship is:
Normal CDF:
This is useful when you know a percentile or cumulative probability and need to find the corresponding value in a normal distribution.
Inverse Normal Calculator Mode
Select Inverse Normal in the calculator, then enter:
- Probability — The cumulative probability or percentile you want to use.
- Mean ((\mu)) — The center of the normal distribution.
- Standard deviation ((\sigma)) — The spread of the distribution.
The calculator returns the corresponding X value, along with its Z-score and percentile.
Example: Find the Score at the 95th Percentile
Suppose test scores have a mean of 75 and a standard deviation of 10. To find the score at the 95th percentile, enter:
- Probability = 0.95
- Mean = 75
- Standard deviation = 10
The 95th percentile corresponds to a Z-score of approximately:
Convert the Z-score to the original score:
This means about 95% of scores are at or below 91.45, assuming the scores follow the specified normal distribution.
Normal CDF vs. PDF
Normal CDF and normal PDF are related, but they answer different questions.
| Normal CDF | Normal PDF |
|---|---|
| Gives cumulative probability | Gives probability density |
| Represents the area to the left of a value | Represents the height of the curve |
| Used to find probabilities and percentiles | Used to describe the shape of a distribution |
| Returns a cumulative area | Returns a density value |
The normal CDF tells you how much probability has accumulated up to a specific value. The normal PDF describes the height of the normal distribution curve at that value.
If you also need to calculate probability density, use our Normal PDF Calculator.
Normal CDF vs. Z-Score
A Z-score and a normal CDF are connected, but they do not represent the same thing.
A Z-score tells you how many standard deviations a value is from the mean:
For example, a Z-score of 1.00 means the value is one standard deviation above the mean. A Z-score itself is not a probability.
The normal CDF takes that position on the distribution and converts it into a cumulative probability. For example, a Z-score of 1.00 corresponds to a CDF of approximately 0.8413, meaning about 84.13% of values are at or below that point.
Normal CDF Formula
The general formula for a normal cumulative distribution function is:
To standardize a value, use:
Here, x is the value of interest, μ is the mean, and σ is the standard deviation.
The standardization step converts the value to a Z-score. The Normal CDF Calculator then handles the numerical CDF calculation automatically, so you do not need to perform a lengthy mathematical calculation by hand.
How to Calculate Normal CDF on a TI-84
You can calculate a normal CDF directly on a TI-84 calculator using the normalcdf( function. This is useful for checking homework answers and solving normal distribution probability problems.
Use the normalcdf( Function
On the TI-84, press:
2nd → VARS → normalcdf(
The calculator uses this format:
For a left-tail probability such as:
with μ = 75 and σ = 10, enter:
The result is approximately:
The value -1E99 is an extremely small number that serves as a practical
approximation of negative infinity for a left-tail calculation.
For a detailed walkthrough, see How to Calculate Normal CDF: Formula, Calculator & Examples.
When Should You Use a Normal CDF Calculator?
A Normal CDF Calculator is useful whenever you need to find a cumulative probability from a normal distribution. Students commonly use it for:
- Finding probabilities below, above, or between values
- Analyzing exam and test scores
- Working with normally distributed heights
- Finding percentiles
- Solving sampling and probability problems
- Answering AP Statistics questions
- Completing college statistics homework
- Solving normal distribution probability questions
It can save time and reduce calculation errors when you already know the distribution’s mean and standard deviation.
Common Normal CDF Calculator Mistakes
Small input errors can produce the wrong probability. Check these common mistakes before accepting your result.
Entering Variance Instead of Standard Deviation
A normal CDF calculator requires the standard deviation, not the variance. If you only have the variance, take its square root first:
For example, a variance of 100 gives a standard deviation of 10.
Using the Wrong Tail
Make sure you select the probability type that matches the question.
- Below finds the area to the left of X.
- Above finds the area to the right of X.
- Between finds the area between two values.
Choosing Below when a question asks for the probability above a value can give you the complement of the answer you need.
Mixing Up Mean and X
The mean describes the center of the distribution. The X value is the specific value where you want to calculate the probability.
For example, if the mean is 75 and you want (P(X\le85)), enter 75 as the mean and 85 as X.
Forgetting That Standard Deviation Must Be Positive
Standard deviation cannot be zero or negative for a valid normal distribution. Enter a positive value for (\sigma).
If your standard deviation is zero or negative, check your data or calculation before using the Normal CDF Calculator.
Confusing CDF With PDF
The CDF gives cumulative probability up to a value. The PDF gives the height or density of the normal curve at a value.
If you need a probability, you generally need an area from the distribution rather than simply the PDF height.
Confusing CDF With Inverse CDF
A regular CDF starts with an X value and finds its cumulative probability:
An inverse CDF starts with a probability and finds the corresponding X value:
Use the inverse normal option when a question asks for a score, measurement, or other value at a specific percentile.
Calculate Normal Probabilities With Confidence
The Normal CDF Calculator makes it easy to find probabilities for a normal distribution. You can calculate values below, above, or between selected X values using the mean and standard deviation.
Whether you are working on exam scores, percentiles, AP Statistics questions, or college statistics homework, the calculator can help you check your normal distribution calculations quickly.
Ready to find your probability? Enter your values in the Normal CDF Calculator above and calculate your result now.
Frequently Asked Questions
A normal CDF calculator finds the cumulative probability for a normal distribution. It can calculate the probability below, above, or between selected values.
First, calculate the Z-score using:
Then use the standard normal CDF to find the cumulative probability for that Z-score.
Normal CDF gives the probability that a normally distributed value is less than or equal to a specific value. It represents the area under the normal curve to the left of that value.
Enter the mean, standard deviation, and X value into a calculator that supports normal CDF. Select the appropriate probability type, then calculate the result.
On a TI-84, press 2nd → VARS → normalcdf(. Enter the lower bound, upper bound, mean, and standard deviation, then press ENTER.
The CDF gives cumulative probability up to a value. The PDF gives the probability density or height of the distribution curve at a value.
Inverse normal CDF starts with a cumulative probability and finds the corresponding X value. It is useful for finding percentiles or cutoff scores.
Yes. A normal CDF calculator can find the probability between two values by subtracting the lower cumulative probability from the upper cumulative probability:
You usually need the mean (μ), standard deviation (σ), and the X value or range you want to analyze.