Normal Distribution Calculator for CDF, Z Scores, and Percentiles
Use this free normal distribution calculator to find normal CDF probabilities, left-tail and right-tail probabilities, area under the normal curve, between-area probabilities, z-scores, percentiles, and raw values from a mean and standard deviation. Set mean 0 and standard deviation 1 to use it as a standard normal distribution calculator, z score normal calculator, or z-table check for Excel NORM.DIST, NORM.S.DIST, NORM.INV, and NORM.S.INV homework.
- Single value probability Use this as a normal probability calculator with a mean, standard deviation, and x value.
- Range probability Find the probability that a normally distributed variable falls between two values on the bell curve.
- Percentile lookup Convert a cumulative probability into the matching z-score, raw x value, and central interval.
- Bell curve visual Use the shaded curve to understand whether the answer is a left tail, right tail, or middle area.
- Standard normal mode Set mean 0 and standard deviation 1 to use the calculator like a z-table, z score normal calculator, or standard normal lookup.
Calculate Normal Distribution Probability
Choose whether you need a single cutoff probability, a probability between two values, or the raw score for a percentile. Enter the mean and standard deviation from your problem; use mean 0 and standard deviation 1 for a standard normal distribution calculator. The result includes z-score conversion, normal probability, formula steps, and a shaded bell curve.
Inputs
Results
For a standard normal distribution, x = 1.96 is about the 97.5th percentile.
The shaded area shows P(X < 1.96).
| Standardize | z = (1.96 - 0) / 1 = 1.96 |
|---|---|
| Probability | P(X < 1.96) = Phi(1.96) = 0.9750 |
| Interpretation | About 97.50% of values are below 1.96. |
Quick Answers for Normal Distribution Searches
Most normal distribution calculator questions come down to choosing the right input type: a cutoff value, an interval, or a percentile. The calculator shows the z-score step and the probability step together.
Normal probability calculator
Enter the mean, standard deviation, and x value to calculate P(X < x), P(X > x), the z-score, and the normal curve density at that point.
Area under normal curve calculator
Use single-value mode for a left or right tail, or between-values mode for the middle area between two cutoffs on the bell curve.
Standard normal distribution calculator
Set mean to 0 and standard deviation to 1. Then enter a z-score as the x value to check standard normal table probabilities.
How to calculate percentile from z score
Use mean 0, standard deviation 1, and enter the z-score as x. The left-tail result P(X < x) is the percentile as a decimal.
Normal Distribution Formula
The calculator standardizes each value first, then uses the standard normal cumulative distribution function.
| Z-score | z = (x - mean) / standard deviation |
|---|---|
| Left tail | P(X < x) = Phi(z) |
| Right tail | P(X > x) = 1 - Phi(z) |
| Between two values | P(a < X < b) = Phi(z_b) - Phi(z_a) |
| Percentile | x = mean + z_p * standard deviation, where z_p is the z-score for the cumulative probability p |
Common Mistakes
- Using a sample standard deviation without explaining that the result is an approximation for the population distribution.
- Forgetting that the standard deviation must be positive.
- Reading P(X < x) as P(X > x), especially in upper-tail homework problems.
- Using a normal model for data that is strongly skewed, bounded, or full of outliers without checking assumptions.
How to Use This Normal CDF Calculator
Normal distribution homework usually becomes easier once you separate the problem into standardizing, choosing the tail, and interpreting the probability. This normal CDF calculator shows each step so you can compare the answer with a z-table, Excel, Google Sheets, SPSS, or R output.
- Identify the raw value x, the mean, and the standard deviation from the question.
- Convert the raw value into a z-score using z = (x - mean) / standard deviation.
- Use Phi(z) for a left-tail probability such as P(X < x).
- Use 1 - Phi(z) for a right-tail probability such as P(X > x).
- Use Phi(z_upper) - Phi(z_lower) for a between-area probability.
- Convert the decimal answer into a percentage and explain it in the context of the problem.
Normal Distribution Examples
These benchmark examples are useful for checking z-table answers, Excel-style normal distribution output, and calculator results.
| Problem | Inputs | Result | Interpretation |
|---|---|---|---|
| Standard normal left tail | mean = 0, sd = 1, x = 1.96 | P(X < 1.96) = 0.9750 | About 97.50% of values are below 1.96. |
| Standard normal right tail | mean = 0, sd = 1, x = 1.96 | P(X > 1.96) = 0.0250 | About 2.50% of values are above 1.96. |
| Between two z-scores | mean = 0, sd = 1, lower = -1, upper = 1 | P(-1 < X < 1) = 0.6827 | About 68.27% of values fall within one standard deviation of the mean. |
| 95th percentile | mean = 0, sd = 1, cumulative probability = 0.95 | z = 1.6449 | The 95th percentile is about 1.645 standard deviations above the mean. |
When This Calculator Helps
Homework and exams
Use it when your problem asks for values below, above, or between cutoffs in a normally distributed variable. It is also helpful for checking z-table and Excel normal curve answers.
Statistics projects
Use it when you already have a reasonable normal model and need quick probabilities for scores, measurements, forecasts, quality-control cutoffs, or sampling distributions.
Normal Distribution FAQs
What does the normal distribution calculator find?
It finds probabilities under a normal curve using the mean, standard deviation, and one or two x values. It also converts percentiles into z-scores and raw values.
Can this be used as a normal probability calculator?
Yes. Enter the mean, standard deviation, and cutoff value to calculate left-tail probability, right-tail probability, density, and the matching z-score.
What is a z-score?
A z-score tells how many standard deviations an x value is above or below the mean. The formula is z = (x - mean) / standard deviation.
When should I use a normal distribution?
Use it when the variable is continuous, roughly bell-shaped, and reasonably symmetric, or when a sampling distribution is approximately normal by the central limit theorem.
Is the normal distribution the same as a z table?
A z table is a printed lookup table for the standard normal distribution. This calculator does the same type of lookup directly.
How do I use this as a standard normal distribution calculator?
Set the mean to 0 and the standard deviation to 1. Then enter your z-score as the x value.
How do I find the area under the normal curve?
Choose a left-tail, right-tail, or between-values calculation. The calculator standardizes each x value into a z-score and then finds the matching area under the bell curve.
How do I find a normal percentile?
Use the percentile mode and enter the cumulative probability as a decimal, such as 0.95 for the 95th percentile. The calculator returns the z-score and raw x value.
How do I calculate percentile from a z-score?
Use mean 0 and standard deviation 1, enter the z-score as the x value, and read P(X < x). That left-tail probability is the percentile as a decimal.
How do I find probability between two normal values?
Use between-values mode, enter the lower and upper cutoffs, and the calculator subtracts the lower cumulative probability from the upper cumulative probability.
Does this calculator match Excel normal distribution functions?
It uses the same normal distribution ideas used by Excel-style NORM.DIST, NORM.S.DIST, NORM.INV, and NORM.S.INV work: standardize x into z, find the cumulative probability, then convert the result as needed. Tiny differences can appear from rounding.
What is the difference between normal and standard normal distribution?
A normal distribution can have any mean and positive standard deviation. The standard normal distribution is the special case with mean 0 and standard deviation 1.
Related Statistics Resources
If your assignment moves from normal probabilities into tests, confidence intervals, or interpretation, these pages are useful next steps.
Need Help Interpreting a Normal Distribution Problem?
Statskan can help with z-scores, normal CDF questions, confidence intervals, hypothesis testing, regression output, Excel results, SPSS output, and full statistics homework interpretation.