Enter a mean (μ), standard deviation (σ), and an x value (or z-score) to calculate probabilities under the normal distribution bell curve. Choose your probability type — left-tail P(X < x), right-tail P(X > x), or two-tailed — and get back the cumulative probability, z-score, and a visual breakdown of the shaded area.
Probability
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Z-Score
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Percentage
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PDF f(x)
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The normal distribution (bell curve) appears naturally in many real-world phenomena — heights, test scores, measurement errors, and more. It's central to statistics because the Central Limit Theorem states that the averages of large samples tend to be normally distributed, regardless of the original population's shape. This makes it the foundation of many statistical tests and confidence intervals.
A standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to the standard normal by computing z-scores. Standard normal tables and calculators use this as their reference distribution.
A z-score measures how many standard deviations a data point is from the mean. It is calculated as z = (x − μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation. A positive z-score means the value is above the mean; a negative z-score means it is below.
A raw score is the original, unstandardized measurement (e.g. a test score of 85). A z-score is that same value expressed in terms of standard deviations from the mean, making it comparable across different distributions. This calculator converts raw scores to z-scores automatically using the mean and standard deviation you provide.
Cumulative probability P(X < x) is the total probability that the random variable X takes a value less than or equal to x. It equals the area under the normal curve to the left of that point. A cumulative probability of 0.84 means 84% of values in the distribution fall below x.
To find the probability between two symmetric values (±z), use the 'Between ±z' option. This calculates P(−|z| < X < |z|), the area under the curve between the two mirrored z-scores. For non-symmetric ranges, compute P(X < upper) − P(X < lower) using two separate left-tail calculations.
The total area under the normal distribution curve always equals 1 (or 100%). This represents the certainty that any outcome must fall somewhere within the distribution. The 68-95-99.7 rule tells us that about 68% of values fall within 1 standard deviation, 95% within 2, and 99.7% within 3.
This calculator goes from x value → probability. If you want the reverse (given a percentile, find x), you would need the inverse normal (quantile) function. The z-score output here can help: for example, a z-score of 1.645 corresponds to the 95th percentile in a standard normal distribution.