Enter a mean (μ), standard deviation (σ), and a value (x) to calculate the Z-score and probability under the bell curve. Choose your probability type — P(X < x), P(X > x), or P(a < X < b) — and see the corresponding area, cumulative probability, and a visual breakdown of where your score falls on the normal distribution.
Probability (Area)
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Probability (%)
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Z-Score
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PDF f(x)
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68% Range (μ ± 1σ)
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95% Range (μ ± 2σ)
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A bell curve, or normal distribution, is a symmetric, bell-shaped probability distribution centered at the mean (μ). It is defined by two parameters: the mean, which sets the center, and the standard deviation (σ), which controls the spread. Many real-world phenomena — test scores, heights, measurement errors — follow this pattern naturally.
A Z-score measures how many standard deviations a value (x) is from the mean: Z = (x − μ) / σ. A Z-score of 0 means the value equals the mean. Positive Z-scores are above the mean; negative Z-scores are below. Z-scores let you compare values from different distributions on a common scale.
The empirical rule states that in any normal distribution, approximately 68% of values fall within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations. This rule is useful for quickly estimating probabilities without detailed calculation.
P(X < x) is the left-tail probability — the area under the curve to the left of your value x, representing the likelihood of getting a result less than x. P(X > x) is the right-tail probability — the area to the right of x. The two always sum to 1, since every outcome must be either less than or greater than x.
To apply a bell curve to test scores, set the mean (μ) to your class average score and the standard deviation (σ) to the spread of scores. Then enter a specific score as x to find what percentage of students scored below or above it. This helps identify cutoffs for letter grades based on the distribution of class performance.
The PDF value f(x) tells you the relative likelihood of the random variable taking the exact value x. Unlike the cumulative probability, the PDF itself is not a probability — it is a density. To get the probability over a range, you integrate the PDF between two points, which is what the area calculations in this calculator do.
The standard normal distribution has a mean of 0 and a standard deviation of 1, written as Z ~ N(0, 1). A general normal distribution has any mean μ and standard deviation σ, written as X ~ N(μ, σ). You convert any general normal value to a standard normal Z-score using Z = (x − μ) / σ, which is what this calculator does internally.
Inverse normal (or quantile) calculation finds the value x corresponding to a given cumulative probability. For example, to find the 90th percentile score, you enter the desired probability (0.90) and work backwards. The formula is x = μ + Z * σ, where Z is the inverse standard normal for your probability. Many grading curves use this to assign score cutoffs.