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Normal Distribution Calculator

Calculate probabilities, PDFs, CDFs, and Z-scores for any normal distribution.

Parameters

P(X ≤ x)
—
Enter values above
PDF f(x)
—
CDF P(≤x)
—
P(X > x)
—
P within ±1σ
68.27%
P within ±2σ
95.45%
Z-Score
—
Percentile
—
f(x) = e^(−(x−μ)²/2σ²) / (σ√2π) P = Φ((x−μ)/σ)
Empirical Rule
68.27% within ±1σ
95.45% within ±2σ
99.73% within ±3σ
Chart: interpretation.
Interpretation
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Chart: cdf at x.
CDF at x
—
xz-scorePDF f(x)P(X≤x)P(X>x)

Range Probability P(a ≤ X ≤ b)

P(a ≤ X ≤ b)
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Uses μ and σ from Calculator tab
MetricValue
Chart: range bell canvas.
P(μ−nσ to μ+nσ) Probability Matrix

Probabilities for symmetric windows of various widths (columns) at different centers (rows). Current inputs highlighted.

Z-Table & Lookup

P(Z ≤ z) for z from −3.5 to +3.5 in 0.1 steps. Current z from Tab 1 highlighted.

Highlights nearest row in the table below
zP(Z ≤ z)P(Z > z)f(z) PDF
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How to Use This Calculator

1

Enter the Parameters

Provide the mean (μ) and standard deviation (σ) of the normal distribution.

2

Specify the X Value

Enter the value of x for which you want to calculate the probability density or cumulative probability.

3

View PDF and CDF

The calculator displays the probability density at x, the cumulative probability P(X ≤ x), and visual shading on the bell curve.

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Formula & Methodology

PDF
f(x) = (1/(σ√(2π))) × e^(−(x−μ)²/(2σ²))
The probability density function gives the height of the bell curve at any point x.
CDF
F(x) = P(X ≤ x) = ∫ f(t) dt from −∞ to x
The cumulative distribution function gives the area under the curve to the left of x.
Empirical Rule
68–95–99.7%
Approximately 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ.

About this calculation · Accuracy & Methodology

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Key Terms

Normal Distribution A symmetric, bell-shaped probability distribution characterized by its mean and standard deviation.
Mean (μ) The center of the distribution; the value around which the bell curve is symmetric.
Standard Deviation (σ) A measure of spread; determines the width of the bell curve.
PDF (Probability Density Function) The function that describes the relative likelihood of a continuous random variable taking a given value.
CDF (Cumulative Distribution Function) The probability that a random variable takes a value less than or equal to x.
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Real-World Examples

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IQ Distribution

Input: μ = 100, σ = 15, x = 130

Result
P(X ≤ 130) = 0.9772

P(X ≤ 130) = 0.9772 — about 97.7% of people have an IQ at or below 130

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Standard Normal Distribution Key Values

z-ScoreP(Z ≤ z)PercentileContext
−2.00.02282.3rdWell below average
−1.00.158715.9thBelow average
0.00.500050thExactly at the mean
+1.00.841384.1stAbove average
+2.00.977297.7thWell above average
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The Normal Distribution: Nature's Bell Curve

Why the Normal Distribution Is So Common

The Central Limit Theorem states that the sum (or average) of many independent random variables tends toward a normal distribution, regardless of the underlying distribution. This is why heights, test scores, measurement errors, and many biological traits approximate a bell curve. The normal distribution is the most important probability distribution in statistics.

The Standard Normal and Z-Scores

Any normal distribution can be transformed into the standard normal (mean 0, standard deviation 1) by subtracting the mean and dividing by the standard deviation. This produces a z-score that allows comparison across different scales. A z-score of 2.0 means the value is two standard deviations above the mean, regardless of the original measurement units.

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Frequently Asked Questions

What is a normal distribution?+

A normal distribution is a symmetric, bell-shaped probability curve where most values cluster around the mean. Approximately 68 percent of values fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three.

How do I find the probability between two values?+

Enter the mean, standard deviation, and your two boundary values. The calculator computes the area under the normal curve between those points, which represents the probability of a random value falling in that range.

What is the empirical rule (68-95-99.7)?+

The empirical rule states that for normally distributed data, about 68 percent of observations fall within 1 standard deviation of the mean, 95 percent within 2, and 99.7 percent within 3. It provides a quick estimate without detailed calculations.

When is data not normally distributed?+

Data is non-normal when it is heavily skewed, has multiple peaks, or contains extreme outliers. Income distributions, insurance claims, and website traffic typically follow skewed or exponential patterns rather than normal ones.

What is the standard normal distribution?+

The standard normal distribution has a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to standard form using z-scores (z = (x - mean) / standard deviation), which allows use of universal probability tables.