Normal Distribution Calculator
Enter a value, mean, and standard deviation to find the z-score and the height of the normal curve at that point. Visualize where a value sits on the bell curve.
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Formula
z = (x−μ)/σ | f(x) = (1/(σ√(2π))) × e^(−½z²)
The z-score standardizes x by subtracting the mean and dividing by the standard deviation. The probability density function (PDF) gives the height of the normal curve at x. The PDF value is not a probability — it is a density. The probability of x falling in a small interval is the PDF value times the interval width. The PDF is highest at the mean and decreases symmetrically in both directions.
How to use the Normal Distribution Calculator
- 1
Enter your value (x)
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Enter your mean (μ)
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Enter your standard deviation (σ)
- 4
Read your results instantly
Results update in real time as you type.
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The normal distribution and its properties
The normal distribution (Gaussian distribution) is the most important probability distribution in statistics. It is bell-shaped, symmetric around the mean, and completely characterized by just two parameters: the mean (μ) and standard deviation (σ). The mean determines the center and the standard deviation determines the width.
The normal distribution arises naturally in many real-world situations: measurement errors, height and weight distributions, IQ scores, and countless other phenomena. By the central limit theorem, the sum of many independent random variables tends toward a normal distribution regardless of their individual distributions.
PDF vs. CDF: density and cumulative probability
The PDF (probability density function) gives the height of the curve at any point x. It answers: how densely packed are values near x? The CDF (cumulative distribution function) gives the probability of observing a value ≤ x. It answers: what fraction of values fall below x?
This calculator computes the PDF. For the standard normal (μ=0, σ=1) at x=0, the PDF value is 1/√(2π) ≈ 0.3989 — the peak of the curve. For CDF values (percentiles), use the Z-Score Calculator on this site, which provides an approximate percentile.
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The standard normal distribution
The standard normal distribution has μ=0 and σ=1. Any normal distribution can be converted to the standard normal by computing z = (x−μ)/σ. This standardization allows probability tables (z-tables) to cover all normal distributions with a single set of values.
For the standard normal, about 68% of values lie within z = ±1, about 95% within z = ±1.96, and about 99.7% within z = ±3. These are the most commonly used critical values in statistics. The default inputs in this calculator (x=1.5, μ=0, σ=1) give z=1.5, placing x in the top 6.7% of the distribution.
Tips & Insights
PDF value is not a probability
The PDF f(x) is a density, not a probability. For continuous distributions, P(X = x) = 0 exactly. Probabilities come from integrating the PDF over an interval. PDF values can exceed 1 for narrow distributions.
Shift and scale to any normal
To work with any normal distribution, convert to z = (x−μ)/σ and use standard normal tables. All the properties of the standard normal apply after this transformation.
The normal is an approximation to many distributions
For large samples, many non-normal distributions (binomial, Poisson, chi-square) are well-approximated by a normal distribution. This is the central limit theorem at work.
Worked Examples
IQ score interpretation
Z-score: +1.00. This IQ is exactly one standard deviation above the mean, corresponding to approximately the 84th percentile.
Manufacturing tolerance check
Z-score: +3.00. This part is three standard deviations above the mean — in the extreme tail. Only about 0.13% of parts would be this far above specification.
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Frequently Asked Questions
What is the normal distribution?
The normal distribution is a symmetric, bell-shaped probability distribution defined by its mean and standard deviation. It is the most widely used distribution in statistics and appears naturally in many physical and social phenomena.
What is the PDF of a normal distribution?
The probability density function (PDF) gives the height of the bell curve at any point. It is highest at the mean and decreases symmetrically. The PDF value itself is not a probability — it is a density.
What is the standard normal distribution?
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be standardized to it using z = (x−μ)/σ, enabling use of standard z-tables.
What is the central limit theorem?
The central limit theorem states that the sampling distribution of the mean of n independent observations approaches a normal distribution as n increases, regardless of the original distribution. This is why the normal distribution is so fundamental.
How do I find the probability that X falls below a value?
You need the CDF, not the PDF. For the standard normal, use a z-table or the Z-Score Calculator on this site, which provides an approximate cumulative percentile given your z-score.
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