Why is the normal distribution called "normal"?

The term reflects how frequently this distribution appeared in 19th century science — it was considered the "normal" or standard form of error in measurements and observations. Carl Friedrich Gauss systematized its properties in 1809, which is why it's also called the Gaussian distribution.

What is the difference between normal distribution and standard normal distribution?

Any normal distribution is defined by its mean μ and standard deviation σ. The standard normal distribution is the special case with μ=0 and σ=1, using Z as the variable. Any normal distribution can be converted to standard normal by computing Z = (x−μ)/σ.

Does the normal distribution apply to small samples?

For the raw data, it may or may not be appropriate — you should check. However, by the Central Limit Theorem, the distribution of sample means from any population becomes approximately normal for samples of n ≥ 30, enabling standard confidence intervals and hypothesis tests.

What is a lognormal distribution and when should I use it?

If the logarithm of your variable is normally distributed, the variable itself is lognormal. Income, stock prices, and biological concentrations often follow lognormal distributions. Use it when your data is positive and right-skewed. Taking logarithms often normalizes such data for analysis.

Normal Distribution Calculator

Parameters

Quick Presets

Mean (μ)

Standard Deviation (σ)

Value (x)

P(X ≤ x)

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Enter values above

PDF f(x)

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CDF P(≤x)

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P(X > x)

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P within ±1σ

68.27%

P within ±2σ

95.45%

Z-Score

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Percentile

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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

Interpretation

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CDF at x

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x	z-score	PDF f(x)	P(X≤x)	P(X>x)

Range Probability P(a ≤ X ≤ b)

Lower bound (a)

Upper bound (b)

P(a ≤ X ≤ b)

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Uses μ and σ from Calculator tab

Metric	Value

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.

Quick Z Lookup

Highlights nearest row in the table below

z	P(Z ≤ z)	P(Z > z)	f(z) PDF

How to Use This Calculator

Enter the Parameters

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

Specify the X Value

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

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.

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

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.

Real-World Examples

Example 1

IQ Distribution

μ = 100, σ = 15, x = 130

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

Example 2

Manufacturing Tolerance

μ = 50 mm, σ = 0.5 mm, x = 51

P(X ≤ 51) = 0.9772 — 97.7% of parts are within this tolerance

Standard Normal Distribution Key Values

z-Score	P(Z ≤ z)	Percentile	Context
−2.0	0.0228	2.3rd	Well below average
−1.0	0.1587	15.9th	Below average
0.0	0.5000	50th	Exactly at the mean
+1.0	0.8413	84.1st	Above average
+2.0	0.9772	97.7th	Well above average

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.

Normal Distribution Calculator

Parameters

Range Probability P(a ≤ X ≤ b)

Z-Table & Lookup

How to Use This Calculator

Enter the Parameters

Specify the X Value

View PDF and CDF

Formula & Methodology

PDF

CDF

Empirical Rule

Key Terms

Real-World Examples

Standard Normal Distribution Key Values

The Normal Distribution: Nature's Bell Curve

Why the Normal Distribution Is So Common

The Standard Normal and Z-Scores

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