STEM Calculators

The Normal Distribution

Statistics • Continuous Random Variables and the Normal Distribution

Written by STEM Calculators Team Published Updated
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If you provide grouped data, the calculator will estimate μ and σ using class midpoints, and will also draw a density histogram (area = 1).

Center of the normal curve; the curve is symmetric about μ.

Controls the spread: smaller σ → taller/narrower curve; larger σ → wider/flatter curve.

If you choose “from data”, μ and σ are computed from the grouped classes (midpoint approximation).

For a continuous distribution, endpoints do not change the probability: \(P(x_1 \le X \le x_2)=P(x_1 < X < x_2)\).

Lower bound (or the single cutoff value for ≤ / ≥ queries).

Upper bound (used for “between” or “outside” queries).

Ready
Enter μ and σ (or paste CSV to estimate them), choose a probability query, then click “Calculate”.

Visualization

The shaded region represents your probability. If you provide grouped data, the bars show density (relative frequency divided by class width) so total bar area is 1.

Family of normal curves

A normal distribution is determined by two parameters: μ (center) and σ (spread). Use the toggles below to see how changing μ or σ changes the curve.

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Frequently Asked Questions

How do I find P(x1 <= X <= x2) for a normal distribution with this calculator?

Choose the between option, enter x1 and x2, and provide mu and sigma (or estimate them from grouped data). The calculator returns the area under the normal curve between the two values.

What is the difference between choosing P(X <= x1) and P(X >= x1)?

P(X <= x1) is the left-tail area up to x1, while P(X >= x1) is the right-tail area beyond x1. The calculator shades the corresponding tail under the curve.

Can this tool estimate mu and sigma from grouped frequency data?

Yes. Paste or upload grouped class data and select frequency or relative frequency, then choose the option to estimate mu and sigma from data using class midpoints.

Why does the histogram use density instead of raw relative frequency?

Density accounts for class width so the total bar area equals 1, matching the idea of probability as area. Density is computed as r_i divided by w_i, where r_i is relative frequency and w_i is class width.

Do inclusive endpoints matter for a normal probability like P(x1 <= X <= x2)?

No. For a continuous normal distribution, the probability at a single point is 0, so including or excluding endpoints does not change the interval probability.

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