Distributions

Bell Curve Explorer

Set the mean and standard deviation of a normal distribution, shade any area under the curve, and read off the probability and z-scores.

Distribution

Mean (μ)

Standard deviation (σ)

Slide the standard deviation (1 to 100)

Find the probability that X is…

From

Show 1, 2 and 3 standard deviation marks

Normal curve

Probability

0.6827

Percentage

68.27%

z-scores

-1 to 1

P(85 < X < 115) = Φ(1) − Φ(-1) = 0.6827 where z = (x − 100) ÷ 15.

Within	Range of X	Probability
±1σ	85 to 115	68.27%
±2σ	70 to 130	95.45%
±3σ	55 to 145	99.73%

How to use this tool

Enter the mean and standard deviation, or slide the standard deviation to see the curve get narrower or wider.
Choose whether you want the probability of being between two values, below a value, or above a value.
The shaded area on the curve is the probability, and the working is shown underneath.
The table shows the share of values within 1, 2, and 3 standard deviations of the mean.

Formula Reference

z-score

z = (x − μ) ÷ σ

Probability between a and b

P(a < X < b) = Φ(z_b) − Φ(z_a)

Probability above a

P(X > a) = 1 − Φ(z_a)

Empirical rule

68% · 95% · 99.7% within 1 · 2 · 3σ

ArticleThe Normal Distribution ArticleZ-Scores Explained CalcNormal Distribution Calculator CalcZ-Score Calculator GlossaryGlossary: Empirical Rule

Frequently Asked Questions

What does the shaded area mean?+

For a continuous distribution, probability is area under the curve. The shaded region is the chance that a randomly chosen value falls in the range you selected. The total area under the whole curve is 1.

What happens when I make the standard deviation bigger?+

The curve gets wider and flatter, because the values are more spread out. The total area stays 1, so a wider curve must be lower. The shape stays symmetric around the mean.

Why is the probability of one exact value zero?+

A single point has no width, so it has no area. That is why the tool asks for a range. For a continuous variable, "below 100" and "at most 100" have the same probability.

How accurate are the probabilities?+

The standard normal CDF is computed with a standard numerical approximation accurate to about seven decimal places, more than enough for the four decimals shown.

Within

Range of X

Probability

±1σ

85 to 115

68.27%

±2σ

70 to 130

95.45%

±3σ

55 to 145

99.73%

How to use this tool

Enter the mean and standard deviation, or slide the standard deviation to see the curve get narrower or wider.

Choose whether you want the probability of being between two values, below a value, or above a value.

The shaded area on the curve is the probability, and the working is shown underneath.

The table shows the share of values within 1, 2, and 3 standard deviations of the mean.