How do I find the probability between two values in a normal distribution?

Convert both values to z scores using z = (x − μ) ÷ σ. Then find the cumulative area below each z score and subtract: Φ(z2) − Φ(z1). The result is the proportion of the normal curve between the two raw values.

What does area under the normal curve mean?

The area under the normal curve represents probability. For example, if the area between two values is 0.68, that means about 68% of observations are expected to fall in that interval under the normal model.

Do I need a z table to use this calculator?

No. The calculator estimates the standard normal cumulative area internally. However, the steps match what you would do with a z table: standardize the raw values, find the left-tail areas, and subtract.

Can the lower and upper values be below the mean?

Yes. Both values can be below the mean, both can be above the mean, or one can be on each side. The z scores will handle the direction automatically. Negative z scores simply mean the raw value is below the mean.

Is this the same as the empirical rule?

No. The empirical rule gives quick approximations for one, two, and three standard deviations from the mean. A normal distribution range calculation is more flexible because it can find the probability between any two values, not only the common 68–95–99.7 intervals.

Education

Normal Distribution Range Calculator

Find the probability that a normally distributed variable falls between two values. The calculator converts raw values to z scores and estimates the normal area between them.

What a normal distribution range calculator is doing

A normal distribution range problem usually asks for the probability that a normally distributed variable falls between two values. The calculator first converts each raw value to a z score, then uses the standard normal curve to estimate the area between those z scores. OpenStax covers this workflow in its lesson on the standard normal distribution.

This page is useful for searches such as “normal distribution between two values,” “normal probability range,” and “find probability between mean and standard deviation values.” The goal is not just to show a percentage, but to show how the raw values turn into z scores.

Normal range formula

z₁ = (lower value − μ) ÷ σ z₂ = (upper value − μ) ÷ σ P(lower < X < upper) = Φ(z₂) − Φ(z₁)

Here, Φ means the cumulative area under the standard normal curve to the left of a z score. In a classroom, that area may come from a z table, software, or a calculator. This page uses a standard approximation so students can check their work quickly.

When the normal model makes sense

The normal distribution is appropriate when the problem states that the variable is normally distributed or when the data are reasonably bell-shaped and symmetric. It should not be assumed automatically for every data set. If the data are strongly skewed, bounded in a way that affects the curve, or full of outliers, a normal probability may not describe the real situation well.

Frequently asked questions

Convert both values to z scores using z = (x − μ) ÷ σ. Then find the cumulative area below each z score and subtract: Φ(z2) − Φ(z1). The result is the proportion of the normal curve between the two raw values.
The area under the normal curve represents probability. For example, if the area between two values is 0.68, that means about 68% of observations are expected to fall in that interval under the normal model.
No. The calculator estimates the standard normal cumulative area internally. However, the steps match what you would do with a z table: standardize the raw values, find the left-tail areas, and subtract.
Yes. Both values can be below the mean, both can be above the mean, or one can be on each side. The z scores will handle the direction automatically. Negative z scores simply mean the raw value is below the mean.
No. The empirical rule gives quick approximations for one, two, and three standard deviations from the mean. A normal distribution range calculation is more flexible because it can find the probability between any two values, not only the common 68–95–99.7 intervals.