# InvNorm calculator: x from an area

Finds the value x with a given area under a normal curve to its left, to its right, or in the center (invNorm), with the z-score and a chart of the shaded area.

- Page: https://www.acalculator.org/statistics/inverse-normal-calculator
- JSON spec: https://www.acalculator.org/statistics/inverse-normal-calculator.json
- Version: aee0535df0b2

## Default answer

Example with the default inputs (Area (probability) 0.95, Area is Left of x, Mean (μ) 0, Standard deviation (σ) 1): An area of 0.95 under this normal curve lies below 1.644853627.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| p | Area (probability) | The area under the normal curve, more than 0 and less than 1, such as 0.95. |
| tail | Area is | Where the area lies: to the left of x, to the right of x, or in the center around the mean. |
| mean | Mean (μ) | The mean of the normal distribution; 0 for the standard normal. |
| sd | Standard deviation (σ) | The standard deviation, more than 0; 1 for the standard normal. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| x | x | The value with the area to its left or right; for a center area, the upper end μ + zσ. |
| z | z-score | How many standard deviations x is from the mean: (x − μ) ÷ σ. |
| lower | Lower end | For a center area: μ − zσ, the low end of the middle area. |
| upper | Upper end | For a center area: μ + zσ, the high end of the middle area. |
| region | The area lies | Below x, above x, or between the two ends. |

## Method

x = μ + σz. Left: Φ(z) = area. Right: 1 − Φ(z) = area. Center: Φ(z) − Φ(−z) = area, with ends μ ± σz. Φ is the standard normal CDF.

## Assumptions

- The values follow a normal distribution with the mean and standard deviation you type.
- The area is a probability, more than 0 and less than 1 (at least 10⁻³⁰⁰).
- Left and right match invNorm(area, μ, σ, LEFT) and RIGHT; center splits the rest of the area evenly between the two tails.

## Worked examples

1. p = 0.95, tail = left, mean = 0, sd = 1 gives x = 1.644854, z = 1.644854. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area from 0 to z: 0.44950 at z = 1.64, 0.45053 at 1.65, 0.47500 at 1.96). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (1.6449 lies between 1.64 and 1.65); NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
2. p = 0.95, tail = center, mean = 0, sd = 1 gives lower = -1.959964, upper = 1.959964, region = between −1.959963985 and 1.959963985. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area from 0 to z: 0.44950 at z = 1.64, 0.45053 at 1.65, 0.47500 at 1.96). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (the area from 0 to 1.96 is 0.475, so the middle 95% runs from −1.96 to 1.96).
3. p = 0.1, tail = right, mean = 100, sd = 15 gives x = 119.223273, z = 1.281552. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
4. p = 0.5, tail = left, mean = 7, sd = 2 gives x = 7, z = 0. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.

## FAQ

### What does invNorm do?

It works backwards from an area to a value. invNorm(0.95, 0, 1) is the z with 95% of the standard normal curve to its left: about 1.6449.

### How do I find x for a mean and standard deviation that are not 0 and 1?

Find z for the area on the standard normal curve, then x = μ + zσ. The top 10% of a scale with mean 100 and SD 15 starts at 100 + 1.28155 × 15 ≈ 119.22.

### What is the difference between left, right and center?

Left finds x with the area below it. Right finds x with the area above it, which is the same as left with 1 − area. Center finds the two ends, μ − zσ and μ + zσ, with the area between them and the rest split evenly between the two tails.

### Why is the middle 95% from −1.96 to 1.96?

The 5% left over splits into 2.5% in each tail. The standard normal table gives an area of 0.475 from 0 to 1.96, so 0.5 + 0.475 = 97.5% lies below 1.96 and 2.5% above it.

### Can the area be 0 or 1?

No. An area of 0 or 1 would need x at minus or plus infinity. The page takes areas more than 0 (at least 10⁻³⁰⁰) and less than 1.

### Is the inverse normal the same as the percentile?

For a left area, yes: invNorm(0.9, μ, σ) is the 90th percentile of the normal distribution with mean μ and SD σ.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, the inverse of the CDF, has no closed form and is computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area under the standard normal curve from 0 to z; add 0.5 for the area below z). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (retrieved 2026-10-02)