Inverse Normal Distribution Calculator

What Does an Inverse Normal Calculator Do?

An inverse normal calculator turns a probability into a cut-off value. You give it an area under the normal curve, such as 0.90, along with the mean and standard deviation, and it returns the value x that has that much of the distribution on one side of it. It runs the usual normal calculation backwards.

A normal CDF lookup starts with a value and ends with a probability: a score of 130 on an IQ test sits at the 97.7th percentile. The inverse starts with the probability and asks where the score is. Which score marks the top 10%? Which two scores fence in the middle 95%? That reverse question is what the tool answers.

Statisticians call the function the quantile function or percent point function of the normal distribution, written Φ^-1(p) for the standard normal. Excel calls it NORM.INV, R calls it qnorm, Python's SciPy calls it norm.ppf, and TI calculators call it invNorm. The page covers all four tail types, because the area you know is not always the area to the left.

How Do You Use This Inverse Normal Calculator?

Pick which area you know, enter it, then set the mean and standard deviation. Leave them at 0 and 1 for a standard normal answer, which is the z-score.

1. Choose the area type. "Left tail" means the area below x. "Right tail" means the area above x. "Middle" means the area between two values placed symmetrically around the mean. "Both tails" means the combined area outside those two values.
2. Enter the area. Type a decimal such as 0.975, or a percentage such as 97.5%. It must be strictly between 0 and 1.
3. Enter the mean and the standard deviation. The standard deviation is the square root of the variance, not the variance itself, and it must be positive.
4. Read the result. One-sided questions give a single x and its z-score. Middle and both-tails questions give a lower and an upper value, which are always the same distance from the mean.

The chart shades the area you entered and marks the cut-off, so you can check that the answer matches your picture of the problem. The step-by-step box shows the z-score first and the conversion to x second, the same two steps you would do by hand with a z-table.

What Is the Inverse Normal Formula?

Every inverse normal calculation has two steps. First find the z-score whose left-tail area equals your probability, then rescale it to your own mean and standard deviation.

z = Φ^-1(p)   and   x = μ + z × σ

Here p is the left-tail area, μ is the mean and σ is the standard deviation. For p = 0.95 the standard normal gives z = 1.6449, so with μ = 100 and σ = 15 the cut-off is 100 + 1.6449 × 15 = 124.67.

The normal CDF has no closed-form inverse, so software uses a numerical method. This calculator uses Wichura's algorithm AS 241 (published in Applied Statistics in 1988), which is accurate to about 16 digits across the whole range. That is the same family of method behind R's qnorm and most statistics libraries.

The other three tail types reduce to the same formula. For a right-tail area p, use the left-tail area 1 − p. For a middle area p, each tail holds (1 − p) / 2, so the upper bound uses a left-tail area of (1 + p) / 2. For a both-tails area p, each tail holds p / 2. Because the bell curve is symmetric, the lower and upper bounds are mirror images: μ − zσ and μ + zσ.

Which Z-Scores Come Up Most Often?

A handful of probabilities account for most inverse normal questions, usually because they are confidence levels or significance levels. These are the standard normal values, checked against SciPy.

Negative values follow from symmetry. A left-tail area of 0.025 gives z = −1.9600, and 0.05 gives −1.6449. The median, left-tail area 0.50, sits at z = 0.

Worked Example: What Score Marks the Top 10%?

IQ scores are scaled to a mean of 100 and a standard deviation of 15. What score do you need to be in the top 10%?

The top 10% is a right-tail area of 0.10, so the left-tail area is 0.90. The standard normal quantile for 0.90 is z = 1.2816. Then x = 100 + 1.2816 × 15 = 119.22.

So a score of about 119.2 or higher puts you in the top 10%. In the calculator, choose "Right tail", enter 0.10, and set the mean to 100 and the standard deviation to 15. It reports x = 119.2233 and z = 1.2816. Most people stumble at the step where a right-tail area turns into a left-tail one, and the calculator does that conversion for you.

Worked Example: Where Is the 90th Percentile?

A test is scored with a mean of 500 and a standard deviation of 100. What score is the 90th percentile?

A percentile is a left-tail area, so there is nothing to convert. The left-tail area is 0.90, z = 1.2816, and x = 500 + 1.2816 × 100 = 628.16.

Someone scoring 628.16 has done better than 90% of test takers. Now run the question the other way, which is what the normal CDF calculator does: feed it 628.16 with the same mean and standard deviation and you get back 0.9000. The two tools are exact inverses of each other, which makes them a good pair for checking your work.

Worked Example: What Range Holds the Middle 95%?

Adult heights in a group average 70 inches with a standard deviation of 3 inches. Between which two heights do the middle 95% of people fall?

The middle 95% leaves 5% in the tails, 2.5% on each side. The lower bound has a left-tail area of 0.025, so z = −1.9600. The upper bound has a left-tail area of 0.975, so z = +1.9600. Then:

Lower = 70 − 1.9600 × 3 = 64.12    Upper = 70 + 1.9600 × 3 = 75.88

About 95% of heights in this group fall between 64.12 and 75.88 inches. Choose "Middle" in the calculator, enter 0.95, mean 70 and standard deviation 3, and it gives both bounds in one go. This is the same ±1.96 standard deviations that sits behind the 95% rule of thumb in the empirical rule.

Worked Example: When Is a Lower Limit Needed?

A bulb maker finds that bulb life is roughly normal with a mean of 1,000 hours and a standard deviation of 50 hours. The warranty should cover only the shortest-lived 2% of bulbs. How many hours should the warranty claim threshold be?

The lowest 2% is a left-tail area of 0.02. The standard normal quantile is z = −2.0537, so x = 1,000 − 2.0537 × 50 = 897.31 hours.

Roughly 2% of bulbs burn out before 897.3 hours. That is also the kind of cut-off that quality teams use to set a minimum acceptable life. The calculation for "the highest 2%" is the mirror image, 1,000 + 2.0537 × 50 = 1,102.69 hours, which you get by switching the area type to right tail.

How Is Inverse Normal Different From a Normal CDF Lookup?

The two tools answer opposite questions about the same bell curve. A normal CDF takes a value and returns the area, and the inverse normal takes an area and returns a value.

If you want to see the full picture first, the normal curve calculator plots the bell curve and shades regions, and the Gaussian distribution calculator shows the density, mean and spread together. Once you know the shape, this page gives you the cut-offs.

How Does the Inverse Normal Give Critical Values?

Critical values for z-tests and confidence intervals are inverse normal results. A 95% confidence interval for a mean uses the z-score that leaves 2.5% in each tail, which is Φ^-1(0.975) = 1.9600. A one-sided test at the 5% level uses Φ^-1(0.95) = 1.6449.

The margin of error is then z × the standard error. If the sample standard deviation is 12 and the sample size is 36, the standard error is 12 / 6 = 2, and the 95% margin of error is 1.96 × 2 = 3.92. That is why the Both tails option matters: a two-sided significance level α means α / 2 in each tail, so entering 0.05 as "Both tails" returns ±1.96 directly.

For small samples with an estimated standard deviation, the t distribution replaces the normal one and the critical value is slightly larger. With a large sample, the two get close, which is why 1.96 is a good approximation for sample sizes above about 100.

What Happens at the Edges: p = 0, p = 1 and Tiny Areas?

The normal distribution never reaches an area of exactly 0 or exactly 1, so no finite value x has a left-tail area of 0 or 1. The quantile at p = 0 is negative infinity and at p = 1 it is positive infinity. The calculator asks for an area strictly between 0 and 1 instead of printing infinity.

Very small areas still work. A left-tail area of 0.000001 gives z = −4.7534, and 0.999999 gives +4.7534. The chart shows the range from −4 to +4 standard deviations, so a cut-off beyond that is marked at the edge with a note rather than drawn off the page.

The standard deviation has to be above zero. A standard deviation of 0 would mean every value equals the mean, and there is no bell curve to take a quantile from.

What Mistakes Should You Avoid?

• Using the wrong tail. An invNorm button gives a left-tail result. "Top 10%" is not 0.10, it is 0.90 on the left. This calculator avoids the trap with the Right tail option.
• Entering the variance. The tool wants the standard deviation. If you have a variance of 225, enter 15. The standard deviation calculator and the variance calculator show the relationship.
• Typing a percentage as a number. Entering 95 instead of 0.95 or 95% is out of range. Percent signs are accepted.
• Treating a middle area as a left-tail area. The middle 95% does not have a left-tail area of 0.95. Each bound sits at 0.025 and 0.975.
• Applying it to skewed data. The results only mean something if the data are close to normal. Check a histogram or the gap between the mean and median in the mean, median and mode calculator first.
• Rounding the z-score too early. Using 1.64 instead of 1.6449 shifts a cut-off by a noticeable amount when the standard deviation is large. Keep the full z until the last step.

How Do You Find the Inverse Normal in Excel, R, Python and a TI Calculator?

The same four-tail logic applies in every tool. Only the function name changes, and each one takes the left-tail area.

For a right-tail area, pass 1 − p, or in R use qnorm(p, lower.tail = FALSE) and in SciPy use norm.isf(p). For a middle area, evaluate the function at (1 − p) / 2 and (1 + p) / 2.

The older Excel functions NORMINV and NORMSINV still work and give the same answers. The step-by-step panel in the calculator prints the exact formula for the values you entered, so you can paste it into a spreadsheet and compare.

How Does the Inverse Normal Connect to Percentiles and Quartiles?

Percentiles are inverse normal values, and the z-score to percentile calculator runs the same conversion in the other direction. The 75th percentile of a normal distribution is the value with a left-tail area of 0.75, which is 0.6745 standard deviations above the mean. The 25th percentile is the same distance below it. For heights with a mean of 170 and a standard deviation of 8, the quartiles are 164.60 and 175.40, and the interquartile range is 10.79, or 1.349 standard deviations.

That 1.349 ratio is a quick normality check. If real data have an interquartile range much larger or smaller than 1.349 times the standard deviation, the data are probably heavier-tailed or lighter-tailed than a normal distribution. The five-number summary calculator gives the quartiles and IQR for a data set, and the coefficient of variation calculator helps when you want to compare spread across data sets with different scales.

The normal distribution is also the large-sample limit for many other models. For instance, a binomial with many trials is approximately normal, which is why the binomial distribution calculator is a close relative of this page. Waiting-time questions such as "how long until the first success" use a different shape, the one on the geometric distribution calculator.

Where Else Does the Inverse Normal Show Up?

Outside the classroom, the same calculation sets thresholds wherever a measurement is roughly bell-shaped and someone needs a cut-off.

• Grading on a curve. If exam marks average 72 with a standard deviation of 9 and the top 15% get an A, the A cut-off is the right-tail 0.15 value: 72 + 1.0364 × 9 = 81.33.
• Quality control. A filling line that targets 500 ml sets its lower alarm limit so that only 1 in 1,000 bottles falls below it. That is the left-tail area 0.001 and a z-score of −3.0902.
• Finance. A one-day 5% value-at-risk figure under a normal model is the 5th percentile of returns. With a mean of 0 and a daily standard deviation of 2%, it is −1.6449 × 2% = −3.29%. Real returns have fatter tails than a normal curve, so treat this as a starting point rather than a risk limit.
• Test design. Setting a pass mark at the 70th percentile of a normally distributed score only needs the left-tail area 0.70, which gives z = 0.5244.

In every case the steps are identical: name the area, say which side it is on, and let the calculator return the value.

Frequently Asked Questions

Sources: Wichura, M. J. (1988), "Algorithm AS 241: The Percentage Points of the Normal Distribution", Applied Statistics 37(3), 477-484. NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 Normal Distribution. SciPy documentation, scipy.stats.norm. R documentation, qnorm.

Related Calculators

Know the value instead of the area? These cover the forward direction and the neighbouring distributions, or browse the full list of free statistics calculators:

Normal CDF CalculatorArea under the normal curve between any two bounds. Normal Curve CalculatorPlot the bell curve and shade any region. Binomial Distribution CalculatorExact, cumulative, and interval binomial probabilities with full statistics. All StatsSolve CalculatorsBrowse the full directory of free statistics tools.