standarddeviationcalculator.net

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Statistics

Inverse normal distribution calculator

Enter an area (probability), say where that area lies, and give the mean and standard deviation. The calculator returns the cut-off value x, or the pair of cut-offs for a central or two-tailed area, together with the z-score.

Between 0 and 1, e.g. 0.95 or 95%.

Use μ = 0 and σ = 1 for a z-score.

x with P(X ≤ x) = 0.95 124.673
z-score1.64485
Area left of the upper cut-off0.95
Mean μ100
Standard deviation σ15
40608010012014016000.0050.010.0150.020.025 124.7

The curve is the probability density; the shaded area is the probability asked for.

Common left-tail areas for this μ and σ
Left-tail areazx
0.91.2816119.223
0.951.6449124.673
0.9751.96129.399
0.992.3263134.895
0.9952.5758138.637
Show the working, step by step

  1. Turn the area into the probability to the left of a cut-off.

    The area to the left is the probability itself: 0.95.

  2. Find z from the inverse standard normal CDF:

    z = Φ⁻¹(0.95) = 1.64485

  3. Convert z back to the scale of X:

    x = μ + zσ = 100 + 1.64485 × 15 = 124.673

Φ⁻¹ is evaluated with Acklam’s rational approximation, accurate to about 1 part in a billion.

The formula

x = μ + zσ, z = Φ⁻¹(pleft)

Φ⁻¹ is the inverse of the standard normal CDF, and pleft is the area to the left of the cut-off. Each area type converts to pleft differently:

left tail A: pleft = A right tail A: pleft = 1 − A central area A: pleft = (1 + A)/2, cut-offs μ ± zσ two tails total A: pleft = 1 − A/2, cut-offs μ ± zσ

Worked example

IQ scores are normal with μ = 100 and σ = 15. What score marks the 95th percentile, so that 95% of people score at or below it?

z = Φ⁻¹(0.95) = 1.64485 x = 100 + 1.64485 × 15 = 124.673

A score of about 124.7 is the 95th percentile. Other settings with the same μ and σ:

  • Top 10% (right tail 0.1): z = Φ⁻¹(0.9) = 1.28155, x = 100 + 1.28155 × 15 = 119.223.
  • Middle 95% (central 0.95): z = Φ⁻¹(0.975) = 1.95996, so the cut-offs are 100 ± 29.3995, that is 70.6005 and 129.399.
  • Two tails totalling 0.05: the same z = ±1.95996 and the same cut-offs, described from the outside.

Reading the result

The shaded area on the chart is the probability you entered, so you can check at a glance that the region is on the side you meant. A left-tail area above 0.5 gives a cut-off above the mean and a positive z; below 0.5 the cut-off falls below the mean. The table under the chart lists the common one-sided percentiles for your μ and σ, useful for building confidence intervals and tolerance limits.

Where you need it

Grading on a curve (the score that puts a student in the top 15%), quality limits (the thickness that only 1% of parts exceed), confidence intervals (the z for 90%, 95% or 99% coverage) and sample-size formulas all need the inverse normal. So do reference ranges in medicine, which are usually the central 95% of healthy values: μ ± 1.95996σ. If your data are clearly skewed, the normal percentiles will be off in the tails, and a lognormal or an empirical percentile is safer.

Common mistakes

  • Entering a right-tail area as a left-tail one. “Top 10%” is a right tail of 0.1, which is the 90th percentile, not the 10th.
  • Using 1.645 for a 95% two-sided interval. Two-sided 95% needs 1.95996; 1.645 is one-sided 95% (or two-sided 90%).
  • Entering a percentage as a whole number. Type 0.95 or 95%, not 95.
  • Entering the variance as σ. If you know σ² = 225, the standard deviation is 15.
Inverse normal distribution calculator: the worked example on this page, with its result and chart
Inverse normal distribution calculator: the worked example above, at a glance.

Common questions

What does the inverse normal distribution calculate?

It runs the normal distribution backwards. A normal calculator takes a value x and returns an area (a probability); the inverse takes the area and returns x. Use it for percentiles, cut-off scores and critical values: “what score puts someone in the top 10%?”

Is this the same as invNorm on a TI-84?

Yes. invNorm(0.95, 100, 15) returns 124.673, the same as the default here. The TI-84 also has a tail setting (LEFT, RIGHT, CENTER); the “area is” drop-down does the same job, plus a two-tailed option. In Excel the function is NORM.INV(0.95, 100, 15).

What is the difference between central and two-tailed?

They are complements. A central area of 0.95 puts 95% in the middle and 2.5% in each tail; a two-tailed area of 0.05 puts 5% in the tails combined. Both give z = ±1.95996 and the same cut-offs. Pick whichever matches how your problem is worded.

How do I get a z-score instead of x?

Set μ = 0 and σ = 1. The answer x is then a z-score. The z-score is also always shown in the results for any μ and σ, since x = μ + zσ.

How accurate is the answer?

Φ⁻¹ is computed with Acklam's rational approximation, which has a relative error below 1.2 × 10⁻⁹ across the whole range, far more precise than any printed z-table.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.