invNorm Calculator (Inverse Normal)

Find the value that cuts off a given area of a normal distribution. This is the TI-84 invNorm(area, μ, σ, tail) function: give it a probability and it returns the matching score — the reverse of normalcdf.

Part of the TI-84 statistics functions guide, which shows when to use invNorm and every other DISTR-menu function.

What invNorm Does

The inverse normal function answers questions like "what score marks the top 10%?" or "what z-value leaves 2.5% in each tail?" It takes an area and returns the point on the horizontal axis where that area ends. Mathematically it is the quantile function, Φ⁻¹, of the normal distribution.

Left: x = μ + σ · Φ⁻¹(area)

Right: x = μ + σ · Φ⁻¹(1 − area)

Center: x = μ ± σ · Φ⁻¹((1 + area) / 2)

How to Use invNorm on a TI-84

  1. Press 2nd then VARS (DISTR).
  2. Choose 3:invNorm(.
  3. Enter the area, μ, and σ. On OS 5.5 or later, also pick the tail: LEFT, CENTER, or RIGHT.
  4. Select Paste and press ENTER.

Older calculators only accept left-tail areas: for a right tail, enter 1 − area; for a central area, enter (1 + area)/2. In Excel or Google Sheets use =NORM.INV(area, μ, σ), or =NORM.S.INV(area) for a z-score.

Worked Example: Top 10% of Test Scores

Scores are normal with μ = 1050 and σ = 200. What score does a student need to be in the top 10%?

  1. The top 10% is a right-tail area of 0.10, so the area to the left is 0.90.
  2. Φ⁻¹(0.90) = 1.2816.
  3. x = 1050 + 1.2816 × 200 = 1306.3.

A score of about 1306 or higher lands in the top 10%. On a TI-84: invNorm(0.10, 1050, 200, RIGHT) or invNorm(0.90, 1050, 200).

Common z Critical Values

90% confidence → invNorm(0.95) = 1.6449

95% confidence → invNorm(0.975) = 1.9600

98% confidence → invNorm(0.99) = 2.3263

99% confidence → invNorm(0.995) = 2.5758

These are the multipliers behind confidence intervals and two-tailed z-tests. For t, chi-square, and F cutoffs, use the critical value calculator.

Frequently Asked Questions

What is the difference between invNorm and normalcdf?

They are inverses. normalcdf takes values and returns an area (probability); invNorm takes an area and returns a value. If normalcdf(-1E99, x, μ, σ) = 0.8, then invNorm(0.8, μ, σ) = x.

Why do I get a negative z-score from invNorm?

Any left-tail area below 0.5 lies below the mean, so its z-score is negative. For example, invNorm(0.05) = −1.6449. If you wanted the upper 5% cutoff, choose the right tail or enter 0.95.

How do I find the z-score for a percentile?

Convert the percentile to a decimal and use a left tail. The 90th percentile is invNorm(0.90) = 1.2816, and the 25th percentile is invNorm(0.25) = −0.6745.

Can the area be 0 or 1?

No. The normal curve never reaches zero height, so an area of exactly 0 or 1 would require an infinite cutoff. Use a value strictly between 0 and 1, such as 0.999.

What does the CENTER tail option mean?

CENTER returns the symmetric pair of values that enclose the given area around the mean. invNorm(0.95, 0, 1, CENTER) returns ±1.96, the classic 95% confidence multiplier.

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