Normal Distribution Calculator (normalcdf & invNorm)

This free normal distribution calculator works exactly like the TI-84 normalcdf, invNorm and normalpdf commands. Find the probability below, above, between or outside two values, turn an area or percentile back into a value, get the height of the bell curve, and convert between z-scores and probabilities. Every answer comes with a shaded bell curve, step-by-step working with your own numbers, the exact TI-84 keystrokes, and the 68–95–99.7 empirical rule for your mean and standard deviation. Results update as you type.

How to Use the Normal Distribution Calculator

  1. Pick a mode. normalcdf gives an area (probability). invNorm gives the value that cuts off an area. normalpdf gives the height of the curve. z-score ↔ probability works on the standard normal curve.
  2. Enter the mean μ and standard deviation σ. Press Standard normal (0, 1) if you are already working with z-scores. Use the standard deviation, not the variance.
  3. Choose the area. In normalcdf, pick P(X ≤ b), P(X ≥ a), between a and b, or outside a and b. Leave a bound blank, or type -inf, inf, -1E99 or 1E99, for an open end, just like on the calculator.
  4. Read the answer. You get the probability, the percentage, the complement and both z-scores, with the region shaded on the curve.
  5. Copy or share. Copy results copies a short summary with the TI-84 command. Copy share link copies a link that reopens the calculator with your numbers, handy for homework help or a class.

normalcdf, invNorm and normalpdf: Which One Do You Need?

Question asks forUseTI-84 syntax
Probability between two valuesnormalcdfnormalcdf(a, b, μ, σ)
Probability less than xnormalcdfnormalcdf(-1E99, x, μ, σ)
Probability greater than xnormalcdfnormalcdf(x, 1E99, μ, σ)
Value for a percentile or areainvNorminvNorm(area, μ, σ)
Height of the curve at xnormalpdfnormalpdf(x, μ, σ)

A common mistake is using normalpdf for a probability. For a continuous variable P(X = x) is always 0, so normalpdf only gives the curve’s height. Use normalcdf for any “less than”, “more than” or “between” question.

How to Use normalcdf on a TI-84 Plus CE

  1. Press 2nd VARS to open the DISTR menu.
  2. Choose 2:normalcdf(.
  3. Fill in lower, upper, μ and σ. For negative infinity type (−) 1 2nd , 9 9, which shows as −1E99.
  4. Highlight Paste and press ENTER twice.

On a TI-83 or an older TI-84 OS there is no input screen, so type normalcdf(60,85,72,9) straight onto the home screen. If you leave out μ and σ, the calculator uses the standard normal (0 and 1). The calculator above shows these keys with your own numbers, plus the ShadeNorm( command to draw the shaded area on the graph screen. You can practise every step on our TI-84 Plus CE calculator online.

invNorm Calculator with LEFT, CENTER and RIGHT Tails

invNorm works backwards from an area to a value. By default the area is to the left of the answer, so the 90th percentile of heights with μ = 170 cm and σ = 8 cm is invNorm(0.9,170,8) ≈ 180.25 cm.

Newer TI-84 Plus CE operating systems add a Tail option. RIGHT treats the area as being to the right of the value. CENTER treats it as the middle area and returns both cut-offs. For example, invNorm(0.95,0,1,CENTER) gives {−1.96, 1.96}, the critical values for a 95% confidence interval. Older models don’t have the Tail option. With those, use 1 − area for a right tail, and (1 + area) / 2 for the upper centre value. The calculator shows both methods.

Normal Distribution Formulas

z-score: z = (x − μ) / σ

Probability density (normalpdf): f(x) = e^(−z²/2) / (σ√(2π))

Cumulative probability (normalcdf): P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a), where Φ is the standard normal CDF.

Inverse (invNorm): x = μ + σ · Φ⁻¹(area)

Our results match statistics software to at least 12 significant figures, even in extreme tails like normalcdf(30, 31, 0, 1) ≈ 4.9 × 10⁻¹⁹⁸, where many online calculators simply show 0.

Worked Example

Test scores are normal with mean μ = 72 and standard deviation σ = 9. What fraction of students score between 60 and 85?

  • z for 60: (60 − 72) / 9 = −1.3333. z for 85: (85 − 72) / 9 = 1.4444.
  • Φ(1.4444) = 0.92569 and Φ(−1.3333) = 0.09121.
  • P(60 ≤ X ≤ 85) = 0.92569 − 0.09121 = 0.8345, so about 83.45% of students.

On a TI-84: normalcdf(60,85,72,9) = 0.8344817820. Press Example in the calculator to load these numbers.

The Empirical Rule (68–95–99.7)

For any normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. The exact figures are 68.27%, 95.45% and 99.73%. The calculator works out these ranges for your μ and σ, so you can quickly check whether an answer makes sense. A value more than 3σ from the mean is unusual, at about 3 in 1,000.

z-Score to Probability (and Back)

In z-score ↔ probability mode, type a z-score to get the left-tail, right-tail, two-tailed or middle area. This replaces a printed z-table and gives more decimal places. Switch to probability → z for critical values, such as z = 1.645 for a one-tailed 5% test or ±1.96 for a two-tailed one. The x → z option standardises a raw score and gives its percentile.

Normal distribution problems often start with data. Use our standard deviation calculator to get μ and σ from a list of numbers, and our linear regression calculator for two-variable data.

Frequently Asked Questions

normalcdf is the normal cumulative distribution function on TI-83 and TI-84 calculators. normalcdf(lower, upper, μ, σ) gives the probability that a normal variable falls between the lower and upper values, which is the area under the bell curve.

Use 1E99 for positive infinity and −1E99 for negative infinity. On the TI-84, the E is 2nd then the comma key. In this calculator you can also leave the box blank or type inf or -inf.

normalcdf starts with values and gives an area (a probability). invNorm starts with an area and gives the value that cuts it off. They are inverses: invNorm(normalcdf(−1E99, x, μ, σ), μ, σ) returns x.

With Tail set to CENTER, the area is the middle part of the curve and invNorm returns the two values that enclose it. invNorm(0.95, 0, 1, CENTER) gives −1.96 and 1.96.

Use normalpdf to graph the bell curve or to find its height at a point. It is not a probability. For probabilities, use normalcdf.

Use normalcdf with μ = 0 and σ = 1. For P(Z < z) use normalcdf(−1E99, z). For P(Z > z) use normalcdf(z, 1E99). Or use the z-score mode of this calculator.

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.