normalcdf Calculator

Compute normalcdf(lower, upper, μ, σ) exactly as a TI-83 or TI-84 does: the probability that a normal random variable falls between two values. Leave a bound blank (or type -1E99 / 1E99) for an open-ended tail.

Going the other way — from an area back to a value? Use the invNorm calculator.

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

What normalcdf Does

normalcdf returns the area under a normal density curve between a lower and an upper bound — which is the probability P(lower ≤ X ≤ upper) for a normally distributed variable X. The name stands for normal cumulative distribution function.

normalcdf(a, b, μ, σ) = Φ((b − μ)/σ) − Φ((a − μ)/σ)

P(X ≤ b) → normalcdf(-1E99, b, μ, σ)

P(X ≥ a) → normalcdf(a, 1E99, μ, σ)

Standard normal: omit μ and σ (defaults 0 and 1)

How to Use normalcdf on a TI-84

  1. Press 2nd then VARS to open the DISTR menu.
  2. Choose 2:normalcdf(.
  3. Enter lower, upper, μ, σ. For negative infinity type (−) 1 2nd , 99 (displayed as −1E99); for positive infinity type 1 2nd , 99.
  4. Select Paste and press ENTER twice.

On a TI-83 or older OS the wizard is absent: type normalcdf(lower, upper, μ, σ) directly with commas. In Excel or Google Sheets the equivalent is =NORM.DIST(b, μ, σ, TRUE) − NORM.DIST(a, μ, σ, TRUE).

Worked Example: Adult Heights

Heights are normally distributed with μ = 170 cm and σ = 10 cm. What share of adults are between 160 and 185 cm?

  1. Enter normalcdf(160, 185, 170, 10).
  2. Standardize: z = (160 − 170)/10 = −1 and z = (185 − 170)/10 = 1.5.
  3. Look up Φ(1.5) = 0.933193 and Φ(−1) = 0.158655.
  4. Subtract: 0.933193 − 0.158655 = 0.774538.

About 77.45% of adults are between 160 and 185 cm. Check the two cumulative values yourself in the z-table.

normalcdf vs. normalpdf

normalpdf(x, μ, σ) returns the height of the density curve at a single point, which is not a probability — for a continuous variable, P(X = x) is always 0. Use normalcdf whenever a question asks for a probability, proportion, or percentage. For probabilities of whole-number counts, use binompdf and binomcdf instead.

Frequently Asked Questions

What does -1E99 mean in normalcdf?

−1E99 is −1 × 10^99, a number so far below the mean that the TI-84 treats it as negative infinity. Use it as the lower bound for 'less than' questions, and 1E99 as the upper bound for 'greater than' questions. On this page you can simply leave that bound blank.

How do I find P(X < x) with normalcdf?

Use normalcdf(-1E99, x, μ, σ). For example, P(X < 120) with μ = 100 and σ = 15 is normalcdf(-1E99, 120, 100, 15) ≈ 0.9088.

How do I find P(X > x) with normalcdf?

Use normalcdf(x, 1E99, μ, σ), or compute 1 − normalcdf(-1E99, x, μ, σ). Both give the same right-tail area.

Why does my calculator give an answer like 2.8665E-7?

That is scientific notation: 2.8665 × 10^−7, or 0.00000028665. It means the region is extremely far into a tail of the distribution, so the probability is tiny but not zero.

Does normalcdf use z-scores or raw values?

Either. With μ and σ supplied it takes raw values and standardizes internally. If you omit μ and σ (or enter 0 and 1), the bounds are treated as z-scores on the standard normal curve.

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