normalpdf Calculator
Compute normalpdf(x, μ, σ) exactly as a TI-83 or TI-84 does: the height of the normal density curve at a single point. It describes how densely values cluster around x — but it is not the probability of x.
Looking for a probability such as P(X < 120)? That is the normalcdf calculator.
Part of the TI-84 statistics functions guide, which shows when to use normalpdf and every other DISTR-menu function.
Related Calculators
normalcdf Calculator
Evaluate normalcdf(lower, upper, μ, σ) exactly like a TI-84, with the shaded area and keystrokes shown.
invNorm Calculator
Find the value or z-score for a given area with invNorm(area, μ, σ) — left, right, or center tail.
Normal Distribution Calculator
Work with the normal curve, cumulative probabilities, and related z-score outputs.
The normalpdf Formula
normalpdf(x, μ, σ) = 1 / (σ√(2π)) × e^(−(x − μ)² / (2σ²))
Maximum height (at x = μ) = 1 / (σ√(2π)) ≈ 0.3989 / σ
The output has units of "probability per unit of x". That is why it can exceed 1: with σ = 0.1 the peak is about 3.99. Areas under the curve — what normalcdf returns — are always between 0 and 1.
How to Use normalpdf on a TI-84
- Press 2nd then VARS (DISTR).
- Choose 1:normalpdf(.
- Enter x, μ, and σ, then select Paste and press ENTER.
The most common classroom use is graphing: enter Y1 = normalpdf(X, μ, σ) and set the window from μ − 4σ to μ + 4σ to draw the bell curve. In Excel use =NORM.DIST(x, μ, σ, FALSE).
Worked Example: Heights
Heights are normal with μ = 170 cm and σ = 10 cm. Evaluate the density at 170 and at 185.
- normalpdf(170, 170, 10) = 1 / (10 × √(2π)) = 0.039894 — the peak.
- At 185, z = 1.5 and e^(−1.125) = 0.324652, so normalpdf(185, 170, 10) = 0.012952.
185 cm is about a third as "dense" as the average height. The probability of a height within ±0.5 cm of 185 is approximately 0.012952 × 1 cm ≈ 1.3% — a density times a width. For exact interval probabilities, use normalcdf(184.5, 185.5, 170, 10).
Frequently Asked Questions
What is the difference between normalpdf and normalcdf?
normalpdf gives the height of the bell curve at one point; normalcdf gives the area under the curve between two points. Only normalcdf returns a probability. If a question asks 'what is the probability' or 'what percentage', use normalcdf.
Why isn't normalpdf the probability that X equals x?
For a continuous variable, the probability of any single exact value is 0 — there is no area above a single point. The density measures how concentrated values are near x; multiplying it by a small interval width approximates the probability of landing in that interval.
Can normalpdf be greater than 1?
Yes. When σ is small the curve is tall and narrow: normalpdf(0, 0, 0.1) ≈ 3.989. Densities can exceed 1; only the total area under the curve must equal 1.
When would I actually use normalpdf?
To graph a normal curve, to compare how likely values near two points are relative to each other, and in likelihood calculations and calculus-based probability courses.
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