What does normalcdf calculate? +
normalcdf calculates the cumulative distribution function (CDF) of a normal distribution — the probability that a normally distributed random variable falls between two values (lower and upper bounds). The result is between 0 and 1. Multiply by 100 for percentage. On a TI-84: normalcdf(lower, upper, μ, σ). This online calculator does the same calculation.
What is the difference between normalcdf and normpdf? +
normalcdf gives the probability (area) between two bounds — the most useful function for answering "what percentage of values fall between X and Y?" normpdf (normal probability density function) gives the height of the bell curve at a specific point — used for graphing the distribution, not for probability calculations. For almost all statistics homework and exam problems, normalcdf is what you need.
How do I use normalcdf for a left-tail probability? +
For P(X ≤ b) — the probability of being below a value — use negative infinity as your lower bound. In this calculator, leave the lower bound field blank or enter a very large negative number. On a TI-84, enter −1E99 as the lower bound: normalcdf(−1E99, b, μ, σ). Example: P(X ≤ 85) with μ=80, σ=10: normalcdf(−1E99, 85, 80, 10) = 0.6915.
How do I use normalcdf for a right-tail probability? +
For P(X ≥ a) — the probability of being above a value — use positive infinity as your upper bound. Leave the upper bound blank or enter a very large positive number. On TI-84: normalcdf(a, 1E99, μ, σ). Alternatively, calculate P(X ≤ a) with the left-tail method and subtract from 1: P(X ≥ a) = 1 − P(X ≤ a).
What are the normalcdf inputs? +
normalcdf takes 4 inputs: (1) Lower bound — the minimum value of your range (use −1E99 for no lower bound). (2) Upper bound — the maximum value (use 1E99 for no upper bound). (3) μ (mu) — the mean of your normal distribution. (4) σ (sigma) — the standard deviation. For the standard normal distribution, use μ=0 and σ=1 and enter Z-scores directly as your bounds.
What is a Z-score? +
A Z-score measures how many standard deviations a value is from the mean: Z = (X − μ) ÷ σ. A Z-score of 0 means the value equals the mean. Z = +1 means 1 standard deviation above mean. Z = −2 means 2 standard deviations below mean. Converting to Z-scores allows you to use the standard normal table (μ=0, σ=1) for any normal distribution.
How accurate is this normalcdf calculator? +
This calculator uses a precise numerical approximation of the complementary error function (erfc) to compute the normal CDF. Results match TI-84 and scientific computing outputs to 6+ decimal places for most inputs. For extreme Z-scores (|Z| > 8), the result approaches 0 or 1 so precisely that floating-point representation makes exact comparison difficult, but the displayed result is accurate for all practical statistics applications.
What is the 68-95-99.7 rule? +
The 68-95-99.7 rule (empirical rule) states that in a normal distribution: 68.27% of values fall within 1 standard deviation of the mean (μ ± 1σ), 95.45% fall within 2 standard deviations (μ ± 2σ), and 99.73% fall within 3 standard deviations (μ ± 3σ). You can verify these with normalcdf: normalcdf(−1, 1, 0, 1) = 0.6827, normalcdf(−2, 2, 0, 1) = 0.9545, normalcdf(−3, 3, 0, 1) = 0.9973.