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Standard Normal Distribution Calculator

Just enter a percentage point x (z-score) and get the standard normal density f(x) plus the lower, upper, and inner cumulative probabilities automatically. A bell-curve graph shades each probability as an area so you can see it at a glance.

Input

The standardized value (z-score). e.g. 1.96, -1, 0

Region to highlight on the graph

Result

Lower cumulative probability P(X ≤ x)

0.9750021049

x = 1.96 (about 97.5002% as a percentage)

-3-2-10123x=1.96

Density f(x)

0.0584409443

Lower P(X≤x)

0.9750021049

Upper P(X≥x)

0.0249978952

Inner P(-x≤X≤x)

0.9500042097

Note: these are approximate probabilities for the standard normal distribution (mean 0, standard deviation 1). They are computed at double precision (roughly 15 significant digits) using the error function (erf).

How it works

  • This tool computes probabilities of the standard normal distribution (mean 0, standard deviation 1) for the percentage point x you enter (a standardized value, or z-score). Enter a standardized deviation score or test statistic as x to find its probability.
  • The probability density f(x) is given by f(x)=(1/√(2π))·e^(−x²/2). This is the height of the curve, not a probability itself; the probability is the area (integral) over a range.
  • The lower cumulative probability P(X≤x) is the area from the left tail up to x, and the upper cumulative probability P(X≥x) is the area from x to the right tail; together they sum to 1. The inner probability P(−x≤X≤x) is the area from −|x| to |x| (the two-sided probability).
  • Probabilities are computed with the error function erf, using the relations Φ(x)=½·erfc(−x/√2), P(X≥x)=½·erfc(x/√2), and inner probability = erf(|x|/√2). The results are double-precision approximations (roughly 15 significant digits).
  • For reference, at x=1.96 the lower probability P(X≤x)≈0.975 and the inner probability P(−x≤X≤x)≈0.95; at x=1 the inner probability ≈0.6827 (±1σ); and at x=2 it ≈0.9545 (±2σ). Use these for significance levels and confidence intervals.
  • Use the toggle below the graph to switch between lower, upper, and inner, and the corresponding area is shaded on the bell curve. When x is large and the tail probability is extremely small, the value is shown in exponential notation.