{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"normal-distribution"},"result":{"entity_type":"calculator","id":"normal-distribution","calculator_id":"normal-distribution","canonical_url":"https://tttkmbb.com/statistics/normal-distribution","name":"Normal Distribution Calculator","title":"Normal Distribution Calculator – Probability Below, Above and Between Values (CDF and PDF)","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_normal_distribution_probability","featured_mcp_tool":false,"description":"Computes cumulative probabilities P(X < x), P(X > x) and P(x < X < x2) and the density for a normal (Gaussian) distribution with a given mean and standard deviation.","use_when":"You need the probability that a normally distributed quantity falls below, above or between given values, or the z-scores of those values.","do_not_use_when":"The variable is a count of successes or events (use binomial-distribution or poisson-distribution), or you need the value at a given probability from real data (use percentile).","inputs":[{"name":"mean","label":"Mean (μ)","type":"number","required":false,"default":0,"description":"Mean of the distribution (default 0 = standard normal).","example":100},{"name":"std_dev","label":"Standard deviation (σ)","type":"number","required":false,"default":1,"exclusive_min":0,"description":"Standard deviation of the distribution (default 1 = standard normal).","example":15},{"name":"x","label":"Value x","type":"number","required":true,"description":"The value at which to evaluate the distribution.","example":85},{"name":"x2","label":"Second value x2","type":"number","required":false,"description":"Optional second value; when given, the probability between x and x2 is also returned.","example":115}],"outputs":[{"name":"z_score","label":"z of x","type":"number","decimals":4,"description":"(x − mean) / std_dev."},{"name":"probability_below","label":"P(X < x)","type":"number","decimals":4,"description":"Cumulative probability up to x."},{"name":"probability_above","label":"P(X > x)","type":"number","decimals":4,"description":"Upper-tail probability beyond x."},{"name":"probability_density","label":"Density f(x)","type":"number","decimals":5,"description":"Value of the probability density function at x (not a probability)."},{"name":"z_score_2","label":"z of x2","type":"number","decimals":4,"description":"(x2 − mean) / std_dev (only when x2 is given)."},{"name":"probability_between","label":"P(x < X < x2)","type":"number","decimals":4,"description":"Probability between the two values, in either order (only when x2 is given)."}],"input_schema":{"type":"object","properties":{"mean":{"description":"Mean of the distribution (default 0 = standard normal).","type":"number","default":0,"examples":[100]},"std_dev":{"description":"Standard deviation of the distribution (default 1 = standard normal).","type":"number","exclusiveMinimum":0,"default":1,"examples":[15]},"x":{"description":"The value at which to evaluate the distribution.","type":"number","examples":[85]},"x2":{"description":"Optional second value; when given, the probability between x and x2 is also returned.","type":"number","examples":[115]}},"additionalProperties":false,"required":["x"]},"output_schema":{"type":"object","properties":{"z_score":{"description":"(x − mean) / std_dev.","type":"number"},"probability_below":{"description":"Cumulative probability up to x.","type":"number"},"probability_above":{"description":"Upper-tail probability beyond x.","type":"number"},"probability_density":{"description":"Value of the probability density function at x (not a probability).","type":"number"},"z_score_2":{"description":"(x2 − mean) / std_dev (only when x2 is given).","type":"number"},"probability_between":{"description":"Probability between the two values, in either order (only when x2 is given).","type":"number"}}},"formula":"z = (x − mean) / std_dev; P(X < x) = Φ(z); P(X > x) = 1 − Φ(z); P(x < X < x2) = |Φ(z2) − Φ(z)|; f(x) = exp(−z² / 2) / (std_dev · √(2π))","method":"Φ is computed from the Abramowitz & Stegun erf approximation (absolute error < 1.5·10⁻⁷); results are exact to the displayed decimals. For a continuous distribution P(X ≤ x) = P(X < x).","sources":[{"name":"NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution","url":"https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm","type":"government","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Normal distribution","url":"https://en.wikipedia.org/wiki/Normal_distribution","type":"reference","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Standard normal, x = 1.96","inputs":{"x":1.96},"expected":{"z_score":1.96,"probability_below":0.975,"probability_above":0.025,"probability_density":0.05844},"url":"https://tttkmbb.com/api/v1/calculate/normal-distribution?x=1.96"},{"name":"IQ 85 to 115 (mean 100, SD 15)","inputs":{"mean":100,"std_dev":15,"x":85,"x2":115},"expected":{"z_score":-1,"probability_below":0.1587,"probability_above":0.8413,"probability_density":0.01613,"z_score_2":1,"probability_between":0.6827},"url":"https://tttkmbb.com/api/v1/calculate/normal-distribution?mean=100&std_dev=15&x=85&x2=115"}],"faq":[{"q":"What is the 68–95–99.7 rule?","a":"For a normal distribution about 68.27 % of values lie within 1 SD of the mean, 95.45 % within 2 SD and 99.73 % within 3 SD; the second example (85–115 with SD 15) reproduces the 68.27 % figure."},{"q":"Why is the density larger than 1 sometimes?","a":"f(x) is a density, not a probability: it can exceed 1 when the standard deviation is small (below 0.399). Probabilities come from areas under the curve, i.e. the CDF outputs."}],"tags":["normal distribution","gaussian","bell curve probability","cumulative distribution","area under the curve"],"related":[{"calculator_id":"z-score","reason":"Standardise a single value and read its percentile."},{"calculator_id":"confidence-interval","reason":"Use normal critical values to build an interval for a mean."}],"links":{"html":"https://tttkmbb.com/statistics/normal-distribution","markdown":"https://tttkmbb.com/statistics/normal-distribution.md","json":"https://tttkmbb.com/statistics/normal-distribution.json","api":"https://tttkmbb.com/api/v1/calculate/normal-distribution","schema":"https://tttkmbb.com/api/v1/calculators/normal-distribution","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:25:00Z","next_actions":[{"tool":"run_calculator","calculator_id":"normal-distribution","reason":"Run Normal Distribution Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/statistics/normal-distribution.md"}}