{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"z-score"},"result":{"entity_type":"calculator","id":"z-score","calculator_id":"z-score","canonical_url":"https://tttkmbb.com/statistics/z-score","name":"Z-Score Calculator","title":"Z-Score Calculator – Standard Score and Normal Percentile of a Value","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_z_score","featured_mcp_tool":false,"description":"Standardises a value against a mean and standard deviation (z = (x − μ) / σ) and reports the corresponding percentile and tail probabilities under the normal distribution.","use_when":"You need how many standard deviations a value lies from the mean, or the percentile a value corresponds to in a normally distributed population (test scores, measurements, quality control).","do_not_use_when":"You need the probability between two values or the density (use normal-distribution), the data are clearly not normal, or you need the empirical percentile rank within an actual data set (use percentile).","inputs":[{"name":"value","label":"Value (x)","type":"number","required":true,"description":"The observation to standardise.","example":85},{"name":"mean","label":"Mean (μ)","type":"number","required":true,"description":"Mean of the population or reference distribution.","example":70},{"name":"std_dev","label":"Standard deviation (σ)","type":"number","required":true,"exclusive_min":0,"description":"Standard deviation of the population or reference distribution.","example":10}],"outputs":[{"name":"z_score","label":"Z-score","type":"number","decimals":4,"description":"(value − mean) / std_dev: distance from the mean in standard deviations."},{"name":"percentile","label":"Percentile","type":"number","unit":"%","decimals":2,"description":"Percentage of a normal population with values below x: 100 × Φ(z)."},{"name":"probability_below","label":"P(X < x)","type":"number","decimals":4,"description":"Standard normal CDF Φ(z)."},{"name":"probability_above","label":"P(X > x)","type":"number","decimals":4,"description":"1 − Φ(z)."},{"name":"interpretation","label":"Interpretation","type":"string","decimals":4,"description":"Plain-language position relative to the mean."}],"input_schema":{"type":"object","properties":{"value":{"description":"The observation to standardise.","type":"number","examples":[85]},"mean":{"description":"Mean of the population or reference distribution.","type":"number","examples":[70]},"std_dev":{"description":"Standard deviation of the population or reference distribution.","type":"number","exclusiveMinimum":0,"examples":[10]}},"additionalProperties":false,"required":["value","mean","std_dev"]},"output_schema":{"type":"object","properties":{"z_score":{"description":"(value − mean) / std_dev: distance from the mean in standard deviations.","type":"number"},"percentile":{"description":"Percentage of a normal population with values below x: 100 × Φ(z). 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When only a sample is available, the sample s is the usual substitute and the percentile becomes approximate, especially for small samples."},{"q":"What is a 'significant' z-score?","a":"By convention |z| > 1.96 corresponds to the outer 5 % of a normal distribution (two-sided) and |z| > 2.576 to the outer 1 %; values beyond |z| = 3 are often flagged as outliers."}],"tags":["z score","standard score","standardize","how many standard deviations","normal percentile"],"related":[{"calculator_id":"normal-distribution","reason":"Probabilities between two values and the density at a point."},{"calculator_id":"percentile","reason":"Empirical percentile rank within an actual list of values."},{"calculator_id":"descriptive-statistics","reason":"Compute the mean and standard deviation from data first."}],"links":{"html":"https://tttkmbb.com/statistics/z-score","markdown":"https://tttkmbb.com/statistics/z-score.md","json":"https://tttkmbb.com/statistics/z-score.json","api":"https://tttkmbb.com/api/v1/calculate/z-score","schema":"https://tttkmbb.com/api/v1/calculators/z-score","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:27:44Z"}