Z-score from normal value calculator
A score of 85 on a test with mean 70 and standard deviation 10 has a z-score of 1.5: one and a half standard deviations above the mean. You also get its absolute size and whether it sits above or below. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 85
- Mean
- 70
- Std
- 10
- Z score
- 1.5
- Abs z
- 1.5
- Direction
- above_mean
A z-score, or standard score, answers a single practical question: how many standard deviations does this observation sit above or below the mean of a normal (or approximately normal) reference distribution? The formula is universal and exact once the three numbers are known: subtract the mean from the value, then divide by the standard deviation. Teachers use it to compare exam marks from tests with different scales; analysts use it to place a reading against a published process mean and sigma; students use it for empirical-rule drills without building a full spreadsheet. This calculator accepts only that classic triple—raw value x, mean μ, and standard deviation σ—and returns the signed z-score, its absolute magnitude, and a simple direction flag that says whether the point is above, below, or exactly at the mean.
What the z-score from a normal value means
Start with a raw observation x and a reference center μ. Their difference x minus μ is the signed deviation from the middle of the distribution: positive when the observation is larger than the mean, negative when it is smaller, and zero when it matches the mean exactly. Dividing that difference by the standard deviation σ rescales the deviation into units of typical spread for that distribution.
After this change of units, a result of zero still means “at the mean,” a result of plus one means “one standard deviation above,” and a result of minus two means “two standard deviations below.” Under a normal model those units connect to familiar coverage rules: roughly two thirds of mass lies within one standard deviation of the mean, about ninety-five percent within two, and almost all within three—so knowing z immediately tells you whether a reading is ordinary or extreme without inventing ad-hoc cutoffs on the original scale.
The special case σ equals zero is rejected on purpose: when there is no spread, the idea of “how many deviations from the mean” has no meaning, and returning Infinity or NaN would break clients that expect a finite score.
Where a single-point z-score calculator earns its keep
Secondary-school and college statistics courses assign endless standard-score drills: convert this exam mark using the published class mean and standard deviation, then read the empirical rule. Quality control and process monitoring convert a sensor reading into z-units against a fixed process mean and sigma so operators can flag excursions beyond two or three sigma without inventing thresholds on the raw engineering scale. Psychometrics and educational measurement map a raw count of correct answers onto a common standard scale when norms are already known from a form or cohort.
Finance and risk teams sometimes express a single return or residual in standard-deviation units for stress summaries and simple anomaly screens when the reference mean and volatility are already estimated elsewhere. If you later need to standardize an entire column, estimate sample sigma with degrees of freedom, or compute percentiles and p-values from a normal table, treat those as separate search intents and landings rather than overloading this single-value z-score-from-normal-value page.
What you can do with it
Compare one exam score to class norms
Enter a student mark with the class mean and standard deviation to obtain a signed z-score that is comparable across tests with different scales.
Flag a process reading against control limits
Map a sensor value to z-units using a fixed process mean and sigma, then treat large |z| as an excursion for review.
Homework and LMS auto-grading
Use a deterministic z = (x − μ) / σ oracle that rejects zero deviation and never invents a score when σ is invalid.
FAQ
What is the formula for the z-score from a normal value?
z = (x − μ) / σ, where x is the raw value, μ is the mean, and σ is the standard deviation. Positive z means above the mean; negative z means below.
Which fields do I need to send?
Provide only one name for the value and only one name for the deviation.
Does this tool estimate the mean or standard deviation from a data list?
No. This calculator is single-point only: you supply μ and σ yourself. For standardizing a full column from sample statistics, use a dataset standardization tool instead.
What does the z-score from normal value calculator cost?
The browser tool is free and runs locally.
What happens if the standard deviation is zero?
The calculation is rejected. With no spread, asking how many standard deviations a value sits from the mean has no meaning, and an infinite score would break whatever uses the result next.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/stat/normal-z-from-x \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":85,"mean":70,"std":10}'const res = await fetch("https://api.kit.forhosting.com/stat/normal-z-from-x", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 85,
"mean": 70,
"std": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/normal-z-from-x",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 85,
"mean": 70,
"std": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/normal-z-from-x", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":85,"mean":70,"std":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":85,"mean":70,"std":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/normal-z-from-x", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 85,
"mean": 70,
"std": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.normal_z_from_x",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
When the deviation is zero the task fails with invalid_input, because division by zero would make every score undefined. The free browser path and the paid API share one pure deterministic module, so homework checks and production scripts never disagree on the same inputs.
What the z-score from a normal value means
This endpoint does not estimate μ or σ from a sample list; it assumes you already know those parameters from a population, a specification, a previous study, or a textbook problem. That single-point contract keeps the request tiny, the arithmetic constant-time, and the JSON shape stable for widgets and auto-graders.
Input contract, response fields, and error behaviour
The mean is required under mean. POST a JSON object with three finite numbers. The raw observation is required as x, or as the alias value if your client already uses that name—send only one of the two. The standard deviation is required under std, with accepted aliases sd, standard_deviation, sigma, and deviation so classroom wording and engineering naming both work; provide only one of those keys. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, NaN, and Infinity fail with invalid_input naming the problem and are not billed. On success the body echoes cleaned x, mean, and std, returns z_score as (x − μ) / σ rounded for stable cross-runtime goldens, returns abs_z as the absolute magnitude of that score, sets direction to above_mean, below_mean, or at_mean, and includes the formula string z = (x − μ) / σ for display next to the answer. Classic checks include x equals eighty-five, mean equals seventy, std equals ten, which yields z equals one point five and direction above_mean; and a value equal to the mean with any positive sigma, which yields z equals zero and direction at_mean. Pair the free browser form for quick homework and demos with the identical API contract when you need scripted checks in LMS plugins or edge workers. Successful API calls cost the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where a single-point z-score calculator earns its keep
A deterministic oracle that returns cleaned JSON speeds grading, demo setup, and student self-check without a spreadsheet.
Technical questions
What happens if the standard deviation is zero?
The request fails with invalid_input. When σ = 0 there is no unit of spread, so the z-score is undefined and the call is not billed.