Standardized moment calculator
For 1, 2, 3, 4, 5, 12 at order 3, the mean is 4.5 and the third central moment is 60; dividing it by the standard deviation cubed gives a positive skewness with no units. Choose any order from 1 to 20 and see every step. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Values
- 1, 2, 3, 4, 5, 12
- Order
- 3
- N
- 6
- Mean
- 4.5
- Central moment
- 60
- Std dev
- 3.5939764421
The standardized moment of order k is a scale-free shape measure: take the k-th central moment of a numeric sample and divide by the population standard deviation raised to the same power. That single ratio answers whether a list is left- or right-skewed (order three), how heavy its tails are relative to a normal baseline (order four), or higher-order shape when textbooks ask for α_k. This standardized moment calculator accepts a list of finite numbers and an integer order k from one through twenty, computes the population mean, the k-th central moment, population standard deviation, and α_k equals μ_k over σ to the k, then returns every intermediate so homework keys and automation agree.
What the standardized moment measures on a list of numbers
Central moments describe how mass sits around the mean, but their units grow with the power of the raw data. The second central moment is a variance in squared units; the third is cubed units; higher powers become hard to compare across variables measured in dollars, milliseconds, or metres. Standardization removes that scale: divide the k-th central moment μ_k by the population standard deviation σ raised to k, and the result α_k is dimensionless.
Then μ_k is the average of those residuals raised to the k-th power, σ is the square root of μ_2, and α_k equals μ_k over σ to the k. Order one is always zero because the average residual is zero by construction. Order two is always one because μ_2 is exactly σ squared.
Order three is the classical moment coefficient of skewness: positive values mean a longer right tail, negative a longer left tail. Order four is raw kurtosis; a normal distribution targets three on this raw scale (excess kurtosis subtracts three separately). Higher orders appear in method-of-moments work and advanced homework.
This page implements the population definition used in many theory texts, not the bias-adjusted sample skewness and excess kurtosis formulas that spreadsheets label SKEW and KURT—those live in a sibling shape calculator.
What you can do with it
Process and sensor screening
Track third-order standardized moment on batches to flag sudden right-skew before it pollutes averages and control charts.
Method-of-moments and research notebooks
Pull α_k for a chosen order without shipping a full stats stack into every micro-service or edge worker.
FAQ
What is the formula for the standardized moment?
α_k equals the k-th population central moment μ_k divided by the population standard deviation σ raised to k. μ is the arithmetic mean; μ_k is the average of (x − μ)^k.
Why do I get an error when every value is the same?
Standard deviation is zero, so dividing by σ^k is undefined.
Is order 3 the same as sample skewness in Excel?
Order 3 is the classical moment coefficient of skewness (population standardization). Excel SKEW uses a bias-adjusted sample formula; use the sibling skewness kurtosis calculator when you need that sample definition.
Is the standardized moment calculator free?
The browser tool is free and runs locally.
What does the standardized moment of order 4 measure?
It is the raw kurtosis, which describes how heavy the tails are. A normal distribution gives 3 on this scale; subtract 3 yourself if you need the excess kurtosis that compares directly with a normal baseline.
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/standardized-moment \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"values":[1,2,3,4,5,12],"order":3}'const res = await fetch("https://api.kit.forhosting.com/stat/standardized-moment", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"values": [
1,
2,
3,
4,
5,
12
],
"order": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/standardized-moment",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"values": [
1,
2,
3,
4,
5,
12
],
"order": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/standardized-moment", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"values":[1,2,3,4,5,12],"order":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"values":[1,2,3,4,5,12],"order":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/standardized-moment", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"values": [
1,
2,
3,
4,
5,
12
],
"order": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.standardized_moment",
"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
Constant series with zero deviation are rejected with invalid_input instead of NaN.
What the standardized moment measures on a list of numbers
Formally, after the arithmetic mean μ equals the sum of the observations divided by n, each residual is x_i minus μ.
How to call the API and what the response contains
Use it from homework checkers that need α_3 or α_4 on a short list, quality screens that watch third-moment skew, method-of-moments pipelines that want a higher order without installing a full statistics package, or any workflow that wants a documented standardized moment rather than a package-specific sample estimator. Send a JSON body with two required fields. values is a non-empty array of finite numbers; aliases numbers, data, list and x are accepted and behave the same. A comma or whitespace separated string is also accepted for simple text forms. order is the moment order k, an integer from one through twenty; aliases k and moment_order work. On success the response includes n as the count of observations, order as the requested k, mean as the arithmetic average, central_moment as the population μ_k, std_dev as the population standard deviation, standardized_moment as α_k, formula as the string α_k equals μ_k over σ to the k, and method set to population_standardized_moment. Failed validation—missing values or order, empty lists, non-finite entries, non-integer order outside one to twenty, lists longer than one hundred thousand points, or a zero standard deviation constant series—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item.
Practical tips, limits, and how this differs from related tools
When teaching, pair a right-skewed example such as one two three four five twelve at order three with the same list at order two (always one) and with a constant series such as seven seven seven to demonstrate the zero deviation error. Prefer this calculator when the assignment or paper literally asks for the standardized moment α_k or the moment coefficient of skewness; prefer the dedicated skewness and kurtosis capability when you need bias-adjusted sample G1 and excess G2 that match Excel SKEW and KURT. Sibling tools in this kit also cover variance, standard deviation, z-scores, and five-number summaries—use those when you want spread or ranking rather than a single shape coefficient. Keep lists within the published size cap so interactive Worker and browser paths stay fast; one hundred thousand finite numbers is ample for classroom sets and most operational samples. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap while students type constant lists by mistake. Prefer sending the raw list once and reading mean, central_moment, std_dev and standardized_moment from one response: one contract, one charge unit, identical results in browser and API. Pin the rounded α_k in your own tests if you build on top of the endpoint so floating-point noise never surprises a regression suite.
Integration use cases
Homework and lab answer keys
Give students a short numeric sample and order k; check their hand-computed standardized moment α_k against a deterministic browser or API result.
Technical questions
What does this standardized moment calculator return?
For a valid list and order k it returns n, order, mean, central_moment (μ_k), std_dev (σ), standardized_moment (α_k = μ_k / σ^k), formula, and method.