Moment coefficient of skewness calculator
For 1, 2, 3, 4, 5 and 12 the mean is 4.5, the third central moment is 60 and the moment coefficient of skewness is above 1, so the sample is labelled right-skewed. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Numbers
- 1, 2, 3, 4, 5, 12
- Quantity
- 6
- Mean
- 4.5
- Variance
- 12.9166666667
- Std dev
- 3.5939764421
The moment coefficient of skewness is the classical Pearson measure of asymmetry for a numeric sample: the third central moment divided by the cube of the population standard deviation. Positive values mean a longer right tail relative to the left, negative values a longer left tail, and values near zero an approximately symmetric profile around the mean. This capability takes a non-empty list of finite numbers, rejects empty lists and constant series where the standard deviation is zero, and returns the mean, population variance, standard deviation, third central moment, moment coefficient of skewness, and a short shape label in one deterministic call.
What the moment coefficient of skewness means on a list of numbers
Descriptive asymmetry statistics answer questions that mean and standard deviation alone cannot. After computing the arithmetic mean μ of n observations, the second central moment m2 is the average of squared deviations from that mean, and the population standard deviation σ is the square root of m2. The third central moment m3 is the average of cubed deviations: positive cubes dominate when large values sit far to the right of the mean, and negative cubes dominate when large deviations sit to the left.
The moment coefficient of skewness is exactly γ1 equals m3 divided by σ cubed, which is the standardized third central moment of the empirical distribution. Because the divisor uses the population scale σ, the result is unitless and comparable across different measurement scales. A right-skewed income or response-time sample tends to produce γ1 greater than zero; a left-skewed ceiling-bounded score set tends to produce γ1 less than zero; a balanced ladder of equally spaced values sits near zero.
This definition is the classical Pearson form taught in many probability and statistics courses. It is deliberately different from the bias-adjusted sample skewness G1 that spreadsheets often label SKEW: G1 multiplies standardized cubes by a finite-sample correction factor and uses the sample standard deviation with n minus one in the variance denominator. When your assignment, paper, or library documentation asks specifically for the moment coefficient of skewness or the standardized third central moment, this page is the matching calculator.
Practical tips, limits, and related searches
When teaching, pair a right-skewed example such as one two three four five twelve with a near-symmetric ladder such as one two three four five and with a constant series such as seven seven seven to demonstrate the zero standard deviation error, and with an empty list to show the empty-array rejection. Prefer the moment coefficient when a textbook or research methods section writes γ1 equals μ3 over σ cubed; prefer a sample skewness and kurtosis tool when you need Excel-style G1 together with excess kurtosis.
Shape labels use a simple plus or minus zero point five threshold for quick reading—they are not formal hypothesis tests or bootstrap confidence intervals. Sibling capabilities in this kit compute sample skewness with kurtosis, mean median mode, or quartiles and fences; this page is specifically for people searching for a moment coefficient of skewness calculator on a plain list of numbers, not a multi-column CSV profile and not a joint kurtosis report.
What you can do with it
Process and sensor screening
Flag batches that suddenly become right-skewed or left-skewed using the classical standardized third moment before they pollute downstream averages.
Finance and risk notebooks
Summarize return series asymmetry with the moment coefficient of skewness without shipping a full stats stack into every micro-service.
FAQ
How is moment skewness defined?
γ1 = m3 / σ³ where m3 is the third central moment (average of cubed deviations from the mean) and σ is the population standard deviation sqrt(m2).
What if every value is the same?
Population standard deviation is zero, so the standardized third moment is undefined.
What if the array is empty?
Provide at least one finite number; constant single-point lists still fail because standard deviation is zero.
Is the moment coefficient of skewness calculator free?
The browser tool is free and runs locally.
Why does my spreadsheet's skewness function give a different number?
Spreadsheets usually report the bias-adjusted sample skewness, which uses n minus one and a correction factor. This calculator gives the classical Pearson moment form, based on the population standard deviation.
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/moment-skewness \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"numbers":[1,2,3,4,5,12]}'const res = await fetch("https://api.kit.forhosting.com/stat/moment-skewness", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"numbers": [
1,
2,
3,
4,
5,
12
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/moment-skewness",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"numbers": [
1,
2,
3,
4,
5,
12
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/moment-skewness", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"numbers":[1,2,3,4,5,12]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"numbers":[1,2,3,4,5,12]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/moment-skewness", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"numbers": [
1,
2,
3,
4,
5,
12
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.moment_skewness",
"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
How to call the API and what comes back
Numeric strings inside the array are coerced; a comma or whitespace separated string is also accepted for simple text forms. Use it from homework checkers that require the Pearson moment form, quality-control screens for asymmetric process data, finance notebooks that report classical skewness of returns, dashboards that need a quick unitless asymmetry flag, or any workflow that wants the standardized third central moment without installing a full statistics package on every machine. Send a JSON body with one required field named numbers: a non-empty array of finite numbers. Aliases values, data, and list are accepted and behave the same. On success the response includes count as n, numbers as an ascending sorted copy for inspection, mean, variance as the population second central moment m2, std_dev as the square root of m2, third_central_moment as m3, moment_skewness as γ1 equals m3 over σ cubed, skew_shape as one of right_skewed, left_skewed, or approximately_symmetric using a simple plus or minus zero point five threshold, plus min and max. Failed validation—missing numbers, an empty list, a zero standard deviation constant series, non-finite entries, or a list longer than one hundred thousand values—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
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 paste empty cells by mistake. Prefer sending the raw list once and reading mean, variance, std_dev, third_central_moment, and moment_skewness from one response instead of calling separate tools: one contract, one charge unit, identical results in browser and API.
Integration use cases
Homework and lab answer keys
Give students a short numeric sample; check their hand-computed Pearson moment coefficient of skewness γ1 = m3/σ³ against a deterministic browser or API result.
Technical questions
What does this moment coefficient of skewness calculator return?
For a valid list it returns count, sorted numbers, mean, population variance, std_dev, third_central_moment, moment_skewness (γ1), skew_shape, min and max.