Pearson mode skewness calculator
With a mean of 12, a mode of 10 and a standard deviation of 4, Pearson's mode skewness is 0.5, a right-skewed shape. Type the three summary numbers, no raw data needed, and see how far the mean sits from the peak. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mean
- 12
- Calculation mode
- 10
- Std
- 4
- Skewness
- 0.5
- Interpretation
- right_skewed
Pearson's first coefficient of skewness, often called the mode skewness coefficient, measures how far the arithmetic mean sits from the mode in units of the standard deviation. The defining formula is simple and universal: subtract the mode from the mean, then divide by sigma. A positive result means the mean is larger than the mode and the bulk of the distribution lies to the left of a longer right tail. A negative result means the opposite left-skewed shape. Zero means the mean and mode coincide under the given scale of spread. This calculator accepts those three summary numbers only—no raw sample is required—so textbooks, reports, and dashboards that already publish mean, mode, and standard deviation can obtain a standardized asymmetry score in one request.
What Pearson mode skewness measures and how the formula works
Start from three descriptive numbers that almost every statistics course introduces in the first weeks: the arithmetic mean, which is the balance point of the data; the mode, which is the most frequent value or the peak of a unimodal density; and the standard deviation, which is the typical distance of observations from the mean. Pearson's first coefficient of skewness asks a single question in standardized units: how many standard deviations separate the mean from the mode? Algebraically that is Sk1 equals mean minus mode, all divided by sigma.
Because the denominator is a scale measure, the coefficient is unitless and comparable across variables measured in different units—test scores, waiting times, income brackets, or sensor readings. When the distribution is symmetric and unimodal, mean and mode coincide (or nearly so), so Sk1 is zero. When a long right tail pulls the mean above the mode, Sk1 is positive and we speak of right skew or positive skew.
When a long left tail pulls the mean below the mode, Sk1 is negative. The magnitude is easy to read: a value near one means the mean–mode gap is about one standard deviation, which is a substantial asymmetry for classroom and reporting purposes. The only hard failure among well-formed numbers is sigma equal to zero: without spread there is no meaningful unit in which to express the mean–mode gap, and the coefficient is undefined.
Where a Pearson mode skewness calculator earns its keep
Business and operations teams often already store those three aggregates in warehouse summary tables for product categories, call-center queues, or lead times, and they want a standardized asymmetry score without reloading every raw event. Quality and process-monitoring groups compare shape across lines or shifts when only roll-up statistics are retained after archival, and mode-based skewness is a familiar classroom bridge to more advanced shape measures.
Content sites that rank for pearson mode skewness calculator benefit from a live tool whose free path matches the paid automation path byte-for-byte on the same pure module. Failed validation does not consume balance. If you later need sample skewness and kurtosis from a raw series, median-based Pearson second coefficient, or full descriptive statistics for every CSV column, treat those as separate search intents and landings rather than overloading this mode-skewness page.
What you can do with it
Homework and exam skewness drills
Students enter mean, mode, and standard deviation from a textbook table and verify Sk1 = (mean − mode) / σ against the answer key without recomputing aggregates by hand.
Dashboard shape from stored summaries
Operations and analytics teams score right or left skew from warehouse roll-ups that already publish mean, mode, and sigma without reloading the full event stream.
LMS and worksheet auto-grading
Mark Pearson first-coefficient items with a deterministic oracle that rejects zero standard deviation and never invents a coefficient when the scale is undefined.
FAQ
What is Pearson's first (mode) coefficient of skewness?
It is Sk1 = (mean − mode) / standard deviation. Positive values indicate right skew (mean above the mode); negative values indicate left skew; zero means mean and mode coincide under the given scale.
Does this compute skewness from a raw data list?
For sample skewness and kurtosis from a numeric series, use a dedicated sample-shape calculator instead.
How do I read a positive or negative result?
A positive value means the mean is above the mode and a long tail stretches to the right. A negative value means the long tail is on the left, and zero means mean and mode coincide.
Why is the gap divided by the standard deviation?
So the result has no units and you can compare shapes across variables measured differently, such as test scores, waiting times or income brackets. A value near 1 already signals a substantial asymmetry.
What happens if the standard deviation is zero?
The coefficient is undefined, because without spread there is no unit in which to express the gap between mean and mode. The calculator rejects the input instead of inventing a number.
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/pearson-mode-skewness \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mean":12,"mode":10,"std":4}'const res = await fetch("https://api.kit.forhosting.com/stat/pearson-mode-skewness", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mean": 12,
"mode": 10,
"std": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/pearson-mode-skewness",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mean": 12,
"mode": 10,
"std": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/pearson-mode-skewness", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mean":12,"mode":10,"std":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mean":12,"mode":10,"std":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/pearson-mode-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
{
"mean": 12,
"mode": 10,
"std": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.pearson_mode_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
When the standard deviation is zero the coefficient is undefined and the task fails with invalid_input instead of inventing a number.
What Pearson mode skewness measures and how the formula works
This endpoint does not estimate mean, mode, or sigma from a raw list; it trusts the three numbers you already computed or were given in a problem statement, then applies the division exactly once. That design matches exam problems, published summary tables, and APIs that store aggregates rather than full samples. Every step is deterministic floating-point math with no random seeds, no wall-clock timestamps, and no network calls, so identical inputs always produce identical JSON.
Input contract, response shape, and error behaviour
A zero or near-zero standard deviation fails with a clear message that Pearson mode skewness is undefined when sigma equals zero. Wire the free browser form for homework and quick sanity checks, then call the identical contract from scripts when you need bulk scoring of published summary rows. POST a JSON object with three required finite numbers: mean, mode, and std. The standard deviation field also accepts the aliases sd, standard_deviation, and sigma so textbook wording and API naming both work; send only one of those aliases per request. 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 field and are not billed. On success the body echoes the cleaned mean, mode, and std; returns skewness as the coefficient Sk1 rounded for stable cross-runtime goldens; restates the formula string Sk1 equals open parenthesis mean minus mode close parenthesis divided by sigma; and attaches a coarse interpretation label: right_skewed when skewness is positive, left_skewed when negative, and symmetric when the cleaned coefficient is zero. Example: mean twelve, mode ten, and standard deviation four yields skewness zero point five because twelve minus ten is two and two divided by four is one half, with interpretation right_skewed. Another check: mean five, mode eight, and standard deviation three yields negative one, left_skewed, because five minus eight is negative three and negative three divided by three is negative one. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where a Pearson mode skewness calculator earns its keep
Secondary-school and college statistics courses constantly assign Pearson skewness drills that start from mean, mode, and standard deviation printed in a table rather than from a full data set; a single trusted oracle that returns cleaned JSON speeds grading, demo setup, and self-check worksheets.
Technical questions
Which inputs does the calculator require?
Three finite numbers: mean, mode, and std (aliases sd, standard_deviation, or sigma). Send only one standard-deviation field. Missing or non-finite values fail with invalid_input.
What happens if the standard deviation is zero?
The task fails with invalid_input. Pearson mode skewness divides by sigma, so the coefficient is undefined when there is no spread and the call is not billed.
What does the Pearson mode skewness calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.