Standard distance of points
Score how compact a cloud of points is with one number: (0,0), (3,4) and (6,0) have their mean center at (3, 1.333333333333) and a standard distance of 3.091206165165. It is the map counterpart of the standard deviation. Free in your browser.
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Standard distance is the spatial counterpart of standard deviation: it measures how tightly a set of map or survey points clusters around their mean center. GIS analysts, urban planners, epidemiologists, and location-intelligence teams use it as a single-number compactness score for store locations, crime incidents, sensor deployments, or wildlife sightings. This calculator accepts planar coordinates (projected eastings and northings, local survey units, or lat/lon treated as planar numbers), finds the mean center as the average of x and of y, then returns the root-mean-square Euclidean distance of every point from that center—the classical Bachi-style standard distance with divisor n. A single point yields zero dispersion; larger clouds inflate the value in the same units as your coordinates. No network calls, no models, and no geodesic approximation unless you project first.
What standard distance measures and how the mean center is defined
In one-dimensional statistics, the standard deviation is the root-mean-square deviation of observations from their arithmetic mean. In the plane, the natural analogue is the standard distance of a point set: replace each observation with a coordinate pair, replace the scalar mean with the mean center, and replace absolute deviation with Euclidean distance. Because the formula is rotationally invariant, rotating the whole cloud about the origin does not change SD, which makes it a pure compactness score rather than a directional ellipse.
Use projected map meters when you need SD in real ground units; treating raw geographic longitude and latitude as planar x and y is convenient for relative ranking of nearby clusters but is not a geodesic radius. When every point coincides, SD is zero—the cloud has no spatial spread. When points form a circle of radius r about the center, SD approaches r as the sample fills the circle.
This capability always uses the classical population form with divisor n (not n minus one), matching GIS textbooks and common spatial-analysis software, so a three-point triangle and a thousand-point survey share one consistent definition.
When standard distance is worth computing
Retail site selection teams score candidate store clusters by how tightly customer points pack around a proposed mean center—lower standard distance often means a walkable trade area. Public-health dashboards compare outbreak compactness across weeks without committing to a full spatial-statistics stack. Wildlife ecologists summarize telemetry bursts as a single dispersion length before running home-range models.
Surveyors and civil engineers validate that control points or stake-out marks stay within an expected tolerance cloud. Teaching platforms need a deterministic oracle for spatial statistics homework that matches the textbook n-divisor formula and rejects empty input instead of inventing a silent zero.
What you can do with it
Retail and service catchment compactness
Score how tightly customer or store locations cluster around their mean center to compare walkable trade areas and site shortlists.
Epidemiology and incident dispersion
Summarize case or crime point clouds as a single spatial standard deviation for week-over-week compactness dashboards.
Survey control and classroom spatial stats
Check that control points stay within a tolerance cloud, or grade standard-distance homework with a deterministic n-divisor oracle.
FAQ
How is standard distance calculated from a set of points?
Find the mean center as the average of all x and all y.
Do I need projected coordinates?
For SD in ground meters, yes—use a projected CRS or local survey units.
What does a standard distance of zero mean?
That the points have no spread at all: every point sits on the same spot. A single point also gives zero, because it coincides with its own mean center.
Why does it divide by n and not by n minus one?
It uses the classical population form found in GIS textbooks and common spatial-analysis software. That way a three-point triangle and a survey of thousands of points share one consistent definition.
Does rotating the points change the result?
No. Turning the whole cloud around the origin leaves the standard distance unchanged, which makes it a pure compactness score. It does not describe direction the way a directional ellipse does.
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/geo/standard-distance \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"points":[{"x":0,"y":0},{"x":3,"y":4},{"x":6,"y":0}]}'const res = await fetch("https://api.kit.forhosting.com/geo/standard-distance", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"points": [
{
"x": 0,
"y": 0
},
{
"x": 3,
"y": 4
},
{
"x": 6,
"y": 0
}
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geo/standard-distance",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"points": [
{
"x": 0,
"y": 0
},
{
"x": 3,
"y": 4
},
{
"x": 6,
"y": 0
}
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geo/standard-distance", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"points":[{"x":0,"y":0},{"x":3,"y":4},{"x":6,"y":0}]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"points":[{"x":0,"y":0},{"x":3,"y":4},{"x":6,"y":0}]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geo/standard-distance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"points": [
{
"x": 0,
"y": 0
},
{
"x": 3,
"y": 4
},
{
"x": 6,
"y": 0
}
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geo.standard_distance",
"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
What standard distance measures and how the mean center is defined
Formally, if your n points are (x_i, y_i), the mean center is mean_x equals the average of all x_i and mean_y equals the average of all y_i. The standard distance SD equals the square root of the average of (x_i minus mean_x) squared plus (y_i minus mean_y) squared over n. That identity is exactly the square root of the sum of the x-variance and y-variance when both variances use the population divisor n.
Input contract, response fields, and validation rules
The service caps the list at ten thousand points to keep worker memory and latency bounded. Example: points (0,0), (3,4), and (6,0) have mean center (3, 4/3) and standard distance equal to the square root of eighty-six over nine. Optional fields are omitted rather than null. POST a JSON body with a required points array of objects, each carrying finite numeric x and y. Aliases locations, coordinates, and vertices are accepted for the array; each point may supply lon, lng, or easting instead of x and lat or northing instead of y. At least one point is required; empty arrays, non-arrays, and missing points fail with invalid_input and are not billed. Each coordinate may arrive as a JSON number or a numeric string; non-finite values, blank strings, and non-numeric tokens fail naming the point index. On success the response echoes n, mean_x, mean_y, and a nested mean_center object with the same coordinates; sum_squared_distance is the raw sum of squared Euclidean deviations before dividing by n; variance_x and variance_y are the population variances of each axis; standard_distance is the primary compactness metric; and formula restates the identity for display next to results. Pair the numeric SD with your own map layer if you need a circle of radius standard_distance drawn about the mean center; this endpoint returns measures only, not GeoJSON or map tiles.
When a standard-distance API earns its place in a pipeline
Project your coordinates first when you need meters; reuse the same points array you already send to bounding-box or convex-hull tools; and pin the rounded standard_distance in automated tests so a future rewrite cannot change the definition unnoticed. The free browser path uses the identical solve module as the API, which keeps marketing widgets and production automation aligned on every edge case from single-point zero to multi-thousand point clouds.
Technical questions
Is standard distance the same as standard deviation?
It is the two-dimensional spatial analogue: RMS Euclidean distance from the mean center, equal to sqrt(variance_x + variance_y) with population variances.
What if I send only one point?
The mean center is that point and standard_distance is 0. Empty arrays and missing points fail with invalid_input and are not billed.
What does the standard distance calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.