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Mathematics · Statistics

Population Variance Calculator

Calculate population variance for six observations.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
Population variance2.916667
Mean3.5
Population standard deviation1.707825

Calculation steps

  1. Mean=3.5.
  2. Average squared deviation=2.9166666666666665.

Understand Population variance

One idea, three depths

Choose how deeply to explain Population variance

Calculate population variance for six observations.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Population variance to answer this question: calculate population variance for six observations? Enter Value 1, Value 2, Value 3, and 3 other inputs; the calculator shows Population variance. For example: 1,2,3,4,5,6 has variance 35/12. The answer tells you Population variance.

Age 15Explain it to a 15-year-oldConnect it to the formula

Variance averages squared distances from the population mean. The rule is σ²=Σ(xᵢ−μ)²/N. Its input values are Value 1, Value 2, Value 3, Value 4, Value 5, Value 6, and the main result is Population variance. For example: 1,2,3,4,5,6 has variance 35/12.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated population variance relation over the valid real-number domain stated below. The implemented relation is σ²=Σ(xᵢ−μ)²/N, evaluated from Value 1, Value 2, Value 3, Value 4, Value 5, Value 6 to produce Population variance. Variance averages squared distances from the population mean. Population variance divides by N; sample variance divides by N−1.

Inputs and valid domain

  • Value 1 must be a finite real number.
  • Value 2 must be a finite real number.
  • Value 3 must be a finite real number.
  • Value 4 must be a finite real number.
  • Value 5 must be a finite real number.
  • Value 6 must be a finite real number.

Important boundary: Population variance divides by N; sample variance divides by N−1.

The formula

σ²=Σ(xᵢ−μ)²/N

How the calculator works through it

It substitutes Value 1, Value 2, Value 3, Value 4, Value 5, Value 6 into the formula and exposes every numerical step above. The main output is Population variance, accompanied by Mean, Population standard deviation.

Read the result correctly

The Population variance is the direct answer to “calculate population variance for six observations.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

1,2,3,4,5,6 has variance 35/12.

Where this model stops being reliable

Population variance divides by N; sample variance divides by N−1.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Sigma symbolsBrowse the mathematics dictionary
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Population variance works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Population variance uses σ²=Σ(xᵢ−μ)²/N. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

  • Averages and representative values

    Representative values help you judge what the Population variance inputs summarise and what the result can legitimately describe.

    Review this foundation about 5 min

Optional enrichment

  • Spread and measurement variation

    Variation is not always part of the Population variance formula, but it helps you judge how stable a reported result may be.

    Review this foundation about 6 min
Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Read Value 1, Value 2, Value 3, Value 4, Value 5, Value 6.
  2. Evaluate the principal relationship: σ²=Σ(xᵢ−μ)²/N.
  3. Return Population variance and check the domain conditions described above.
Python
            from math import *

def population_variance(v1, v2, v3, v4, v5, v6) -> float:
    return ((((((((v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0))) + ((v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) / 6.0)

assert abs(population_variance(1, 2, 3, 4, 5, 6) - 2.9166666666666665) < 1e-6 * max(1.0, abs(2.9166666666666665))
Current calculator valuesUpdates when you change an input above.
C
            #include <assert.h>
#include <math.h>

double population_variance(double v1, double v2, double v3, double v4, double v5, double v6) {
    return ((((((((v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0))) + ((v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) / 6.0);
}

int main(void) {
    const double expected = 2.9166666666666665;
    const double actual = population_variance(1, 2, 3, 4, 5, 6);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
Current calculator valuesUpdates when you change an input above.
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double population_variance(double v1, double v2, double v3, double v4, double v5, double v6) {
    return ((((((((v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0))) + ((v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) / 6.0);
}

int main() {
    constexpr double expected = 2.9166666666666665;
    const double actual = population_variance(1, 2, 3, 4, 5, 6);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Current calculator valuesUpdates when you change an input above.
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double population_variance(double v1, double v2, double v3, double v4, double v5, double v6)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global population_variance
section .text

population_variance:
    push rbp
    mov rbp, rsp
    sub rsp, 928
    movsd [rbp-8], xmm0
    movsd [rbp-16], xmm1
    movsd [rbp-24], xmm2
    movsd [rbp-32], xmm3
    movsd [rbp-40], xmm4
    movsd [rbp-48], xmm5
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-160], xmm0
    movsd xmm0, [rbp-160]
    addsd xmm0, [rbp-24]
    movsd [rbp-152], xmm0
    movsd xmm0, [rbp-152]
    addsd xmm0, [rbp-32]
    movsd [rbp-144], xmm0
    movsd xmm0, [rbp-144]
    addsd xmm0, [rbp-40]
    movsd [rbp-136], xmm0
    movsd xmm0, [rbp-136]
    addsd xmm0, [rbp-48]
    movsd [rbp-128], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-168], xmm0
    movsd xmm0, [rbp-128]
    divsd xmm0, [rbp-168]
    movsd [rbp-120], xmm0
    movsd xmm0, [rbp-8]
    subsd xmm0, [rbp-120]
    movsd [rbp-112], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-224], xmm0
    movsd xmm0, [rbp-224]
    addsd xmm0, [rbp-24]
    movsd [rbp-216], xmm0
    movsd xmm0, [rbp-216]
    addsd xmm0, [rbp-32]
    movsd [rbp-208], xmm0
    movsd xmm0, [rbp-208]
    addsd xmm0, [rbp-40]
    movsd [rbp-200], xmm0
    movsd xmm0, [rbp-200]
    addsd xmm0, [rbp-48]
    movsd [rbp-192], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-232], xmm0
    movsd xmm0, [rbp-192]
    divsd xmm0, [rbp-232]
    movsd [rbp-184], xmm0
    movsd xmm0, [rbp-8]
    subsd xmm0, [rbp-184]
    movsd [rbp-176], xmm0
    movsd xmm0, [rbp-112]
    mulsd xmm0, [rbp-176]
    movsd [rbp-104], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-296], xmm0
    movsd xmm0, [rbp-296]
    addsd xmm0, [rbp-24]
    movsd [rbp-288], xmm0
    movsd xmm0, [rbp-288]
    addsd xmm0, [rbp-32]
    movsd [rbp-280], xmm0
    movsd xmm0, [rbp-280]
    addsd xmm0, [rbp-40]
    movsd [rbp-272], xmm0
    movsd xmm0, [rbp-272]
    addsd xmm0, [rbp-48]
    movsd [rbp-264], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-304], xmm0
    movsd xmm0, [rbp-264]
    divsd xmm0, [rbp-304]
    movsd [rbp-256], xmm0
    movsd xmm0, [rbp-16]
    subsd xmm0, [rbp-256]
    movsd [rbp-248], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-360], xmm0
    movsd xmm0, [rbp-360]
    addsd xmm0, [rbp-24]
    movsd [rbp-352], xmm0
    movsd xmm0, [rbp-352]
    addsd xmm0, [rbp-32]
    movsd [rbp-344], xmm0
    movsd xmm0, [rbp-344]
    addsd xmm0, [rbp-40]
    movsd [rbp-336], xmm0
    movsd xmm0, [rbp-336]
    addsd xmm0, [rbp-48]
    movsd [rbp-328], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-368], xmm0
    movsd xmm0, [rbp-328]
    divsd xmm0, [rbp-368]
    movsd [rbp-320], xmm0
    movsd xmm0, [rbp-16]
    subsd xmm0, [rbp-320]
    movsd [rbp-312], xmm0
    movsd xmm0, [rbp-248]
    mulsd xmm0, [rbp-312]
    movsd [rbp-240], xmm0
    movsd xmm0, [rbp-104]
    addsd xmm0, [rbp-240]
    movsd [rbp-96], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-432], xmm0
    movsd xmm0, [rbp-432]
    addsd xmm0, [rbp-24]
    movsd [rbp-424], xmm0
    movsd xmm0, [rbp-424]
    addsd xmm0, [rbp-32]
    movsd [rbp-416], xmm0
    movsd xmm0, [rbp-416]
    addsd xmm0, [rbp-40]
    movsd [rbp-408], xmm0
    movsd xmm0, [rbp-408]
    addsd xmm0, [rbp-48]
    movsd [rbp-400], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-440], xmm0
    movsd xmm0, [rbp-400]
    divsd xmm0, [rbp-440]
    movsd [rbp-392], xmm0
    movsd xmm0, [rbp-24]
    subsd xmm0, [rbp-392]
    movsd [rbp-384], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-496], xmm0
    movsd xmm0, [rbp-496]
    addsd xmm0, [rbp-24]
    movsd [rbp-488], xmm0
    movsd xmm0, [rbp-488]
    addsd xmm0, [rbp-32]
    movsd [rbp-480], xmm0
    movsd xmm0, [rbp-480]
    addsd xmm0, [rbp-40]
    movsd [rbp-472], xmm0
    movsd xmm0, [rbp-472]
    addsd xmm0, [rbp-48]
    movsd [rbp-464], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-504], xmm0
    movsd xmm0, [rbp-464]
    divsd xmm0, [rbp-504]
    movsd [rbp-456], xmm0
    movsd xmm0, [rbp-24]
    subsd xmm0, [rbp-456]
    movsd [rbp-448], xmm0
    movsd xmm0, [rbp-384]
    mulsd xmm0, [rbp-448]
    movsd [rbp-376], xmm0
    movsd xmm0, [rbp-96]
    addsd xmm0, [rbp-376]
    movsd [rbp-88], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-568], xmm0
    movsd xmm0, [rbp-568]
    addsd xmm0, [rbp-24]
    movsd [rbp-560], xmm0
    movsd xmm0, [rbp-560]
    addsd xmm0, [rbp-32]
    movsd [rbp-552], xmm0
    movsd xmm0, [rbp-552]
    addsd xmm0, [rbp-40]
    movsd [rbp-544], xmm0
    movsd xmm0, [rbp-544]
    addsd xmm0, [rbp-48]
    movsd [rbp-536], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-576], xmm0
    movsd xmm0, [rbp-536]
    divsd xmm0, [rbp-576]
    movsd [rbp-528], xmm0
    movsd xmm0, [rbp-32]
    subsd xmm0, [rbp-528]
    movsd [rbp-520], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-632], xmm0
    movsd xmm0, [rbp-632]
    addsd xmm0, [rbp-24]
    movsd [rbp-624], xmm0
    movsd xmm0, [rbp-624]
    addsd xmm0, [rbp-32]
    movsd [rbp-616], xmm0
    movsd xmm0, [rbp-616]
    addsd xmm0, [rbp-40]
    movsd [rbp-608], xmm0
    movsd xmm0, [rbp-608]
    addsd xmm0, [rbp-48]
    movsd [rbp-600], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-640], xmm0
    movsd xmm0, [rbp-600]
    divsd xmm0, [rbp-640]
    movsd [rbp-592], xmm0
    movsd xmm0, [rbp-32]
    subsd xmm0, [rbp-592]
    movsd [rbp-584], xmm0
    movsd xmm0, [rbp-520]
    mulsd xmm0, [rbp-584]
    movsd [rbp-512], xmm0
    movsd xmm0, [rbp-88]
    addsd xmm0, [rbp-512]
    movsd [rbp-80], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-704], xmm0
    movsd xmm0, [rbp-704]
    addsd xmm0, [rbp-24]
    movsd [rbp-696], xmm0
    movsd xmm0, [rbp-696]
    addsd xmm0, [rbp-32]
    movsd [rbp-688], xmm0
    movsd xmm0, [rbp-688]
    addsd xmm0, [rbp-40]
    movsd [rbp-680], xmm0
    movsd xmm0, [rbp-680]
    addsd xmm0, [rbp-48]
    movsd [rbp-672], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-712], xmm0
    movsd xmm0, [rbp-672]
    divsd xmm0, [rbp-712]
    movsd [rbp-664], xmm0
    movsd xmm0, [rbp-40]
    subsd xmm0, [rbp-664]
    movsd [rbp-656], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-768], xmm0
    movsd xmm0, [rbp-768]
    addsd xmm0, [rbp-24]
    movsd [rbp-760], xmm0
    movsd xmm0, [rbp-760]
    addsd xmm0, [rbp-32]
    movsd [rbp-752], xmm0
    movsd xmm0, [rbp-752]
    addsd xmm0, [rbp-40]
    movsd [rbp-744], xmm0
    movsd xmm0, [rbp-744]
    addsd xmm0, [rbp-48]
    movsd [rbp-736], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-776], xmm0
    movsd xmm0, [rbp-736]
    divsd xmm0, [rbp-776]
    movsd [rbp-728], xmm0
    movsd xmm0, [rbp-40]
    subsd xmm0, [rbp-728]
    movsd [rbp-720], xmm0
    movsd xmm0, [rbp-656]
    mulsd xmm0, [rbp-720]
    movsd [rbp-648], xmm0
    movsd xmm0, [rbp-80]
    addsd xmm0, [rbp-648]
    movsd [rbp-72], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-840], xmm0
    movsd xmm0, [rbp-840]
    addsd xmm0, [rbp-24]
    movsd [rbp-832], xmm0
    movsd xmm0, [rbp-832]
    addsd xmm0, [rbp-32]
    movsd [rbp-824], xmm0
    movsd xmm0, [rbp-824]
    addsd xmm0, [rbp-40]
    movsd [rbp-816], xmm0
    movsd xmm0, [rbp-816]
    addsd xmm0, [rbp-48]
    movsd [rbp-808], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-848], xmm0
    movsd xmm0, [rbp-808]
    divsd xmm0, [rbp-848]
    movsd [rbp-800], xmm0
    movsd xmm0, [rbp-48]
    subsd xmm0, [rbp-800]
    movsd [rbp-792], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-904], xmm0
    movsd xmm0, [rbp-904]
    addsd xmm0, [rbp-24]
    movsd [rbp-896], xmm0
    movsd xmm0, [rbp-896]
    addsd xmm0, [rbp-32]
    movsd [rbp-888], xmm0
    movsd xmm0, [rbp-888]
    addsd xmm0, [rbp-40]
    movsd [rbp-880], xmm0
    movsd xmm0, [rbp-880]
    addsd xmm0, [rbp-48]
    movsd [rbp-872], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-912], xmm0
    movsd xmm0, [rbp-872]
    divsd xmm0, [rbp-912]
    movsd [rbp-864], xmm0
    movsd xmm0, [rbp-48]
    subsd xmm0, [rbp-864]
    movsd [rbp-856], xmm0
    movsd xmm0, [rbp-792]
    mulsd xmm0, [rbp-856]
    movsd [rbp-784], xmm0
    movsd xmm0, [rbp-72]
    addsd xmm0, [rbp-784]
    movsd [rbp-64], xmm0
    mov rax, 0x4018000000000000
    movq xmm0, rax
    movsd [rbp-920], xmm0
    movsd xmm0, [rbp-64]
    divsd xmm0, [rbp-920]
    movsd [rbp-56], xmm0
    movsd xmm0, [rbp-56]
    leave
    ret
Current calculator valuesUpdates when you change an input above.
MATLAB
            function result = population_variance(v1, v2, v3, v4, v5, v6)
    result = ((((((((v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0))) + ((v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) / 6.0);
end
Current calculator valuesUpdates when you change an input above.
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[v1_, v2_, v3_, v4_, v5_, v6_] := ((((((((v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v1 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0))) + ((v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v2 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v3 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v4 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v5 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) + ((v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)) * (v6 - ((((((v1 + v2) + v3) + v4) + v5) + v6) / 6.0)))) / 6.0);
Current calculator valuesUpdates when you change an input above.

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Calculate variance-to-mean dispersion index from observed count variance and observed count mean.

Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations

Standards, reading and academic references

Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.

Introductory Statistics 2e

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Cite this book
APA 7
Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
MLA 9
Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
Chicago author-date
Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.

OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

Organizational author: MW SysArc. Published:. Canonical page:https://math.mwsysarc.com/statistics/population-variance.

APA 7

MW SysArc. (2026, July 21). Population Variance Calculator. MW SysArc Tools. https://math.mwsysarc.com/statistics/population-variance

MLA 9

MW SysArc. “Population Variance Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/statistics/population-variance. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Population Variance Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/statistics/population-variance.

Harvard

MW SysArc (2026) ‘Population Variance Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/statistics/population-variance (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_population_variance_2026,
  author = {{MW SysArc}},
  title = {Population Variance Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/statistics/population-variance},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Population Variance Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/statistics/population-variance
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Population variance do?

Calculate population variance for six observations.

How does the Population variance work?

The calculator applies σ²=Σ(xᵢ−μ)²/N. Variance averages squared distances from the population mean.

What can I learn from the Population variance?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

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