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Mathematics · Statistics

Standard Deviation Calculator

Calculate population and sample standard deviation for six values.

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 standard deviation2.134375
Sample standard deviation2.33809
Mean4.666667

Calculation steps

  1. Mean = 4.666666666666667.
  2. Sum of squared deviations = 27.333333333333332.
  3. Population σ = √(27.333333333333332 ÷ 6) = 2.1343747458109497; sample s = √(27.333333333333332 ÷ 5) = 2.3380903889000244.

Understand Standard deviation

One idea, three depths

Choose how deeply to explain Standard deviation

Calculate population and sample standard deviation for six values.

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

Imagine using Standard deviation to answer this question: calculate population and sample standard deviation for six values? Enter Value 1, Value 2, Value 3, and 3 other inputs; the calculator shows Population standard deviation. For example: For 2, 4, 4, 4, 5, 9, the population standard deviation is about 2.13. The answer tells you Population standard deviation.

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

Standard deviation measures typical distance from the mean. Sample deviation uses N − 1 to estimate population spread from a sample. The rule is Population σ = √[Σ(x − 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 standard deviation. For example: For 2, 4, 4, 4, 5, 9, the population standard deviation is about 2.13.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated standard deviation relation over the valid real-number domain stated below. The implemented relation is Population σ = √[Σ(x − μ)² ÷ N], evaluated from Value 1, Value 2, Value 3, Value 4, Value 5, Value 6 to produce Population standard deviation. Standard deviation measures typical distance from the mean. Sample deviation uses N − 1 to estimate population spread from a sample. Choose population or sample deviation according to what the entered data represents.

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: Choose population or sample deviation according to what the entered data represents.

The formula

Population σ = √[Σ(x − 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 standard deviation, accompanied by Sample standard deviation, Mean.

Read the result correctly

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

A worked check

For 2, 4, 4, 4, 5, 9, the population standard deviation is about 2.13.

Where this model stops being reliable

Choose population or sample deviation according to what the entered data represents.

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 symbolsStandard deviationVarianceZ-scoreBrowse 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 Standard deviation works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Standard deviation uses Population σ = √[Σ(x − 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 Standard deviation 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 Standard deviation 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: Population σ = √[Σ(x − μ)² ÷ N].
  3. Return Population standard deviation and check the domain conditions described above.
Python
            from math import *

def standard_deviation(v1, v2, v3, v4, v5, v6) -> float:
    return sqrt(((((((((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(standard_deviation(2, 4, 4, 4, 5, 9) - 2.1343747458109497) < 1e-6 * max(1.0, abs(2.1343747458109497))
Current calculator valuesUpdates when you change an input above.
C
            #include <assert.h>
#include <math.h>

double standard_deviation(double v1, double v2, double v3, double v4, double v5, double v6) {
    return sqrt(((((((((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 = 2.1343747458109497;
    const double actual = standard_deviation(2, 4, 4, 4, 5, 9);
    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 standard_deviation(double v1, double v2, double v3, double v4, double v5, double v6) {
    return std::sqrt(((((((((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 = 2.1343747458109497;
    const double actual = standard_deviation(2, 4, 4, 4, 5, 9);
    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 standard_deviation(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 standard_deviation
section .text

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

APA 7

MW SysArc. (2026, July 21). Standard Deviation Calculator. MW SysArc Tools. https://math.mwsysarc.com/statistics/standard-deviation-calculator

MLA 9

MW SysArc. “Standard Deviation Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/statistics/standard-deviation-calculator. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Standard Deviation Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/statistics/standard-deviation-calculator.

Harvard

MW SysArc (2026) ‘Standard Deviation Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/statistics/standard-deviation-calculator (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_standard_deviation_2026,
  author = {{MW SysArc}},
  title = {Standard Deviation Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/statistics/standard-deviation-calculator},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Standard Deviation Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/statistics/standard-deviation-calculator
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Standard deviation do?

Calculate population and sample standard deviation for six values.

How does the Standard deviation work?

The calculator applies Population σ = √[Σ(x − μ)² ÷ N]. Standard deviation measures typical distance from the mean. Sample deviation uses N − 1 to estimate population spread from a sample.

What can I learn from the Standard deviation?

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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