Befolkningsvarians Kalkylator

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What Is Population Variance?

Population variance, written σ², describes the spread of a group when you have measured every member of it. There is nothing to estimate, so you take the average of the squared deviations from the mean exactly as it is: divide by N, the number of values. Typical examples are the marks of every student in one class, the weight of every parcel in a single shipment you weighed in full, or one company's monthly sales for a completed year. The key question is not how big your data set is, but whether it is the whole group you want to describe.

Worked Example: A Complete Tutorial Group

A tutor has exactly six students and wants to describe how spread out their test scores are. These six scores are the entire group, so population variance applies.

Data (marks):70, 85, 75, 90, 65, 95

  1. Mean. x̄ = 480 / 6 = 80
  2. Deviations and squares. Subtract the mean from each value and square the result:
xᵢxᵢ − x̄(xᵢ − x̄)²
70−10100
85525
75−525
9010100
65−15225
9515225
Sum0700
  1. Sum of squares. Σ(xᵢ − x̄)² = 700
  2. Population variance. σ² = 700 / 6 = 116.6667; population standard deviation σ = √116.6667 = 10.8012

Population variance σ² = 700 / 6 ≈ 116.67, so the population standard deviation is σ ≈ 10.80 marks. If the same six scores were treated as a sample, s² = 700 / 5 = 140; the gap between the two answers is largest for small groups like this one.

Formula

Population variance uses the population mean μ and divides by N:

Population Variance

σ² = Σ(xᵢ − μ)² / N
  • σ²Population variance
  • xᵢEach value in the population
  • μPopulation mean
  • NNumber of values in the population

Population Standard Deviation

σ = √σ²
  • σPopulation standard deviation, in the original units

How to Calculate Population Variance Step by Step

  1. Confirm you have the entire population.Every member of the group you are describing must be in the data.
  2. Find the population mean (μ).Add all values and divide by N.
  3. Compute deviations.Subtract μ from each value.
  4. Square each deviation.Calculate (xᵢ − μ)².
  5. Add the squared deviations.Σ(xᵢ − μ)² is the sum of squares.
  6. Divide by N.σ² = Σ(xᵢ − μ)² / N.

More Population Variance Examples

1All Students in a Class

Data:78, 82, 91, 67, 85, 73, 88, 79
Solution:Mean μ = 80.375. Sum of squared deviations = 435.875.
Answer:Population variance (σ²) = 435.875 / 8 ≈ 54.48

2Every Product in a Small Batch

Data:10.2, 10.0, 10.1, 10.3, 10.0
Solution:Mean μ = 10.12. Sum of squared deviations = 0.068.
Answer:Population variance (σ²) = 0.068 / 5 = 0.0136

3Complete Inventory Count

Data:150, 200, 175, 180, 195
Solution:Mean μ = 180. Squared deviations: 900, 400, 25, 0, 225. Sum of squares = 1550.
Answer:Population variance (σ²) = 1550 / 5 = 310

4All Employees' Years of Experience

Data:2, 5, 8, 3, 12, 1, 7
Solution:Mean μ ≈ 5.43. Sum of squared deviations ≈ 89.71.
Answer:Population variance (σ²) ≈ 89.71 / 7 ≈ 12.82

How to Use This Population Variance Calculator

Switch the calculator to Population, then:

  1. Enter every value in the population, separated by commas, spaces or line breaks.
  2. Check that the result card says Population Variance (σ²).
  3. Read σ², the population standard deviation σ and the mean μ.
  4. Open the calculation steps to see the division by N written out with your numbers.

What the Calculator Shows You

  • The exact population variance and standard deviation, with no correction applied.
  • The sample figure for the same data, so you can see the difference if you are unsure which applies.
  • Every deviation and squared deviation.
  • Calculations run in your browser; your numbers are not sent anywhere.

When to Use Population Variance

A single class or team

A teacher describing the spread of marks for the class they actually taught.

Census data

Statistics about a population that has been counted in full.

Inspected-in-full batches

When every unit in a lot has been measured, not just a few.

Completed periods

Describing last year's twelve monthly figures as they are, without predicting anything.

Probability distributions

The variance of a die roll or another fully specified distribution is a population variance.

Population or Sample? How to Decide

Size does not decide it. What matters is which group your conclusions are about:

It depends on your question

If you are describing "all students in Section A", Section A's marks are the population. If you are describing "all students at the university", Section A is a sample.

Describing versus predicting

Last year's monthly sales are a population if you only want to describe last year. If you are using them to say something about future months, treat them as a sample.

Same data, two answers

For the tutorial group above, σ² ≈ 116.67 but s² = 140. With 600 values instead of 6 the two would be almost identical.

When in doubt, use sample variance

Most real data is a sample of something larger, and the n − 1 version will not understate the spread.

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Vanliga Frågor: Befolkningsvarians Kalkylator

Allt du behöver veta om varians.

Vad är populationsvarians?

Befolkningsvarians (σ²) mäter den exakta spridningen av datapunkter från populationsmedelvärdet (μ). Formeln är σ² = Σ(xᵢ − μ)² / N, där N är det totala antalet datapunkter i populationen. Till skillnad från sampelvarians, divideras den med N (inte N−1) eftersom du har all data — ingen uppskattning eller korrigering behövs.

När ska jag använda populationsvarians?

Använd populationsvarians när din datamängd inkluderar alla medlemmar i populationen du studerar. Exempel: alla elever i en klass, varje produkt i en batch, fullständig folkräkningsdata, alla transaktioner under en tidsperiod. Om du har ett urval (delmängd) från en större population, använd urvalsvarians istället. Nyckelfrågan är: representerar dina uppgifter hela gruppen?

Vad är populationsvariansen på 2, 4, 4, 4, 5, 5, 7, 9?

För 2, 4, 4, 4, 5, 5, 7, 9: Medelvärde (μ) = 40/8 = 5. Avvikelser: −3, −1, −1, −1, 0, 0, 2, 4. Kvadrat: 9, 1, 1, 1, 0, 0, 0, 3, 2, s. varians (σ²) = 32/8 = 4. Populationens standardavvikelse (σ) = √4 = 2.

Vad är skillnaden mellan σ² och s²?

σ² (populationsvarians) dividerar med N och använder populationsmedelvärdet μ. s² (urvalsvarians) dividerar med n−1 och använder urvalets medelvärde x̄. Urvalsvariansen är alltid något större än populationsvariansen för samma datauppsättning. Denna skillnad (Bessels korrigering) är viktigast för små datamängder och blir försumbar för stora N.

Kan populationsvariansen vara noll?

Ja, populationsvarians är lika med noll när alla värden är identiska. Om varje datapunkt i populationen har samma värde (t.ex. {7, 7, 7, 7}), är varje avvikelse från medelvärdet noll, vilket gör att σ² = 0. Detta indikerar nollvariabilitet – ingen spridning alls. Varje skillnad mellan värden ger en positiv varians.

Hur hittar man populationsvarians i Excel?

I Excel, använd =VAR.P(intervall) för populationsvarians. Exempel: =VAR.P(A1:A20). För populationens standardavvikelse, använd =STDEV.P(intervall). I Google Sheets fungerar samma funktioner. I Python: np.var(data) (standard är populationsvarians i NumPy). I R: var(x) * (n-1)/n eftersom R:s var() är standard på sampelvarians.