(Sample variance for this dataset would be –)
A Population Variance Calculator (also known as a True Variance Utility, Sigma Squared Calculator, or Uncorrected Variance Analyzer) measures the exact spread or dispersion of a complete, finite dataset. It calculates the average of the squared mathematical differences from the population mean (μ).
In census data analysis, corporate revenue auditing, grading small classroom exams, and any scenario where you possess 100% of the data points for the group you are studying, Population Variance (σ2) is the correct metric. Because you have the entire population, you calculate the exact, true variance by dividing by the total number of observations (N). If you only had a small subset of the data (a sample), you would be forced to use Sample Variance and divide by n - 1 (Bessel’s Correction).
Our free online Population Variance Calculator provides instant tracking across all absolute dispersion metrics:
- Population Variance Formula (σ2):
σ2 = ∑ (xi - μ)2 ÷ N - Population Mean (μ): The exact absolute mathematical average of all data points.
- Sum of Squares (SS): The total absolute squared deviation from the mean
∑ (xi - μ)2. - Population Standard Deviation (σ): Instantly outputs the square root of the variance
√σ2to return the dispersion to its original units of measurement.
Master Population Variance Reference Table (Complete Classroom Exam: N = 5 Students)
The table below tracks a complete, finite population (a small classroom of exactly 5 students). Because we have all 5 scores, this is a Population, not a sample. We calculate the exact mean, individual deviations, and total population variance (Dataset: 70, 75, 80, 85, 90):
| Student (Population Member) | Exam Score (xi) | Deviation from Mean (xi – μ) | Squared Deviation (xi – μ)2 |
|---|---|---|---|
| Student 1 | 70 | 70 – 80 = -10 | (-10)2 = 100 |
| Student 2 | 75 | 75 – 80 = -5 | (-5)2 = 25 |
| Student 3 (Matches Mean) | 80 | 80 – 80 = 0 | (0)2 = 0 |
| Student 4 | 85 | 85 – 80 = +5 | (+5)2 = 25 |
| Student 5 | 90 | 90 – 80 = +10 | (+10)2 = 100 |
| POPULATION METRICS | Mean μ = 80.0 | Total Size N = 5 | Sum of Squares (SS) = 250 |
| POPULATION VARIANCE (σ2) | – | SS ÷ N | 250 ÷ 5 = 50.0 |
Step-by-Step Population Variance (σ2) Calculation
To calculate the true population variance for the 5 complete exam scores (N = 5: 70, 75, 80, 85, 90):
Step 1 (Find Population Mean μ): (70 + 75 + 80 + 85 + 90) ÷ 5 = 400 ÷ 5 = 80.0
Step 2 (Subtract Mean from Each): -10, -5, 0, +5, +10
Step 3 (Square the Deviations): (-10)2 = 100, (-5)2 = 25, 0, 25, 100
Step 4 (Calculate Sum of Squares SS): 100 + 25 + 0 + 25 + 100 = 250.0
Step 5 (Divide by N for Variance): σ2 = 250.0 ÷ 5 = 50.0
Step 6 (Optional: Population SD): σ = √50.0 ≈ 7.071
Because we had 100% of the students in the class, we divide exactly by N=5, resulting in a Population Variance of 50.0.
Statistical Comparison: Population Variance vs. Sample Variance
The single most common mistake in statistics is using the wrong variance denominator. Below is a comparative chart detailing when to use Population versus Sample variance:
| Metric Type | Symbol | Denominator | When To Use It |
|---|---|---|---|
| Population Variance | σ2 (Sigma Squared) | Divide by N | When you have exactly 100% of the data points for the group you are studying. |
| Sample Variance | s2 (s Squared) | Divide by (n – 1) | When you only have a subset of data and are trying to estimate the population parameters. |
History & Mathematics: 1918 Ronald Fisher
Ronald Fisher & The Formalization of “Variance”
While mathematicians like Carl Friedrich Gauss used “mean squared error” throughout the 1800s, the specific statistical term “variance” was coined and formalized by British statistician Ronald Fisher in his 1918 paper, “The Correlation Between Relatives on the Supposition of Mendelian Inheritance.” Fisher cemented Population Variance (σ2) as the absolute foundational metric for his subsequent development of ANOVA (Analysis of Variance), which remains one of the most important statistical modeling tools in human history.
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Frequently Asked Questions (FAQ)
Why does Population Variance divide by N instead of N – 1?
Because you have 100% of the data. The Mean (μ) is the exact true mean of the population. Therefore, calculating the exact mathematical average of the squared deviations requires simply dividing by the total count (N). You only divide by n - 1 (Bessel’s Correction) when dealing with a sample, because a sample mean is an imperfect estimate that inherently underestimates true population variance.
What is the relationship between Population Variance and Standard Deviation?
Population Variance (σ2) is in squared units (e.g., if you are measuring weight in pounds, the variance is in “squared pounds,” which is impossible to visualize). The Population Standard Deviation (σ) is simply the square root of the variance, converting the dispersion metric back into the original, interpretable unit of pounds.
When should I absolutely use Population Variance?
Use Population Variance when the dataset is the entirety of what you care about. For example, if a teacher wants to know the variance of their specific 30-student classroom, they use Population Variance. They do not care about estimating the variance of all students in the country.