Variance: Formula, Interpretation, SPSS, Python, R and Excel Guide

What Is Variance?

Variance is a measure of statistical dispersion. It tells us how spread out the values are around the mean. The idea is simple: subtract the mean from each value, square the difference, add the squared differences and divide by the appropriate denominator. The result is the average squared distance from the mean.

Variance is important because many statistical methods are built from squared deviations. Standard deviation is the square root of variance. ANOVA literally means analysis of variance. Regression residual diagnostics often use squared errors. Tests of equal variance, such as Levene Test, Brown-Forsythe Test and Cochran C Test, also depend on the concept of variance.

In this guide, the main variable is G3 final grade. The output shows that G3 has sample variance = 10.4371 and standard deviation = 3.2307. The variance is useful mathematically, while the standard deviation is easier for readers because it returns the spread to the original grade scale.

Variance Formula

There are two common variance formulas: sample variance and population variance. The difference is the denominator. Sample variance divides by n − 1, while population variance divides by N.

Sample Variance Formula

s² = Σ(xᵢ − x̄)² / (n − 1)

where:
s² = sample variance
xᵢ = each observed value
x̄ = sample mean
n = sample size

Population Variance Formula

σ² = Σ(xᵢ − μ)² / N

where:
σ² = population variance
xᵢ = each population value
μ = population mean
N = population size

For the G3 final grade data, sample variance and population variance are very close because the sample size is large, N = 649. In smaller samples, the difference between the two formulas can be more visible.

Null and Alternative Hypothesis for Variance

Variance is usually reported as a descriptive statistic, so a hypothesis is not always required. However, a hypothesis becomes important when you compare a variance to a target value or compare variance between two or more groups.

Single-Variance Hypothesis

Two-Group Variance Hypothesis

Dataset and Variables Used

The worked example uses the student performance data structure. The main variable is G3 final grade, and the selected comparison variables are G1, G2, age and absences. A group-level variance comparison is also shown for G3 by sex.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified SPSS, Python and R Results

The variance results show that different variables have different levels of spread. The grade variables G1, G2 and G3 are on the same 0 to 19 scale, so comparing their variances is meaningful. Absences are measured on a different count scale, so their larger variance should be interpreted as a sign of wider absence-count dispersion, not automatically as a larger educational effect.

Variance Summary for Selected Numeric Variables

Share of Selected Variance

Group Variance Summary for G3 by Sex

Python Charts and Interpretation

1. Variance: Distribution Around the Mean

This chart explains variance visually. The solid vertical line marks the G3 mean at about 11.91. The dashed vertical lines mark approximately one standard deviation below and above the mean. Since the standard deviation is about 3.23, the dashed lines fall near 8.68 and 15.14. Most G3 values cluster around the central grade range, but there are also low scores near zero and higher scores near 18 or 19. Variance is calculated by taking the squared distance of every G3 value from the mean. Values close to 11.91 contribute little to variance, while values far from 11.91 contribute a lot. This is why the low-score values near zero strongly increase the total variance.

2. Variance: Comparison Across Numeric Variables

This chart ranks the selected numeric variables by sample variance. Absences has the largest variance, 21.54, which means absence counts are highly spread out. G3 has the next largest variance, 10.44, followed by G2 = 8.49 and G1 = 7.54. Age has the smallest variance, 1.48, because most students are within a narrow age range. The chart is useful because it quickly shows which variables are more dispersed. However, it should be interpreted with scale awareness: absences and grades are measured differently, so the largest variance does not automatically mean the strongest educational importance.

3. Variance and Standard Deviation

This chart compares variance with standard deviation. Variance is always in squared units, while standard deviation is the square root of variance and returns spread to the original scale. For example, G3 variance is 10.44, but its standard deviation is 3.23. Absences variance is 21.54, but its standard deviation is 4.64. This is why standard deviation is often easier to explain in reports. A standard deviation of 3.23 grade points is more understandable than a variance of 10.44 squared grade points. Still, variance is important because many statistical formulas and models are based on squared deviations.

4. Variance: Largest Squared-Deviation Contributors

This chart shows why variance is sensitive to extreme values. Cases with G3 values far away from the mean create very large squared deviations. The largest displayed contributors have G3 = 0, a deviation of about -11.906 from the mean and a squared deviation of about 141.753. Because the deviation is squared, these cases contribute much more to total variance than scores near the mean. This does not mean the cases should automatically be deleted. Instead, it means they should be understood as important contributors to dispersion. If these values are valid grades, they are part of the real spread of the dataset.

5. Variance: Group Comparison by Sex

This chart compares G3 variance by sex. Female students have sample variance = 9.76, while male students have sample variance = 11.03. Descriptively, male G3 scores are slightly more spread out. Female students have a higher mean G3 score in the summary table, but variance answers a different question: it measures dispersion, not average performance. Because no formal variance equality p-value is attached, this chart should be reported as descriptive. A formal test would require Levene, Brown-Forsythe or an F-test depending on the assumptions and reporting purpose.

R Validation Charts and Interpretation

The R charts validate the same results using a different plotting style. They confirm the distribution around the mean, the ranking of variable variances, the relationship between variance and standard deviation, the influence of large squared deviations and the group-level variance comparison.

1. R Distribution Around the Mean

The R chart confirms that G3 scores cluster around the mean of about 11.91 but also include values far from the center. The dashed standard-deviation reference lines help readers understand how variance is connected to spread. Values outside the central range add more to total variance because their deviations from the mean are larger and then squared.

2. R Variance Comparison Across Numeric Variables

The R variance comparison confirms the Python ranking. Absences show the largest variance, followed by G3, G2, G1 and age. The chart reinforces the key result that G3 is the most variable of the three grade variables, while age is the least variable among the selected variables.

3. R Variance and Standard Deviation

The R chart confirms that variance values are larger because they are in squared units, while standard deviation is lower because it is the square root of variance. This is an important teaching point. Variance is powerful for statistical formulas, but standard deviation is usually clearer for practical interpretation.

4. R Largest Squared-Deviation Contributors

The R contributor chart confirms the same idea as the Python chart: a small number of values far from the mean can contribute strongly to total variance. Because the deviations are squared, extreme low or high values have much larger influence than values near the mean. This is why variance should be interpreted with a distribution plot or boxplot whenever possible.

5. R Group Variance Comparison

The R group comparison confirms that male students have slightly higher G3 variance than female students. This validates the descriptive group result. For formal inference, this chart should be followed by an equality-of-variance test such as Levene Test or Brown-Forsythe Test.

SPSS Output Interpretation

The SPSS output link supports the variance workflow by providing descriptive statistics and output documentation. The key values used in this guide are the sample variances, population variances, standard deviations, ranges and group-level variance summaries.

Main SPSS-Style Interpretation

For G3, the sample variance is 10.4371. This means the average squared deviation from the G3 mean is about 10.44. The standard deviation is 3.2307, which means the typical spread is about 3.23 grade points from the mean. The minimum is 0, maximum is 19 and interquartile range is 4. These values show that most grades are concentrated around the center, but a few very low scores create large squared deviations.

SPSS-Ready Reporting Table

How to Calculate Variance in Python, R, SPSS and Excel

Python Method

In Python, pandas calculates sample variance with var() by default because it uses ddof = 1. Population variance uses ddof = 0.

import pandas as pd
import numpy as np

df = pd.read_csv("student-por.csv", sep=";")

selected = ["G1", "G2", "G3", "age", "absences"]

summary = []
for col in selected:
    x = df[col].dropna()
    summary.append({
        "variable": col,
        "n": x.size,
        "mean": x.mean(),
        "sample_variance": x.var(ddof=1),
        "population_variance": x.var(ddof=0),
        "standard_deviation": x.std(ddof=1),
        "minimum": x.min(),
        "maximum": x.max(),
        "range": x.max() - x.min(),
        "interquartile_range": x.quantile(0.75) - x.quantile(0.25)
    })

summary_df = pd.DataFrame(summary)
print(summary_df)

# Group variance for G3 by sex
group_summary = df.groupby("sex")["G3"].agg(
    n="count",
    mean="mean",
    sample_variance=lambda s: s.var(ddof=1),
    standard_deviation=lambda s: s.std(ddof=1),
    minimum="min",
    maximum="max"
)

print(group_summary)

R Method

In R, the var() function calculates sample variance. Use a custom formula if you need population variance.

df <- read.csv("student-por.csv", sep = ";")

selected <- c("G1", "G2", "G3", "age", "absences")

variance_summary <- data.frame(
  variable = selected,
  n = sapply(df[selected], function(x) sum(!is.na(x))),
  mean = sapply(df[selected], mean, na.rm = TRUE),
  sample_variance = sapply(df[selected], var, na.rm = TRUE),
  population_variance = sapply(df[selected], function(x) {
    x <- x[!is.na(x)]
    sum((x - mean(x))^2) / length(x)
  }),
  standard_deviation = sapply(df[selected], sd, na.rm = TRUE),
  minimum = sapply(df[selected], min, na.rm = TRUE),
  maximum = sapply(df[selected], max, na.rm = TRUE)
)

variance_summary$range <- variance_summary$maximum - variance_summary$minimum

print(variance_summary)

# Group variance for G3 by sex
aggregate(G3 ~ sex, data = df, function(x) c(
  n = length(x),
  mean = mean(x),
  sample_variance = var(x),
  standard_deviation = sd(x),
  minimum = min(x),
  maximum = max(x)
))

SPSS Method

In SPSS, variance can be requested through DESCRIPTIVES, FREQUENCIES or EXAMINE. The following syntax calculates descriptive statistics for the selected variables and group summaries for G3 by sex.

DESCRIPTIVES VARIABLES=G1 G2 G3 age absences
  /STATISTICS=MEAN STDDEV VARIANCE MIN MAX.

EXAMINE VARIABLES=G3 BY sex
  /PLOT NONE
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL
  /PDF DOCUMENTFILE="D:\DATA ANALYSIS\A Basic Descriptive Statistics Guides\Variance\SPSS_Output\Variance-SPSS-Output.pdf".

Excel Method

Excel has direct variance formulas. Use VAR.S for sample variance and VAR.P for population variance.

How to Report Variance

Variance can be reported in a short descriptive sentence or in a table. Because variance is in squared units, it is usually helpful to report standard deviation together with variance.

Common Mistakes When Interpreting Variance

Related Guides

These related Salar Cafe guides help you continue from descriptive variance to assumption testing, group comparison and formal reporting.

Downloads and Resources

The following hosted resources support this variance guide. The chart links show the Python and R visual outputs used in the interpretation sections, and the SPSS PDF provides the software output for the variance workflow.

External References

• UCI Machine Learning Repository: Student Performance dataset
• pandas DataFrame.var documentation
• R statistics documentation for variance and covariance functions
• Microsoft Excel VAR.S function