What Is Variance? How to Calculate, Read Results, and Distinguish It from Standard Deviation
What Is Variance
Variance is a descriptive measure of how far values are dispersed around the mean. When variance is small, observations tend to stay close to the mean. When variance is large, the data are more spread out, and the gaps between observations may be substantial.
In quantitative research, you will usually see variance in SPSS's Descriptive Statistics table alongside N, Minimum, Maximum, Mean, and Std. Deviation. Variance is expressed in the square of the original measurement unit. For example, if income is measured in million VND, its variance is expressed in squared million VND. Variance is useful for calculations, but standard deviation is usually easier to interpret when you write your thesis.
For a population containing N observations, the population variance is calculated as follows:
σ² = Σ(xᵢ - μ)² / N
For a sample containing n observations, the formula usually used is:
s² = Σ(xᵢ - x̄)² / (n - 1)
Here, xᵢ is each observed value, x̄ is the sample mean, μ is the population mean, N is the population size, and n is the sample size. Dividing by n - 1 gives a more appropriate estimate of population variance from sample data.
For example, the three values 2, 4, and 6 have a mean of 4. The sum of the squared deviations from the mean is (2 - 4)² + (4 - 4)² + (6 - 4)² = 8. The population variance is 8 / 3, while the sample variance is 8 / 2. You can also review the variance formula if you need to check each hand-calculation step.
Why Variance Matters in Quantitative Research
Variance tells you whether data are relatively uniform or dispersed. This is background information to check before you run regression, test means, or interpret other statistics. A variable with variance equal to 0 means that everyone gave the same response. That variable provides no information for distinguishing between observations.
In a Likert questionnaire, variance helps you detect an item for which responses are too concentrated. For example, almost every respondent may select level 5 for one item. The item's mean may be high, but its variance is very small, which shows that the item creates little difference between respondents.
Variance is also related to standard deviation through the formula standard deviation = the square root of variance. Because taking the square root returns the measure to the original unit, standard deviation is more often reported in descriptive tables. You should not treat variance as an independent pass or fail score. Read it alongside the scale, mean, value range, and purpose of the analysis.
When two variables use the same scale, the variable with the larger variance has greater dispersion. If the two variables use different units, however, comparing their variances directly can be misleading. A variable measured in thousand VND and a variable measured on a Likert scale have variances expressed in different units.
Variance also appears in several other statistical concepts. In regression, you are concerned with the part of the dependent variable's variation explained by the model through R². In factor analysis, total variance explained describes how much variation is retained by the factors, but this is a different measure from the variance of an individual variable. Check the exact table name and column name in the output before drawing a conclusion.
What Variance Is Acceptable
There is no single general threshold for the question, “What is a good variance?” Variance depends on the scale, value range, coding method, and characteristics of the sample. A large variance does not automatically mean that the data are wrong. A small variance does not prove that the scale is good.
Check the data in this order: does the variable contain enough distinct values, are the Minimum and Maximum reasonable, does the mean fall within the scale range, are any values entered incorrectly, and is an outlier driving the result. If a Likert variable from 1 to 5 has very small variance, inspect the frequency of each response level. If a continuous variable has large variance, check its measurement unit and distribution before removing any observation.
The table below lists related reference points commonly used when evaluating quantitative data. It is not a table of variance thresholds, because methodological sources do not specify one general variance value for every variable.
| Item to check | Reference point | Interpretation | Source |
|---|---|---|---|
| Variance of one variable | No general threshold | Evaluate it according to the scale, value range, and research purpose | No fixed threshold applies |
| Variance equal to 0 | 0 | The variable has no variation and should be considered for removal from an appropriate analysis | No fixed threshold applies |
| Standard deviation | Square root of variance | Returns dispersion to the same unit as the variable | No fixed threshold applies |
| Observations per item when designing a scale | 5 to 10 observations per item | A reference point for data planning and factor analysis | (Hair et al., 2010) |
| Regression sample size | From 50 + 8m, where m is the number of independent variables | A reference point when planning regression | (Tabachnick and Fidell, 2013) |
The final two rows concern data and sample size, not variance thresholds. They are shown together so you do not confuse “what variance is acceptable” with “how large should the sample be.” If you need to determine sample size for a finite population, you can consult median for the descriptive statistics section and handle sample selection separately using the appropriate formula.
How to Read Variance in the Output
In SPSS, open Analyze > Descriptive Statistics > Descriptives. Move the variable you want to check into Variable(s), select Options, tick Mean, Std. deviation, Variance, Minimum, and Maximum, then select Continue and OK. SPSS will generate the Descriptive Statistics table.
Illustrative output
The table below is illustrative output, not the result of a real study. The column names follow the way SPSS commonly displays the Descriptive Statistics table.
| Descriptive Statistics | N | Minimum | Maximum | Mean | Std. Deviation | Variance |
|---|---|---|---|---|---|---|
| SAT1 | 180 | 1.00 | 5.00 | 3.72 | 0.846 | 0.716 |
| SAT2 | 180 | 1.00 | 5.00 | 3.65 | 0.921 | 0.848 |
| SAT3 | 180 | 2.00 | 5.00 | 3.81 | 0.733 | 0.537 |
| Valid N (listwise) | 180 |
The SAT1 row means that there are 180 valid observations. The minimum is 1, the maximum is 5, and the mean is 3.72. The standard deviation is 0.846 and the variance is 0.716. In calculation terms, 0.846² is approximately 0.716.
Read the Valid N (listwise) row carefully. If you run several variables and choose listwise deletion, SPSS keeps only rows with complete data for every variable. As a result, N may differ between rows when missing values are handled in another way.
If the variance of SAT3 is smaller than the variance of SAT2, responses to SAT3 are more concentrated around the mean. You cannot yet write that SAT3 is better than SAT2. This statistic describes dispersion. It does not evaluate the reliability or validity of the scale.
In the Frequencies table, you can also see the count and percentage at each response level. Go to Analyze > Descriptive Statistics > Frequencies, move the variable into the analysis box, and select Display frequency tables. This step is necessary when variance is very small or when Minimum and Maximum do not match the expected scale.
What to Do When Variance Looks Unusual
First, do not delete a variable simply because its variance is larger or smaller than that of another variable. Check the original data, coding, and collection context in the sequence below.
Check entry and coding errors
Compare Minimum and Maximum with the questionnaire. If the scale runs only from 1 to 5 but the output contains 55, 0, or 99, you may have entered a value incorrectly or be using a missing-response code. Correct the original data file or code missing values properly, then run the analysis again.
Also check reverse-coded variables. If an item needs reverse coding but has not been recoded, its variance may not look unusual by itself, but it can distort Cronbach's Alpha and the analyses that follow. Keep a record of the data version before and after the correction so you can explain what changed.
Check the distribution and outliers
Use Analyze > Descriptive Statistics > Explore to inspect additional charts, the median, and unusual values. You can open outlier to review how to identify observations that differ substantially from most of the data.
Remove an outlier only when you have a clear data-based reason, such as a value outside the possible range or a response form showing signs of an entry error. Do not delete observations in bulk just to make variance smaller. If you keep the outlier, record how it affects the descriptive and subsequent analyses.
Review the variable and scale design
If an item has almost no variation because every respondent selected the same level, the cause may be the wording of the item, the surveyed population, or the real-world context. Removing the item may be reasonable in some analyses, but you need to state the reason and check the effect on the model.
EFA, Cronbach's Alpha, and regression have their own criteria. Variance does not replace Corrected Item-Total Correlation, Factor Loading, VIF, or p-value. For example, Corrected Item-Total Correlation from 0.3 upward is a commonly used reference point according to (Nunnally and Bernstein, 1994), but that reference point is not a variance threshold.
Variance Versus Standard Deviation
Variance and standard deviation both measure dispersion around the mean. The difference is their unit and interpretation. Variance is the mean of the squared deviations, while standard deviation is the square root of variance.
If variable SAT1 has variance 0.716, its standard deviation is approximately 0.846. Variance is convenient for calculations and for constructing statistical measures. Standard deviation is easier to present because it remains in the unit of the original variable.
Do not confuse variance with the range. The range is Maximum - Minimum and uses only the two boundary values. Variance uses every observation, so it reflects overall dispersion more fully, although it is still affected by outliers.
The mean and median are also different measures. The mean describes the central position by dividing the total value by the number of observations. The median is the middle value after the data have been ordered. When the distribution is skewed or contains outliers, review the mean, median, standard deviation, and variance together.
Common Errors
The first error is searching for one general number that proves variance is “acceptable.” This ignores the measurement unit and scale range. With a Likert scale from 1 to 5, variance cannot be interpreted in the same way as the variance of income or purchase frequency.
The second error is reading Std. Deviation as Variance. If you need variance, check the Variance column. If the table contains only Std. Deviation, variance is the square of standard deviation, with a small rounding difference because SPSS displays a limited number of decimal places.
The third error is using variance as a substitute for a reliability test. A variable can have a reasonable variance and still fail to fit with the other variables in the same scale. Run the appropriate procedure, such as Analyze > Scale > Reliability Analysis, when evaluating Cronbach's Alpha.
The fourth error is deleting an outlier without saving the original data version. The committee may ask why the number of observations changed. Create a new variable or file version, record the checking criterion, and report the number of observations before and after processing.
The final error is comparing variances for variables measured in different units. In that situation, absolute size has limited meaning. Use an appropriate alternative measure or present each variable within its measurement context.
Frequently asked questions
What is variance in statistics?
Variance measures how far values are dispersed around the mean. It is calculated as the mean of the squared deviations from the mean. Larger variance generally means that the data are more widely dispersed, but it does not mean that the variable is better.
What is a good variance?
There is no single variance threshold for every study. Review variance together with Minimum, Maximum, Mean, standard deviation, distribution, and scale. When a variable has variance equal to 0, the data have no variation and should be checked before the variable is included in the analysis.
How are variance and standard deviation different?
Standard deviation is the square root of variance. Variance is expressed in squared units, while standard deviation uses the same unit as the original variable. This is why standard deviation is usually easier to read in a descriptive table.
Why is the variance in SPSS not exactly the absolute square of standard deviation?
The two values are usually close after standard deviation is squared. A small difference can result because SPSS displays a rounded standard deviation, while the variance calculation uses a more precise value internally.
Does large variance mean that the data are wrong?
Large variance only shows that the data are widely dispersed. Check the unit, valid range, outliers, and data-entry method. Remove an observation or variable only when there is a clear methodological and data-based reason, not simply to make the statistic look better.
Open your SPSS file, run Analyze > Descriptive Statistics > Descriptives, check Variance, Std. Deviation, Minimum, Maximum, and N together, then record the variables that need review before running the next analysis. If you want to process .sav or .csv data using the correct workflow, M4 data analysis handles this step.