What Is Sample Variance, How to Calculate It, and How to Read the Result

What Is Sample Variance

You may have a dataset containing evaluation scores, income, usage time, or the average score of a scale. Sample variance tells you how far the values in the sample are spread around the mean. A larger variance means the data are more widely dispersed, while a smaller variance means the observations are closer to the mean.

The word “sample” indicates that this variance is calculated from part of a population. For a sample with n observations, the usual formula is:

s² = Σ(xᵢ - x̄)² / (n - 1)

Here, xᵢ is each observed value, x̄ is the sample mean, n is the number of observations, and s² is the sample variance. The denominator is n - 1, not n. Using n - 1 provides a more appropriate estimate of population variance from sample data.

For example, with the values 2, 4, and 6, the sample mean is 4. The squared deviations are 4, 0, and 4, respectively. The sum of squared deviations is 8, so the sample variance is 8 / (3 - 1) = 4. You can also see the variance calculation formula if you need to calculate each step by hand.

In SPSS, you usually do not calculate this formula manually. When you run descriptive statistics, SPSS displays variance in the Descriptive Statistics table, under the Variance column. You still need to understand the denominator and the meaning of the measure so that you do not misinterpret the output.

What Sample Variance Means in Quantitative Research

Sample variance is a measure of dispersion. It answers the question of whether the observations are close to the mean or noticeably different from one another. It usually appears alongside Mean, Std. Deviation, Minimum, and Maximum in the descriptive statistics output.

If you measure satisfaction on a 1 to 5 Likert scale, variance shows whether respondents’ answers are concentrated around one rating. Low variance may indicate that most respondents selected similar levels. High variance indicates more dispersed opinions, such as one group selecting 1 or 2 while another group selects 4 or 5.

Variance is expressed in squared units. If a variable is measured in points, its variance is expressed in squared points. Therefore, when you want to report dispersion in the same unit as the original variable, also read the standard deviation, which is the square root of the variance.

In a thesis, sample variance helps you describe the data before running regression, mean tests, or other analyses. It can also reveal variables with very little variation. A variable for which nearly everyone gives the same answer may be less useful for explaining differences between cases.

Variance does not, by itself, tell you whether the data are good or bad. A large variance is not enough to conclude that there is a problem with the questionnaire. You need to examine the scale, valid value range, mean, standard deviation, chart, and analytical purpose. Variance also does not tell you whether the data are normally distributed.

When reading a descriptive table, place variance next to standard deviation. If the variance is 0.64, the standard deviation is 0.8. This is often easier to interpret because 0.8 remains in the original unit of the variable. If you want another measure that is less affected by extreme values, you can also compare the median.

What Sample Variance Is Acceptable

There is no universal threshold for sample variance in the form of “from this value upward, it is acceptable.” Variance depends on the measurement scale, unit, value range, and research context. Income variance may be very large because the unit is currency, while the variance of a 1 to 5 Likert variable is usually much smaller.

Therefore, do not remove a variable simply because its variance is large, and do not conclude that a scale is acceptable simply because its variance is small. For variance, the useful question is whether the value is reasonable within the measurement range and reflects the actual data.

If your proposal or supervisor requires a specific criterion, use the criterion for your study and explain its source. Do not apply the threshold for Cronbach's Alpha, KMO, or factor loading to variance. Those thresholds belong to different measures and support different decisions.

For a scale with multiple items, you can first examine the variance of each item and then calculate the mean or composite score according to the study procedure. If an item has variance equal to 0, it cannot distinguish between respondents and should be checked before you include it in the next analysis.

How to Read Sample Variance in the Output

In SPSS, go to Analyze > Descriptive Statistics > Descriptives. Move the variable you want to check into the Variable(s) box, select Options, tick Mean, Std. deviation, Variance, Minimum, and Maximum, then click Continue and OK.

The Descriptive Statistics table displays each variable on a separate row. The N column is the number of valid observations used by SPSS. Mean is the average, Std. Deviation is the standard deviation, Variance is the variance, and Minimum and Maximum are the lowest and highest values.

The table below is illustrative output, not the result of a real study. The variable names and values are constructed so that you can identify the position of Variance in the SPSS output.

For the HL1 row, N = 180 means that SPSS has 180 valid values. Mean = 3.61 is the average score. Std. Deviation = 0.824 shows dispersion in Likert-score units. Variance = 0.679 is the square of the standard deviation, which you can check with 0.824², approximately 0.679.

HL2 has a variance of 0.812, which is larger than HL1. This indicates that the HL2 answers are more dispersed in the illustrative sample. You cannot conclude from this alone that HL2 is a poor-quality item. Check the frequency, item wording, corrected item-total correlation, and the purpose of the scale as well.

HL3 has N = 178, which is lower than the other two variables. The Valid N (listwise) = 178 row shows that, when the variables are considered together, 178 cases have complete data. The difference in N is a sign that you should check missing values before reporting the results.

If you use Analyze > Descriptive Statistics > Frequencies, you can select Statistics and request Mean, Variance, and Std. deviation. If you want to examine several variables by group, Explore also allows you to check descriptive statistics and charts. The menus differ, but the principle for reading the Variance column remains the same.

What to Do When Sample Variance Seems Unacceptable

First, identify the criterion behind your description of the result as “unacceptable.” If you only notice that variance is larger or smaller than the variance of another variable, that is not enough reason to alter the data. Work through the checks below in order.

Check Coding and the Value Range

Open Variable View and check Type, Values, Missing, and the valid value range. A Likert variable from 1 to 5 that contains 55, 99, or -1 may have an unusually high variance. Compare these codes with the original data codebook before making any changes.

Check Missing Values

SPSS calculates variance from valid observations. If missing values have been entered as ordinary numbers, the result will be distorted. You can declare missing codes in Variable View, or process them through Transform > Recode into Same Variables after saving a copy of the original dataset.

Check Outliers

A few extreme values can cause variance to increase sharply. Use Analyze > Descriptive Statistics > Explore, move the variable into Dependent List, select the boxplot, and examine the flagged cases. You can read more about outlier to distinguish an entry error from an extreme but genuine observation.

If the value is an entry error, correct it using the questionnaire or original data. If it is a genuine value, do not delete it simply to make the variance look better. Record the detection rule, the reason for removal, and the number of affected cases. A reviewer is more likely to accept a justified decision than a result table that has been made to look clean.

Review Variables With Very Little Variation

If nearly all respondents selected the same level, review the sampling process, the wording of the item, and the way the data were entered. Once data collection is complete, you can report this pattern and consider its effect on the analysis. Do not create additional values or change answers to increase variance.

Choose an Appropriate Analysis

Large variance does not automatically require a different statistical test. The choice between regression, t-test, ANOVA, and a PLS-SEM model must be based on the research question, variable type, and relevant assumptions. If your supervisor asks you to remove an item, save both the pre-processing and post-processing versions so that you can explain the decision.

Sample Variance Versus Standard Deviation

Sample variance and standard deviation both describe dispersion, but they do not use the same unit. Variance is the square of standard deviation. Standard deviation is the square root of variance.

For example, if variance equals 0.81, standard deviation equals 0.90. When a variable is measured on a 1 to 5 scale, a standard deviation of 0.90 is easier to interpret than a variance of 0.81 because it remains in the scale’s unit. In an SPSS table, you usually report Mean and Std. Deviation in the descriptive section, while Variance is used when you need deeper analysis or checking.

You also need to distinguish sample variance from variance in general. Population variance uses n in the denominator when you have the entire population. Sample variance uses n - 1 when the data are only a sample from the population. SPSS usually displays sample variance in descriptive statistics, so do not compare it with a formula divided by n and conclude that the software is wrong.

Variance differs from range. Range is the largest value minus the smallest value, so it depends heavily on the two endpoints. Variance uses the deviation of every observation from the mean. Variance also differs from coefficient of variation, which is used to compare relative dispersion when variables have different means and units.

Common Errors

Confusing Variance With Standard Deviation

Many students look at the Variance column and write in the thesis that the data have a standard deviation equal to that value. Check the column name carefully. Std. Deviation is the standard deviation, while Variance is the variance.

Using n in the Formula

This is common when calculating by hand or using Excel. If you are calculating sample variance, check the function and denominator. In Excel, VAR.S is for sample variance, while VAR.P is for population variance. The two functions return different results when n is small.

Looking for One Universal Acceptable Threshold

There is no single variance value that fits every variable. Comparing the variance of two variables is meaningful only when you understand that they use the same scale, unit, and measurement context.

Removing an Outlier Only Because Variance Is Large

An outlier may be an error, but it may also come from a valid respondent. Check the original data, detection rule, and effect of the case. If you remove it, record the reason and the number of observations removed.

Ignoring the Number of Valid Observations

A variance calculated from 180 observations is different from a variance calculated from 80 observations. Always report N, especially when variables have different numbers of missing values.

Frequently asked questions

What is sample variance, and what is it used for?

Sample variance measures how much the values in a sample are dispersed around the mean. It is used in descriptive statistics and is the basis for calculating standard deviation, but it does not decide by itself whether a variable is acceptable.

What sample variance is acceptable?

There is no universal threshold for sample variance. You need to examine the measurement unit, value range, number of observations, mean, and standard deviation. For a Likert scale, also check the answer distribution and whether the variable is concentrated at one level.

How are sample variance and standard deviation different?

Variance is the square of standard deviation. If variance equals 0.64, standard deviation equals 0.8. Standard deviation is usually easier to interpret because it keeps the original variable’s unit.

Why is my manually calculated variance different from SPSS?

The usual causes are using n instead of n - 1, including missing-value codes in the calculation, or calculating from a different number of observations than SPSS. Compare N, valid values, and the formula before concluding that the software is wrong.

Should I remove a variable when its variance is too large?

Not yet. First check entry errors, missing codes, the value range, and outliers. Remove a variable only when there is support from the scale design, original data, or methodological requirements, and then record the decision in your data-processing procedure.

How to write this in your thesis

You can write: “Descriptive statistics show that the variable [variable name] has a mean of [Mean], a standard deviation of [Std. Deviation], and a variance of [Variance] based on [N] valid observations. These values are used to describe the level of data dispersion.” Replace the placeholders with the actual values in the Descriptive Statistics table. Do not use the illustrative values from this article.

Open your SPSS file, run Analyze > Descriptive Statistics > Descriptives, compare N, Mean, Std. Deviation, and Variance, then record the variables with missing codes or unusual values separately. If you need to run the analysis on your own .sav or .csv file and check how to report it, see DoThesis M4 data analysis.