Variance Formula and How to Read the Results in SPSS
What is the variance formula
Variance tells you how far the values in a dataset are dispersed around the mean. When variance is small, the observations tend to lie close to the mean. When variance is large, the data are more widely dispersed. In an SPSS file, variance appears as Variance, while standard deviation appears as Std. Deviation.
You can also review the Statistics topic if you are working with several descriptive statistics. For a sequence of values x₁, x₂, ..., xₙ, you first need to calculate the arithmetic mean:
x̄ = (x₁ + x₂ + ... + xₙ) / n
The population variance formula is:
σ² = Σ(xᵢ - μ)² / N
Here, μ is the population mean and N is the number of elements in the population. When the data are only a sample taken from a larger population, the sample variance formula is:
s² = Σ(xᵢ - x̄)² / (n - 1)
The denominator is where people most often get confused. Population variance is divided by N, while sample variance is divided by n - 1. In thesis research, survey data are usually treated as a sample of a broader group, so software generally reports sample variance.
For example, with three observations, 2, 4, and 6, the mean is 4. The sum of squared deviations is (2 - 4)² + (4 - 4)² + (6 - 4)² = 8. Population variance is 8 / 3 = 2,67, while sample variance is 8 / 2 = 4. The results differ because the two formulas answer different questions.
If you want to focus specifically on how the formula is presented, you can refer to the variance formula article. This article focuses on understanding the statistic and using it when reading quantitative data.
Why the variance formula matters in quantitative research
In quantitative research, variance helps you describe the amount of variation in an observed variable or outcome variable. When reporting a variable such as income, satisfaction, purchase intention, or a scale mean, you usually present the mean together with the standard deviation. Variance appears less often in explanations written for readers because its unit is squared, but it is still used in many statistical calculations.
Variance has the same basic role as standard deviation in describing dispersion. In regression, for example, variation in the dependent variable affects how the model explains the data. In ANOVA, variance between groups is compared with variance within each group. In factor analysis, variance is used when examining how much variation is explained by the factors.
When reading Likert survey data, you should not treat variance as an independent pass or fail criterion. A variable with very low variance means that most respondents selected the same level. This can make the variable less useful for distinguishing between respondents, but it is not enough to conclude that the item must be removed. You also need to examine the distribution, item content, corrected item-total correlation, and the purpose of the analysis.
Variance can also help you detect responses with little variation. If every respondent selects 4 or 5, variance may be zero or close to zero. By contrast, large variance may occur because the respondents genuinely differ, or because the data contain an outlier. When you suspect an unusual value, review the guidance on outliers instead of deleting an observation simply because the variance is large.
What variance value is acceptable
There is no common threshold that determines whether variance passes or fails. The appropriate value depends on the scale, measurement unit, range, and research objective. You should therefore avoid writing that variance must be greater than a fixed number unless you have a theoretical basis or a requirement specified by the study.
Several related criteria are commonly used to evaluate data and scales. They are shown below. These are thresholds for related statistics, not direct variance thresholds.
| Statistic or criterion | Practical interpretation | Canonical source |
|---|---|---|
| Variance | There is no common pass threshold; interpret it according to the scale and data distribution | No fixed threshold applies |
| Standard deviation | Read it together with the mean and measurement range; do not judge it from one number alone | No fixed threshold applies |
| Corrected Item-Total Correlation | From 0.3 upward is commonly considered appropriate when evaluating an item | (Nunnally and Bernstein, 1994) |
| Cronbach's Alpha | From 0.7 upward is commonly accepted; an exploratory scale may be considered from 0.6 | (Nunnally, 1978); (Hair et al., 2010) |
| KMO | From 0.5 upward when assessing whether EFA can be conducted | (Kaiser, 1974) |
| Total Variance Explained | Total explained variance from 50% upward in an appropriate context | (Hair et al., 2010) |
| Factor loading | From 0.5 upward is commonly used as a reference point | (Hair et al., 2010) |
In the table above, Total Variance Explained is different from the variance of one variable. Total Variance Explained shows what percentage of the variation in the item set is explained by the retained factors. You must not use the 50% figure as a threshold for the Variance column in a descriptive statistics table.
Likewise, large variance does not automatically prove that the data are good, and small variance does not automatically prove that the data are poor. Examine the mean, standard deviation, minimum, maximum, and charts. If you are assessing a scale, run the Cronbach's Alpha and EFA procedures in the appropriate order.
How to read the variance formula in SPSS output
In SPSS, open Analyze > Descriptive Statistics > Descriptives. Move the variables you want to describe into Variable(s), select Options, check Mean, Std. deviation, Variance, Minimum, and Maximum, then select Continue and OK. SPSS will produce the Descriptive Statistics table.
The table below is illustrative output, not the result of a real study. Suppose that HL1 measures satisfaction on a Likert scale from 1 to 5 and has 120 valid observations.
| Descriptive Statistics | N | Minimum | Maximum | Mean | Std. Deviation | Variance |
|---|---|---|---|---|---|---|
| HL1 | 120 | 1.00 | 5.00 | 3.62 | 0.842 | 0.709 |
| Valid N (listwise) | 120 |
In the HL1 row, Mean = 3.62 shows the average score across 120 responses. Std. Deviation = 0.842 shows the amount of dispersion in the original measurement unit. Variance = 0.709 is approximately the square of the standard deviation because 0.842² is approximately 0.709.
You can also run Analyze > Descriptive Statistics > Frequencies. Select the variable, click Statistics, check the statistics you need, then select Continue and OK. This option also produces a frequency table, which is useful when you want to see how many respondents selected each level from 1 to 5.
If a variable has many missing values, check the N column before interpreting the result. Two variables may both have a Mean of 3.6, but if one is based on 120 observations and the other on only 87 observations, their comparison does not have the same level of reliability. You also need to check how missing values were coded. If a missing value was entered as 0, both variance and mean will be distorted.
The relationship between the two statistics is standard deviation = the square root of variance. Variance therefore has a squared unit, while standard deviation remains in the original unit. In the results chapter, standard deviation is usually easier to read and is commonly reported together with the Mean.
What to do when the variance formula does not meet expectations
First, identify the criterion behind the claim that the variance is unacceptable. If you only see that variance is large or small, do not remove the item immediately. Check the following steps in order.
Check the data and coding
Open Variable View to check the variable type, value labels, and response codes. For a Likert scale, the levels must be coded consistently. If one variable uses 1 to 5 but some rows use 99 to represent missing data, declare 99 as missing or handle it according to the coding rule you established.
Examine the distribution and unusual values
Run Analyze > Descriptive Statistics > Explore. Move the variable into Dependent List, open Plots, and select the appropriate charts. Check the minimum, maximum, boxplot, and observations that appear unusual. Remove an observation only when there is a clear data-related reason, such as an entry error or failure to meet the survey conditions, and record that decision.
Check the content of the item
An item may have low variance because its content is too predictable or because the entire sample genuinely holds the same view. Compare the item with the questionnaire and hypotheses. Do not alter the data to make the variance look better. If the item is unsuitable in terms of content, discuss removing it with your supervisor and state the basis for the decision clearly.
Rerun the related statistics
If your goal is to assess a scale, run Analyze > Scale > Reliability Analysis, move the items measuring the same concept into Items, select Statistics, and check Scale if item deleted and Correlations. Review Cronbach's Alpha, Corrected Item-Total Correlation, and the change in Alpha when an item is deleted. Then, if required by the research design, move on to EFA.
EFA output is rarely tidy the first time you run it. Removing an item, rerunning the analysis, and recording each deletion round is normal, but each decision must be based on the criteria stated in the methods section. Do not delete large numbers of observations or items simply to force variance into a preferred range.
Distinguishing the variance formula from standard deviation
Variance and standard deviation both measure dispersion, but you read them differently. Variance is the mean of the squared deviations from the mean. Standard deviation is the square root of variance.
In the illustrative data above, variance is 0.709 and standard deviation is 0.842. If the scale runs from 1 to 5, standard deviation of 0.842 remains in Likert score units, so it is easier to interpret. You can review the standard deviation article to compare how the two statistics are read.
The median is a different concept. The median is the middle value after the data are sorted, while variance measures dispersion around the mean. When the data are strongly skewed or contain an outlier, the median may represent the centre better than the mean, but that does not make the median a measure of dispersion. See median if you are deciding between Mean and Median.
Another common confusion is the variance of one variable versus Total Variance Explained in EFA. The variance of one variable describes one data column. Total Variance Explained describes how much variation the extracted factors explain for the item group. The two numbers appear in different output tables and cannot replace each other.
Common errors
The first error is using the population formula for sample data without explaining how the sample was selected. When calculating variance in Excel or with a calculator, determine whether your data cover the entire population or represent a research sample. In a survey thesis, the formula must be consistent with both the software and the methods section.
The second error is calling Variance = 0.709 the standard deviation. The standard deviation in the example is 0.842. You can check this quickly by taking the square root of the variance.
The third error is setting a rigid threshold for variance. Variance has no common cutoff like Cronbach's Alpha or KMO. The assessment must be connected to the measurement range, mean, distribution, and purpose of the analysis.
The fourth error is ignoring missing values. If code 99 is entered as a valid value, Mean and Variance will no longer represent a 1 to 5 scale. Check N and the frequency table before copying the result into Chapter 4.
The fifth error is deleting an outlier simply because you want a lower standard deviation. Every data adjustment must have a basis, be recorded, and be defensible. Without a data-related reason, keeping the observation and reporting the relevant limitation is usually safer.
Frequently asked questions
How do the sample and population variance formulas differ?
Population variance divides the sum of squared deviations by N, while sample variance divides it by n - 1. The appropriate formula depends on whether the data cover the entire population or are a sample used to make inferences about a larger population.
What variance value is acceptable?
There is no single variance threshold that applies to every study. Read variance together with Mean, Std. Deviation, minimum, maximum, and the scale context. For scale assessment, also review Cronbach's Alpha, Corrected Item-Total Correlation, and the EFA results instead of looking only at Variance.
Why is variance equal to 0?
Variance is zero when every observation has the same value. On a Likert scale, this may happen when all respondents select the same level or when the data were entered incorrectly, such as when one value was accidentally filled down an entire column. Check the frequency table and the original data file.
Is variance the same as standard deviation?
The two statistics are related but different. Standard deviation is the square root of variance, so variance has a squared unit while standard deviation has the same unit as the original variable. When writing a data description, you usually present Mean and Std. Deviation.
Do you need to report variance in a thesis?
This depends on your department's guidelines, your supervisor, and the type of analysis. For descriptive statistics, Mean and Std. Deviation are usually easier to interpret. If the results table or methods section requires Variance, report the column name correctly and briefly explain how it was calculated.
Which formula does SPSS use for variance?
For descriptive statistics based on sample data, SPSS generally displays sample variance, meaning that the sum of squared deviations is divided by n - 1. You should still check the methods documentation and the number of valid observations, especially when the data contain missing values.
Open Analyze > Descriptive Statistics > Descriptives in your file, select Mean, Std. deviation, Variance, Minimum, and Maximum, then check the N column before writing your interpretation. If you need to run and interpret .sav or .csv data directly, you can use M4 data analysis from DoThesis.