What Is the Symbol for Variance? How to Read and Calculate It in SPSS
What Is the Symbol for Variance
You open a descriptive statistics table and see a Variance column, or you come across the symbols s², S², or σ² in a textbook. These forms all refer to variance, a measure of how far the values in a dataset are spread around the mean. The larger the variance, the farther the data tend to lie from the mean. The smaller the variance, the more closely the observations are concentrated around the mean.
In quantitative research, you need to distinguish between two cases. σ² usually denotes the population variance, while s² usually denotes the sample variance. When your survey data come from a group of respondents, you are usually working with sample variance.
The sample variance formula is:
s² = Σ(xᵢ - x̄)² / (n - 1)
Here, xᵢ is each observation, x̄ is the sample mean, and n is the number of observations. The denominator n - 1 appears when you estimate population variance from a sample. If you need to see each substitution step, you can refer to the article variance formula.
For a population, the formula is usually written as:
σ² = Σ(xᵢ - μ)² / N
μ is the population mean and N is the population size. When you run SPSS, you usually do not need to calculate variance by hand. The important points are knowing which statistic the software is reporting, whether the data come from a sample or a population, and whether that variance fits your research question.
What the Variance Symbol Means in Quantitative Research
Variance measures dispersion, but it does not directly tell you whether a scale is reliable or whether a hypothesis is supported. This is where many students get confused when they see a large number in the Descriptive Statistics table.
If you measure satisfaction with a Likert scale from 1 to 5, variance tells you whether the responses are concentrated at one level or spread across several levels. A small variance may indicate that respondents selected similar levels. However, a small variance can also appear because an item is too easy, the sample is too homogeneous, or respondents repeatedly selected the same type of response.
A large variance indicates clearer differences between observations. In some analyses, this variation is necessary because finding relationships between variables is difficult when everyone gives exactly the same answer. A large variance does not automatically prove that the data are good. You still need to check data entry, extreme values, variable coding, and the data collection context.
Variance is also the basis for calculating standard deviation. Standard deviation is the square root of variance, so the two statistics are closely related. You can also read the article standard deviation when the output contains both the Std. Deviation and Variance columns.
In descriptive statistics, variance helps you describe the data. In regression, ANOVA, and more advanced models, issues related to unequal variance may affect how you interpret the results. For that reason, read variance together with Mean, Std. Deviation, the minimum, the maximum, and the number of valid observations instead of treating it as an isolated number.
What Variance Value Is Acceptable
There is no single general threshold that makes variance acceptable for every topic. Variance depends on the measurement scale, unit, value range, and analysis objective. The variance of income measured in millions of đồng will have a very different scale from the variance of a Likert variable ranging from 1 to 5.
When you read a variable description table, use variance to understand dispersion rather than treating it as the only criterion for deleting an item. A variable with low variance is not necessarily a poor variable. A variable with high variance does not necessarily need to be removed. Your decision must follow the research design and the relevant tests.
| Point to check | Cautious interpretation | Source or basis |
|---|---|---|
Variance is large or small | Compare values within the same scale and unit. Do not apply one general threshold | No general threshold in the permitted source list |
| Variance equals 0 | All observations have the same value, so the variable has no variation | Direct inference from the definition of variance |
| Very small variance | Check whether the variable was entered or coded incorrectly, or whether the sample is too homogeneous | Check the actual data |
| Large variance | Check outliers, measurement units, and the value range before drawing a conclusion | Check the actual data |
| Standard deviation | The square root of variance, and often easier to interpret because it uses the same unit as the variable | Algebraic relationship between the two statistics |
| Variance in a statistical test differs | You may need to examine variance assumptions depending on the type of test | Determine from the specific analysis procedure |
| Likert scale | Variance describes dispersion. It does not replace Cronbach's Alpha or EFA | (Cronbach, 1951), (Hair et al., 2010) |
Thresholds such as Cronbach's Alpha of 0.7 or above are used to assess scale reliability according to (Nunnally, 1978), while a corrected item-total correlation of 0.3 or above follows (Nunnally and Bernstein, 1994). These are thresholds for different statistics. They are not thresholds for deciding whether Variance is acceptable.
If an item has a variance of 0, check it immediately. When every respondent selects the same code, that item provides no information for distinguishing between cases. This situation often relates to a poorly designed item, a survey filter, a coding error, or duplicated data.
How to Read the Variance Symbol in SPSS Output
In SPSS, the quickest approach is to run descriptive statistics for the numerical variables you need to check. Open the .sav file and select Analyze > Descriptive Statistics > Descriptives. Move the variables into the Variable(s) box, click Options, select Mean, Std. deviation, Variance, Minimum, and Maximum, then click Continue and OK.
SPSS will create a Descriptive Statistics table. The Variance column contains the variance. The Std. Deviation column contains the standard deviation. The N column shows the number of observations used in the calculation, usually the number of valid cases for that variable.
The table below is illustrative output, not the result of a real study. The numbers are constructed so that you can identify the correct location and column names in SPSS output.
| Descriptive Statistics | N | Minimum | Maximum | Mean | Std. Deviation | Variance |
|---|---|---|---|---|---|---|
| HL1 | 182 | 1.00 | 5.00 | 3.74 | 0.912 | 0.832 |
| HL2 | 182 | 1.00 | 5.00 | 3.61 | 1.041 | 1.084 |
| HL3 | 181 | 2.00 | 5.00 | 3.88 | 0.768 | 0.590 |
| Valid N (listwise) | 181 |
For the HL1 row, the illustrative variance is 0.832 and the standard deviation is 0.912. The two values match approximately because 0.912² ≈ 0.832. Do not read 0.832 as a percentage, and do not say that HL1 “achieves 83.2%”. It is a squared quantity expressed in the squared unit of the variable.
The HL3 row has N = 181, which is lower than the other two variables. This indicates that at least one case has missing data for HL3 or was excluded from that variable's calculation. If you are preparing to run Cronbach's Alpha or EFA, find the reason for the difference in the number of observations before reporting the results.
When the output includes a Frequencies table, you can also view variance in the Statistics section if you selected this statistic for display. For an Explore table, use Analyze > Descriptive Statistics > Explore. Move the variable into Dependent List, click Statistics if you need additional descriptive measures, and then inspect the generated results.
If you run regression, a Descriptive Statistics table may appear in the same output, but the variance there remains a descriptive statistic. You need to read the Coefficients, ANOVA, and Model Summary tables separately to draw conclusions about the model. The variance of one variable does not replace the p-value, regression coefficient, or R².
What to Do When the Variance Symbol Looks Problematic
First, stop deleting a variable simply because its Variance is high or low. Work through the checks in order so that you do not alter the data merely to make the output look better.
Check Coding and Variable Type
Open Variable View and inspect Measure, Values, Missing, and the variable labels. For a Likert variable, check that codes from 1 to 5 were entered correctly. A common error is having some rows use code 0 or ina, which can distort the variance.
If a variable is reverse-coded, check whether you recoded it before calculating the scale score. The usual path is Transform > Recode into Different Variables. Keep the original variable, create a new variable, and record the recoding rule in the syntax file or your analysis log.
Check Missing Values and the Number of Observations
Compare N across variables. If one variable has an unusually low number of observations, open Analyze > Descriptive Statistics > Frequencies to inspect the distribution of codes. Do not automatically replace every missing value with the mean, because this can change dispersion and affect later results.
Check Outliers and Data-Entry Errors
If the variance is large, inspect Minimum, Maximum, boxplots, and values outside the range permitted by the questionnaire. You can refer to the article outlier to distinguish a valid extreme value from an entry error.
A respondent with genuinely high income should not be deleted simply because that value increases the variance. You need a clear methodological reason, such as a value outside the range allowed by the questionnaire or a record that violates a screening condition. If you exclude a case, record the case code, reason, and results before and after the treatment.
Rerun the Analysis After a Valid Reason Has Been Established
After checking the data, you can run Analyze > Descriptive Statistics > Descriptives again. If your objective is scale assessment, continue with the appropriate Cronbach's Alpha, EFA, or PLS-SEM procedure for your thesis. A variable having a different variance from the other variables is not enough to decide whether to keep or remove it.
If the data genuinely have no variation, a software option in SPSS cannot create new information. You need to report the data limitation and discuss the appropriate treatment with your supervisor. Editing original responses to make the variance look better creates serious risk when the committee asks about the raw data.
Distinguishing the Variance Symbol from Standard Deviation
Variance and standard deviation both measure dispersion, but their units and interpretation differ. Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance.
For example, if the variance of a score variable is 0.832, the corresponding standard deviation is approximately 0.912. Variance uses a squared unit, while standard deviation uses the same unit as the original variable. For that reason, Std. Deviation is often more intuitive in a descriptive table intended for readers.
You also need to distinguish variance from the mean. Mean shows the central level of the data, while Variance shows dispersion around that centre. Two groups can have the same mean but completely different variances. Conversely, two groups can have similar variances but different means.
Variance also differs from the median. The median is the middle value after the data are ordered, and it is useful for describing central position when the data are skewed or contain outliers. You can also read median if you are comparing Mean, Median, and the data distribution.
When reading a paper or output, do not confuse S² and σ² with S and σ. A symbol without the exponent 2 usually denotes standard deviation, while a symbol with the exponent 2 denotes variance. In SPSS, you do not identify the statistic from a mathematical symbol. You identify it from the column names Variance and Std. Deviation.
Common Errors
The first error is searching for one fixed number that determines whether variance is acceptable. This ignores the scale and unit of measurement. Compare variables within the same context and read the purpose of the analysis.
The second error is interpreting variance as a percentage. Variance = 1.20 does not mean that the data vary by 120%. If you want to describe dispersion, use a sentence that reflects the correct unit and statistic.
The third error is confusing variance with R². R² shows the proportion of variation in the dependent variable explained by a model in regression or PLS-SEM. In PLS-SEM, R² levels are often interpreted according to (Hair et al., 2011), but R² is a model statistic, not the Variance column in descriptive statistics.
The fourth error is deleting a variable because its variance is low without checking the questionnaire. If everyone answered alike because the variable reflects a genuinely stable characteristic, deleting it requires a clear argument. If the cause is an incorrect code, correct the original data or exclude the record according to a rule established in advance.
The fifth error is ignoring the number of valid observations. A variance calculated from different numbers of observations should not be compared mechanically. Always read the N column, check missing values, and state how the data were handled in the methods chapter.
How to write this in your thesis
You can adapt the following sentence when reporting the result: “[Variable code] had a mean of [Mean] and a standard deviation of [Std. Deviation], while its variance was [Variance]. These values were used to describe the dispersion of the data and were not used as the sole criterion for deleting the variable.” Replace the placeholders with the values in your file and state the number of valid observations.
Frequently asked questions
What is the variance symbol in statistics?
The population variance is usually represented by σ², while sample variance is usually represented by s². In SPSS, you identify variance by the Variance column in the Descriptive Statistics table rather than by a mathematical symbol.
What variance value is acceptable?
There is no general threshold that makes variance acceptable for every variable and topic. You need to consider the scale, value range, unit, number of observations, and analysis objective. A variance of 0 indicates no variation, while a large variance requires further checking and is not, by itself, a basis for deleting a variable.
Is variance the same as standard deviation?
The two statistics are related but not identical. Standard deviation is the square root of variance and retains the unit of the original variable. Therefore, Variance and Std. Deviation can both describe dispersion, but you must not use one name in place of the other.
Why is the variance in SPSS very large?
Check the measurement unit, value range, coding, missing values, and outliers. If the large value is valid, do not delete it merely to make the output look better. If it is an entry error or a code outside the questionnaire range, correct it using the original data and save the reason for the correction.
Should I remove a variable with low variance?
Not necessarily. Low variance only shows that the responses are concentrated. Check whether the variable was entered with the same code, whether its design is too uniform, and whether it fits the research construct. The decision to remove it should also consider Cronbach's Alpha, corrected item-total correlation, EFA, or the appropriate measurement model.
Is it better to calculate variance by hand or run SPSS?
Calculating variance by hand is useful for understanding the formula and checking a small dataset. With many variables and rows, SPSS reduces calculation errors and provides Mean, Std. Deviation, Variance, Minimum, Maximum, and N together. You should still understand the formula so you can detect data-entry errors or misread output.
Open your SPSS file, run Analyze > Descriptive Statistics > Descriptives, compare N, Mean, Std. Deviation, and Variance, then record the cases that need checking before you run the next analysis. If you need to run these steps on your own .sav or .csv data, use M4 analysis by DoThesis.