What Is Covariance? How to Read and Analyze It in SPSS

What Is Covariance

You will usually encounter covariance when reading descriptive statistics, correlation analysis, regression, or data matrices in SPSS. This statistic shows whether two variables tend to change together or in opposite directions. For example, when weekly study hours increase and exam scores also tend to increase, the covariance between the two variables may be positive. When one variable increases while the other decreases, the covariance may be negative.

For two variables X and Y, sample covariance is commonly calculated with the following formula:

[ Cov(X,Y)=\frac{\sum_{i=1}^{n}(X_i-\bar X)(Y_i-\bar Y)}{n-1} ]

Here, (X_i) and (Y_i) are the values for one observation, (\bar X) and (\bar Y) are the means, and n is the number of observations. The denominator n - 1 is used when covariance is calculated from a sample. When you calculate it for an entire population, the denominator is usually n.

The direct interpretation comes from the product ((X_i-\bar X)(Y_i-\bar Y)). When X and Y are both above their means or both below their means, this product is positive. When one value is above its mean and the other is below its mean, the product is negative. Adding these products across all observations shows the general pattern in which the two variables vary together.

You can also review the variance formula to see how variance and covariance are related and how they differ. Variance examines one variable in relation to itself, whereas covariance examines the joint variation between two variables.

What Covariance Means in Quantitative Research

Covariance answers a specific question: do two variables tend to change in the same direction? You can usually read the sign of the statistic as follows:

• Positive covariance: large values of X tend to occur with large values of Y, and small values of X tend to occur with small values of Y.
• Negative covariance: large values of X tend to occur with small values of Y, or the other way around.
• Covariance close to 0: there is no clear linear pattern showing that the two variables vary together.

You should not use covariance alone to conclude that a relationship is strong or weak. Its magnitude depends on the measurement units of both variables. If you change X from metres to centimetres, the covariance changes as well, even though the underlying data and the direction of the relationship between X and Y have not changed.

In quantitative analysis, covariance commonly appears in three situations. First, it describes the joint variation between variables. Second, it is part of the calculation of the Pearson correlation coefficient. Third, it is used in matrix calculations for regression, factor analysis, and some multivariate models.

If your goal is to interpret the strength of association in a way that is easy to compare across pairs of variables, the correlation coefficient is usually more suitable because it standardizes covariance using the standard deviations of the two variables. You can read more about standard deviation to understand why the same covariance may have a different meaning when the measurement scale changes.

Covariance can also help you identify data problems. Very large covariance between a pair of variables may simply reflect different measurement units, such as one variable measured in đồng and the other in million đồng. An unusual covariance may also be related to an outlier, an entry error, or a subgroup of observations with different behaviour.

What Covariance Value Is Acceptable

There is no universal cutoff for deciding whether a covariance value is acceptable. This is easy to confuse when you are looking for a number similar to Cronbach's Alpha, KMO, or factor loading. A value of 10, 100, or 0,1 cannot be judged on its own without knowing the measurement units, the variance of each variable, and the purpose of the analysis.

The table below helps you choose the right way to read the result. The thresholds shown are thresholds for related statistics, not mandatory cutoffs for covariance.

The more useful questions are therefore whether the sign of the covariance matches the hypothesis, whether the value is affected by the measurement units, and what the correlation coefficient or regression model shows. If both variables use a 1 to 5 Likert scale, comparison is easier than when one variable measures income and the other measures the number of purchases.

You also need to distinguish covariance from hypothesis testing. Covariance describes a common pattern of variation. It does not automatically provide a p-value that allows you to conclude that the relationship is statistically significant. When you need to make a statement about correlation, read the Correlations table and the Sig. column in SPSS.

How to Read Covariance in the Output

SPSS does not always display a table named exactly Covariance Matrix when you run basic analyses. Depending on the procedure, covariance may appear in a Covariance matrix, a Parameter Estimates table, or behind other calculations. Read the table name and column names instead of guessing from an isolated number.

The example below is illustrative output, not the result of a real study. Suppose you have two composite variables: weekly study hours and semester grade point average. You run Analyze > Correlate > Bivariate, select Pearson, and enable the option to display the covariance matrix if your version of SPSS provides that option.

On the main diagonal, 18,420 is the variance of Weekly study hours and 0,214 is the variance of Semester grade point average. The two symmetrical cells outside the diagonal are the covariance between the two variables, which is 1,860 here. The positive value shows that the two variables tend to increase in the same direction in the illustrative sample.

You should not conclude that 1,860 represents a strong relationship simply because it is larger than 0,214. The two variables have different units and different levels of dispersion. To assess the relationship on a standardized scale, read the Pearson coefficient in the Correlations table.

In the illustrative example, Pearson Correlation is 0,296 and Sig. 2-tailed is 0,012. If your study sets the significance level at 5%, you can describe a positive linear association that is statistically significant in the illustrative sample. This conclusion comes from the correlation table and the p-value, not from covariance alone.

When you are reading regression output, do not look for covariance in the Coefficients table and interpret it as a regression coefficient. B is the unstandardized coefficient, Beta is the standardized coefficient, t is the test statistic, and Sig. is the p-value. Covariance is only one component in the calculations that produce the regression estimates.

What to Do When Covariance Does Not Look Acceptable

Because covariance has no fixed acceptable cutoff, you should not delete a variable simply because its value is negative, small, or large. Work through the checks in order so that you do not alter the data without a defensible reason.

Check the Measurement Units and Data Entry

Start by checking the units used for both variables. If one variable records money in đồng and the other records money in million đồng, the covariance will be strongly affected by the way the units are entered. Also check unusually large values, decimal separators, and cases where 500000 was entered instead of 5000000.

Check Reverse-Coded Variables and Coding

If your questionnaire contains a reverse-coded item and you have not recoded it, the relationship may appear with the opposite sign. For a 1 to 5 Likert scale, a reverse-coded item usually needs to be transformed according to the relevant rule before you calculate a composite score. Check the questionnaire and the study's coding scheme. Do not change the sign simply to make the covariance positive.

Check Outliers and Missing Data

A few extreme observations can pull covariance substantially upward or downward. Examine the scatterplot, the minimum and maximum values, and the number of valid observations. You can refer to the article on outliers to check unusual data before deciding whether to retain or remove anything.

Removing an outlier requires a clear reason, such as an entry error or a value outside the survey population. Do not delete a row simply because it makes the result less favourable to the hypothesis. If you remove data, record the criterion and the number of observations before and after processing.

Use Correlation for Comparison

If your goal is to compare the strength of association across several pairs of variables, report Pearson Correlation or Spearman Correlation according to the data conditions. Covariance can remain in the calculations or appendix, but the main interpretation should use the statistic that matches your research question.

Distinguishing Covariance from Variance and Standard Deviation

Covariance, variance, and standard deviation are all related to dispersion, but they answer different questions. Variance shows how much one variable is dispersed around its mean. Standard deviation is the square root of variance and uses the same units as the original variable. Covariance shows how two variables vary together.

You can read the article on variance when you need to analyse one variable, or the article on median when you want to compare central values in data with a skewed distribution. The table below summarizes the differences:

A quick check is to look at the diagonal of the covariance matrix. The cell where a variable intersects with itself is that variable's variance. The cells outside the diagonal are the covariances between each pair of variables. If you standardize the off-diagonal cells using the standard deviations of the two variables, you obtain the Pearson correlation.

Covariance also differs from a regression coefficient. A regression coefficient represents the expected change in the dependent variable when the independent variable increases by one unit, while other variables are held constant. Covariance only describes joint variation and does not control for other variables.

Common Errors

The first error is looking for a universal cutoff for covariance and then declaring the result acceptable or unacceptable. This ignores the measurement units and the dispersion of each variable. Examine the sign, measurement context, correlation, and purpose of the analysis.

The second error is calling covariance correlation. The two statistics are related by their formulas, but they do not use the same scale. Correlation is usually easier to report because it falls between -1 and 1, whereas covariance has no fixed range of that kind.

The third error is interpreting covariance as a causal relationship. Positive covariance does not prove that X causes Y. To test a causal hypothesis, you need an appropriate model, control variables where necessary, and a report of the coefficient and p-value.

The fourth error is ignoring reverse-coded items, outliers, and missing data. An unexpected negative sign may result from incorrect coding, but it may also reflect the theory or the behaviour of the sample. Check the data before making changes.

You can write the result in your thesis using the following template: “The analysis shows that the covariance between [variable X] and [variable Y] is [value], with a [positive/negative] sign, indicating that the two variables [vary in the same/opposite direction] in the sample. Because covariance depends on the measurement units, the study also uses [Pearson Correlation/appropriate test] to assess the strength of the association and its statistical significance.”

Frequently asked questions

What is covariance and what is it used for?

Covariance is a statistic that describes the direction in which two variables vary together. A positive sign indicates a tendency to move in the same direction, a negative sign indicates a tendency to move in opposite directions, and a value close to 0 indicates no clear common linear pattern.

What covariance value is acceptable?

Covariance has no universal acceptable cutoff because it depends on the measurement units and dispersion of the two variables. You should not use a large or small value on its own to conclude that a relationship is good or bad. Also examine the correlation coefficient, p-value, and research hypothesis.

Does negative covariance mean that the data contain an error?

Negative covariance does not automatically mean that the data are incorrect. It may reflect an inverse relationship that is consistent with the theory. However, check reverse-coded items, coding, measurement units, and outliers if the negative sign is unexpected.

How does covariance differ from correlation?

Covariance depends on the measurement units of the two variables, whereas correlation is standardized and is therefore easier to compare across pairs of variables. Covariance shows the direction of joint variation, while correlation shows both the direction and strength of the linear association on a scale from -1 to 1.

Where is the covariance table in SPSS?

Depending on the procedure, SPSS may display a covariance matrix in tables related to correlation, regression, or parameter estimation. When you run the analysis, read the table name carefully and compare the diagonal cells with the off-diagonal cells. If your goal is to test an association, you also need to examine the Correlations table and the Sig. column.

How to write this in your thesis

Open your data file again, identify the two variables being compared, check their coding and measurement units, and then read covariance together with Pearson Correlation instead of treating one number as the entire result. If you need to run the analysis on your own .sav or .csv file, M4 analysis from DoThesis performs this step with SPSS and SmartPLS.