What Is the Geometric Mean? How to Calculate and Interpret It in SPSS
What Is the Geometric Mean?
The geometric mean, also called the geometric mean, is calculated by multiplying positive values together and then taking the nth root, where n is the number of observations. If there are n values, (x_1, x_2, ..., x_n), the formula is:
[ G = \sqrt[n]{x_1 \times x_2 \times ... \times x_n} ]
A more practical method for a dataset with many numbers is to take the exponential of the arithmetic mean of the logarithms:
[ G = e^{\frac{\ln(x_1)+\ln(x_2)+...+\ln(x_n)}{n}} ]
The important condition is that every value used in the calculation must be greater than 0. If the series contains a zero or a negative number, the usual geometric mean formula cannot be applied directly. When you open your data file, check the scale coding, missing values, and measurement units before choosing this statistic.
For example, with the three values 2, 8, and 32, the geometric mean is the cube root of 512, which equals 8. The arithmetic mean of the same series is 14. The results differ because the geometric mean reflects multiplicative relationships or relative rates of change, while the arithmetic mean reflects the total amount divided equally across the observations.
If you are checking other statistical concepts for your thesis, open the Statistics category to compare how each statistic is selected.
Why the Geometric Mean Matters in Quantitative Research
The geometric mean is suitable when values are connected through ratios, growth rates, change factors, or several successive multiplicative stages. For example, an investment that rises by 10% in the first period and then falls by 10% in the second period does not return exactly to its starting level because the two effect factors are multiplied. In this situation, the geometric mean of the growth factors is usually more meaningful than the arithmetic mean of the percentage changes.
In survey data, the geometric mean is rarely used to aggregate Likert items scored from 1 to 5 directly. For observed items measuring agreement, you usually report the arithmetic mean, standard deviation, Cronbach's Alpha, EFA, or statistics that match your model. If you are working with a scale, see what is a scale before calculating anything so that you understand the nature of the variable.
The geometric mean may appear in the following situations:
- Calculating the average growth rate across several periods.
- Combining ratio-based indicators, such as price indexes or efficiency coefficients.
- Describing right-skewed data in which a few very large observations pull the arithmetic mean upward.
- Calculating a representative value for quantities that have a multiplicative relationship.
- Comparing relative changes between groups or periods.
This statistic does not automatically show that your research model is good or that a hypothesis is supported. It describes a dataset in a way that matches the nature of the variable. When you decide to use the geometric mean, explain clearly why the data follow a ratio-based or successive multiplicative relationship.
One point to remember is that the geometric mean is usually less than or equal to the arithmetic mean when both are calculated from the same set of positive values. The larger the gap between the two statistics, the more likely the data are skewed or contain substantial differences between observations. You should not, however, identify outliers from this difference alone. To check unusual observations, read the guide on outliers and inspect the chart, frequency table, and the treatment rule you selected.
What Geometric Mean Value Is Acceptable?
The geometric mean does not have a general cutoff such as “0.7 or above is acceptable.” Whether a value is appropriate depends on the variable being measured, the data units, and the research question. For example, the geometric mean of a growth rate may be interpreted as a percentage, while the geometric mean of time or cost retains the original unit.
The table below helps you decide based on the nature of the data. The sources in the table support the corresponding methodological criteria. They are not universal cutoffs for every geometric mean.
| Situation to check | Decision rule | Methodological source |
|---|---|---|
| The variable contains zero or negative values | Do not calculate the usual geometric mean; check the coding and measurement method | No general geometric-mean cutoff in the source list |
| The data represent a rate or growth factor across several periods | You may use the geometric mean when multiplication across periods is meaningful | No general geometric-mean cutoff in the source list |
| The data are Likert observed items | Do not choose the geometric mean simply because the software allows it; consider the measurement objective and scale aggregation method | (Nguyễn Đình Thọ, 2011) |
| You need to check whether very large values are distorting the variable | Compare the geometric mean, arithmetic mean, median, and chart | (Field, 2013) |
| You need to report scale reliability | Use an appropriate reliability statistic rather than replacing it with the geometric mean | (Cronbach, 1951) and (Nunnally, 1978) |
The better question is therefore not “what geometric mean value is acceptable?” It is “is the geometric mean appropriate for this variable and for the interpretation required by the study?” If your supervisor asks for a specific cutoff, state whether it comes from the variable definition, methodological literature, or a convention in your field. Do not take one number from the Internet and apply it to every dataset.
How to Read the Geometric Mean in SPSS Output
SPSS does not always display a separate table named “Geometric Mean.” Depending on the procedure you run, the statistic may appear in a Descriptive Statistics table, an Explore output, or a customized summary table. Check the variable name, the number of valid observations, and the measurement unit before interpreting the value.
To view basic descriptive statistics, open Analyze > Descriptive Statistics > Explore. Move the variable you want to check into Dependent List, select Statistics, keep the descriptive statistics you need, then click Continue and OK. If you want to calculate a geometric mean for one variable, you can first create a logarithmic variable through Transform > Compute Variable, calculate the mean of the logarithmic variable, and then convert it back using the exponential function.
The following example assumes that you have a variable named TANGTRUONG containing positive factors. The numbers only show how to read the output and are not results from a real study.
Illustrative output, Descriptive Statistics table after aggregating the data:
| N | Minimum | Maximum | Mean | Std. Deviation | Geometric Mean |
|---|---|---|---|---|---|
| 120 | 0.82 | 1.45 | 1.084 | 0.116 | 1.078 |
In this table, N = 120 means that 120 valid observations entered the calculation. Minimum and Maximum show the lower and upper bounds of the data. Mean = 1.084 is the arithmetic mean, while Geometric Mean = 1.078 is the geometric mean. If the values are growth factors, a geometric mean of 1.078 may be converted into a relative increase of approximately 7.8% per period, but only when 1.000 represents no change and the period unit has been defined clearly.
You also need to check Missing or the number of observations excluded from the calculation. If SPSS uses listwise deletion for several variables, N may decrease because one missing variable is enough to exclude the entire row. Record your missing-value rule because the same dataset can produce different results under different observation-deletion rules.
If the result was calculated from logarithms, check that the intermediate variable has a recognizable name, such as LN_TANGTRUONG. Do not delete it before comparing the results. A common mistake is to calculate the arithmetic mean of percentage changes and then label that result the geometric mean.
What to Do When the Geometric Mean Is Not Appropriate
The geometric mean is not judged in the same way as a factor loading or p-value. If the result does not fit your research objective, work through the following checks.
Check the Positive-Value Requirement
Use a Frequencies table or Explore to find zero values, negative numbers, and fake missing-value codes such as 99, 999, or -1. These codes may represent invalid responses but still be read by SPSS as real numbers. Correct them in Variable View or create a separate processed variable, then run the analysis again using the rule documented in your methods section.
Check the Units and Transformation
Determine whether the data are recorded as percentages, factors, or original values. If an 8% increase is entered as 8 instead of 1.08, the geometric mean will have a different meaning. Standardize the coding before calculating, and record the formula in syntax or a processing log so that you can explain it later.
Compare the Arithmetic Mean and Median
If the geometric mean is much lower than the arithmetic mean, check the distribution and outliers. The median may help you describe the central value with less sensitivity to extreme observations. Do not delete large values simply to make the two means closer. Every data-removal decision needs a reason and a criterion stated in advance.
Change the Statistic When the Research Question Does Not Fit
If the data are a Likert scale and the goal is to describe agreement, the arithmetic mean is usually easier to interpret. If the data are right-skewed time, income, or cost values, the median or logarithm may be more suitable depending on the model. When you change the statistic, update the methods section, results table, and interpretation so that they remain consistent.
A reviewer can accept a change of statistic when you explain the nature of the variable, the condition for applying the statistic, and the methodological reason. Running several statistics and selecting only the “prettiest” result without reporting the selection process reduces the credibility of the results chapter.
How the Geometric Mean Differs from the Arithmetic Mean and Median
The arithmetic mean is calculated by dividing the sum of the values by the number of observations. It is suitable when each additional unit has an additive meaning, such as a total score or number of hours. The arithmetic mean is sensitive to a few very large values.
The median is the value in the middle after the data are ordered. When the number of observations is even, the median is usually determined from the two middle values. This statistic is useful when the data are skewed or contain outliers, but it does not represent a multiplicative relationship across periods.
The geometric mean is suitable for ratios and multiplicative factors. It answers a question about a typical relative change rather than a typical amount added. The following table uses a small example so that you can see the difference.
| Illustrative data | Arithmetic mean | Median | Geometric mean |
|---|---|---|---|
| 2, 8, 32 | 14.00 | 8.00 | 8.00 |
| 1, 2, 100 | 34.33 | 2.00 | 5.85 |
| 0.8, 1.0, 1.2 | 1.00 | 1.00 | 0.986 |
No statistic is always better than the other two. Choose according to the mechanism that generated the data and the way readers need to understand the result. With a Likert questionnaire, do not call the average of observed-item scores a geometric mean when the actual calculation is an arithmetic mean.
Another source of confusion appears when writers use “average value” to refer to every measure of central tendency. In your thesis, write Mean, Median, or Geometric Mean explicitly. This tells the reader exactly which calculation was performed.
Common Mistakes
Using the Geometric Mean for Data with Zero or Negative Values
This is a mathematical-condition error, not an SPSS formatting error. Check why the value exists before applying a logarithmic transformation. Do not replace zero with an arbitrary small number unless you have a clear methodological reason.
Confusing Percentages with Growth Factors
A 20% increase corresponds to a factor of 1.20, while a 20% decrease corresponds to a factor of 0.80. If you enter 20 and -20 directly into a geometric-mean calculation, the result can no longer be interpreted as a growth rate.
Calling the Arithmetic Mean the Geometric Mean
The statistic's name must match the formula. Reopen Transform or the syntax to confirm the calculation rather than relying only on the number shown in the output table.
Setting an “Acceptable” Cutoff for the Geometric Mean
A number cannot establish by itself that a statistic is appropriate. Answer three questions: what does the variable measure, what is its unit, and why are the relationships between values additive or multiplicative?
Deleting Outliers Because the Two Means Differ
The difference may be a real feature of the distribution. Check moderator or related analyses only when your model actually contains a moderating variable. Do not use them to explain a descriptive table without a model-based reason.
Frequently asked questions
What is the geometric mean, and when should I use it?
The geometric mean is the nth root of the product of n positive values. Use it when the data represent ratios, growth factors, or multiplicative effects across several stages. For ordinary Likert scores, reconsider the measurement objective before selecting this statistic.
What geometric mean value is acceptable?
There is no general cutoff for the geometric mean. A result is appropriate when the data conditions, formula, measurement unit, and research objective are consistent. Explain the logic behind your choice instead of labeling one isolated number acceptable or unacceptable.
How do I calculate the geometric mean using the example formula?
With the data 2, 8, and 32, multiply the three values to obtain 512, then take the cube root. The result is 8. With many observations, using logarithms reduces the risk of numerical overflow and is more convenient in SPSS.
How does the geometric mean differ from the arithmetic mean and median?
The arithmetic mean is based on the sum, the median is based on the middle position after sorting, and the geometric mean is based on the product of positive values. The geometric mean is more suitable for ratio data, while the arithmetic mean is commonly used to describe scores or total quantities.
What should I do if SPSS does not show a Geometric Mean column?
Create a logarithmic variable through Transform > Compute Variable, calculate its mean, and use the exponential function to convert the result back to the original scale. Check zero values, negative numbers, missing values, and data units before drawing a conclusion.
Should I use the geometric mean for a Likert scale?
Usually, you should not choose it simply as a replacement for the arithmetic mean. Base the decision on the way the scale was constructed, the analysis objective, and the methodological guidance for your study. If you are testing a scale, the geometric mean does not replace Cronbach's Alpha, EFA, outer loading, or validity tests.
How to write this in your thesis
Open your SPSS file, check for zero values, negative numbers, and missing-value codes, then record the arithmetic mean, median, and geometric mean separately so that you can compare them in the descriptive table. If you need to run the analysis on your own .sav or .csv file, use the M4 data analysis module.