Sample Size Calculation in Quantitative Research: Formulas and Selection
What is sample size calculation
Sample size calculation is the process of determining the minimum number of observations a study needs for its data to support the intended analysis. An observation is usually one questionnaire respondent, household, business, or other research unit, depending on your topic.
You need to separate three concepts that are often mixed up: the study population, the research sample, and the sample size. The population is the entire group about which you want to draw conclusions. The sample is the part selected from that population. Sample size, usually written as n, is the number of actual units in that sample. If you survey 250 customers, the sample size is 250, even if you initially sent the questionnaire to 300 people and removed 50 invalid responses.
The formula depends on your research objective. For estimating a proportion in a large population, you can use a formula of the following form:
n = Z² × p × (1 - p) / e²
In this formula, Z is the value corresponding to the confidence level, p is the expected proportion of the characteristic being estimated, and e is the allowable error. When the population is finite and its size is known, the formula needs to be adjusted for that population size. The finite-population sample size formula is commonly presented according to Yamane (1967).
If your study uses regression, EFA, or PLS-SEM, the required number also depends on the number of variables, the number of observed variables, the number of paths, and the testing objective. You should therefore not take one formula and apply it to every topic. You can review the background explanation in what is sample size, then compare it with your model and analysis method.
Why sample size calculation matters in quantitative research
Sample size directly affects whether your data can be used. A sample that is too small can produce unstable estimates, make regression coefficients difficult to interpret consistently, and reduce the ability of a test to detect a real relationship. A larger sample usually gives your results a stronger basis, but it does not automatically fix a weak questionnaire, an incorrectly selected population, or dishonest responses.
When you calculate sample size, you are answering at least four questions. First, who does the study survey. Second, is the population size known. Third, do you need to estimate a proportion, compare groups, or test a relationship model between variables. Fourth, is the final analysis descriptive statistics, regression, EFA, CFA, AMOS, or SmartPLS.
For a scale with multiple observed variables, a commonly used rule is 5 to 10 observations for each observed variable (Hair et al., 2010). For example, if your questionnaire has 28 observed variables, this rule gives a reference range of 140 to 280 valid observations. This is one basis for planning, not permission to ignore the analysis objective or sampling design.
For linear regression, Tabachnick and Fidell propose a minimum sample size rule based on the number of independent variables, commonly written as n ≥ 50 + 8m, where m is the number of independent variables (Tabachnick and Fidell, 2013). If the model has 6 independent variables, this rule gives a reference minimum of 98 observations. You still need to consider the number of groups, control variables, and your plan for removing invalid responses.
A correctly calculated sample size can still produce weak conclusions if you collect data from the wrong respondents. Read more about sampling methods and state clearly how you approached respondents in the methods chapter.
What sample size is sufficient
There is no single number that is sufficient for every study. State the formula, the parameters used, the minimum number of observations after cleaning, and the expected number of questionnaires to distribute. The table below shows common reference points and their sources, so you can explain your decision to your supervisor.
| Situation or criterion | Reference level | How to use it in your study | Source |
|---|---|---|---|
| Proportion in a large population | Calculate using Z² × p × (1-p) / e² | Use when the objective is to estimate a proportion and you have specified the confidence level and error | (Cochran, 1977) |
| Finite population with known size N | Use the formula adjusted for N | Use when you have a list or a clear estimate of the population size | (Yamane, 1967) |
| Each observed variable | 5 to 10 observations | Multiply the number of observed variables by 5 or 10 to obtain a reference range | (Hair et al., 2010) |
| Regression with m independent variables | n ≥ 50 + 8m | Check the minimum level before running regression | (Tabachnick and Fidell, 2013) |
| Factor analysis | Evaluate together with the number of variables, factor loadings, and data quality | Use as an additional basis, not as a replacement for checking KMO | (Comrey and Lee, 1992) |
| Scale Alpha | 0.7 or above is commonly accepted | This is a criterion for a scale after you have data, not a sample size formula | (Nunnally, 1978) |
For example, you have 24 observed variables, 5 independent variables, and plan to run regression. The observed-variable rule gives a range of 120 to 240 observations. The regression rule gives a minimum of 90 observations. In this case, 120 is the lower reference point between the two calculations, but you should still set a higher collection target to cover incomplete and invalid responses.
If you need 200 valid responses and expect 15% of responses to be removed, the number of questionnaires to distribute can be calculated as 200 / (1 - 0,15), approximately 236 questionnaires. Report two separate figures: the valid sample size required for analysis and the number of questionnaires to collect. Do not call 236 the final sample size if only 200 observations remain after cleaning.
How to read sample size information in output
Sample size calculation usually does not appear as a separate table in SPSS. You calculate it before the survey, then check the actual number of observations in the data file and output. In SPSS, open the file and go to Analyze > Descriptive Statistics > Frequencies, then move an identification variable or a coded variable into the analysis field. The Statistics and Frequency Table tables help you check the number of valid observations.
For regression, go to Analyze > Regression > Linear. Move the dependent variable to Dependent and the independent variables to Independent(s), then run the model. The Model Summary table has an N column in some summary results, or the number of observations may appear in the output notes. If the variables contain missing values, the number used in regression may be lower than the number of rows visible in Data View.
The table below is illustrative output, not the result of a real study. The column names follow SPSS output so you know where to look.
Illustrative output in SPSS
| Output or information to read | Illustrative value | Meaning when checking sample size |
|---|---|---|
Valid N in Descriptive Statistics | 238 | There are 238 valid observations for the variable being checked |
Missing N in Descriptive Statistics | 12 | There are 12 missing values for the variable being checked |
| Number of independent variables in the model | 5 | Use this to calculate m in the regression rule |
50 + 8m level | 90 | Reference minimum based on the number of independent variables |
| Number of observed variables in the questionnaire | 24 | A range of 120 to 240 under the 5 to 10 times rule |
N used in Model Summary | 221 | This is the actual number of observations entered into the regression model |
In this example, 238 is the valid number when checking one variable, while 221 is the number used in regression because some rows have missing values for related variables. You must report 221 when describing the regression model. Using 238 as the N for every analysis would make the descriptive and regression tables inconsistent.
If you are using SmartPLS, check the .csv file, column names, and number of rows before running Calculate. SmartPLS cannot turn a file with too few observations into a sufficiently strong sample. With PLS-SEM, you also need to assess outer loading, CR, AVE, HTMT, VIF, and the paths in the model. The section on sample size in SPSS is useful when you need to check N, missing values, and SPSS output tables directly.
What to do when the sample size is insufficient
First, identify the basis against which you are falling short. If you are below a target of 300 but still above a defensible minimum, you should not immediately conclude that the data are unusable. If the number of observations is below the rule you selected, proceed in the following order.
Check the observation unit and valid responses
Check whether you counted the same respondent twice, included incomplete responses, or left test rows in the file. Review response time, required items, and repeated response patterns. After this step, record the number of questionnaires distributed, the number returned, the number removed, and the final N.
Compare the calculation with the actual model
Count the observed variables and independent variables in the final model correctly. Do not reduce the number of variables simply to make the formula look better if that conflicts with the theoretical model or registered hypotheses. If an item is removed from a scale, state the reason and rerun the analysis using a consistent procedure.
Collect additional responses if access is still possible
If you still have time and access to respondents, collecting more data is the easiest option to explain. Keep the respondent criteria, questionnaire introduction, and cleaning procedure unchanged. Do not switch to a different population halfway through simply because the new group is easier to reach.
Report the limitation if you cannot collect more
When you cannot increase N, present the formula used, the number of valid observations, the reason for the insufficient sample, and the possible effect on generalizability. Your supervisor may ask you to rerun the model, narrow the scope of the conclusions, or add stability checks. Do not use software operations to hide an insufficient sample size.
Distinguishing sample size calculation, sample size, and sampling method
Sample size calculation is the decision about how many observations are needed. What is the sample size of a dataset is often used to explain the number of elements in a specific dataset. The two expressions are closely related, but one emphasizes the process of setting a target and the other emphasizes the number available after collection.
Sample size is also different from the sampling method. Random, convenience, quota, and stratified sampling answer the question, “Who is selected and how are they selected?” Sample size answers the question, “How many units are needed?” A sample of 200 people can still be biased if every respondent comes from a convenient group that does not represent the population.
Sample size is not the same as scale reliability. Cronbach's Alpha is calculated after you have data and run the scale reliability test. Sample size is also not one fixed N for the entire thesis, because individual analyses may remove different rows with missing values.
Common mistakes
The first mistake is copying a formula without explaining its parameters. When you write the formula, state the population, confidence level, error, expected proportion, or the way you selected a substitute value. If you use the rule of 5 to 10 times the number of observed variables, include both the number of variables and the multiplication.
The second mistake is treating the number of questionnaires distributed as the analysis N. N must be the number of valid observations used in the relevant table or model. If SPSS removes rows because of missing values, report the actual N in the output.
The third mistake is calculating several formulas and then choosing the lowest number without explanation. You can present several bases, but you need to state which basis best matches the main objective and why the final level is sufficient for the planned analysis.
The fourth mistake is assuming that a large sample will fix incorrect sampling. If the questionnaire reaches the wrong respondents, adding more respondents does not automatically make the sample more representative. The final mistake is removing large numbers of responses or variables only to meet a threshold. Each removal needs a predefined criterion, a trace in the file, and a defensible reason.
Frequently asked questions
What is sample size calculation, and when should I do it?
It is the process of determining the minimum number of observations you need before collecting data. You should do it after identifying the respondents, model, observed variables, and planned analysis method, then update the valid number after cleaning the data.
What sample size is sufficient for a thesis?
There is no universal level for every thesis. Compare the research objective, number of observed variables, number of independent variables, and analysis method. The rule of 5 to 10 observations for each observed variable and the 50 + 8m rule for regression are commonly used bases, but you must state the source and how you applied it.
Can I use the Cochran formula for every study?
Use the formula only when it matches the objective of estimating a proportion and the assumptions of the formula. If your study focuses on EFA, regression, or PLS-SEM, add a basis that reflects the model and number of variables. A formula does not replace checks of sample and data quality.
What should I do if the calculated sample size is insufficient?
First check duplicate rows, incomplete responses, missing values, and removal criteria. If N is still below the selected basis, prioritize collecting additional responses using the same sampling criteria. If you cannot collect more, report the limitation and discuss it with your supervisor instead of deleting data arbitrarily or changing the formula after seeing the results.
Is sample size the number of questionnaires distributed?
The analysis sample size is the number of valid observations remaining in the data used for a particular analysis. The number of questionnaires distributed needs to be higher than this target to cover incomplete responses, unsuitable responses, and missing values. Present the two figures separately in your thesis.
How to write this in your thesis
Open your data file now, record the number of observed variables, the number of independent variables, the valid N, and the sample size basis you are using. Then compare each output before writing the methods chapter. If you need to run checks on your own .sav or .csv file, you can use the M4 data analysis module.