How to Calculate Standard Deviation and Read SPSS Results
What is standard deviation
You open your data file and see the Std. Deviation column, but you are not sure what the number says about your sample. Standard deviation, commonly written as SD or Std. Deviation, measures how far values are dispersed around the Mean. The smaller the SD, the more closely the observations cluster around the Mean. The larger the SD, the more widely the data are spread.
Calculating standard deviation starts with four steps: calculate the Mean, subtract the Mean from each value, square the deviations, and then take the average using the appropriate formula before taking the square root. For a population, the formula is:
σ = √[Σ(xᵢ - μ)² / N]
For a survey sample, the formula usually used is:
s = √[Σ(xᵢ - x̄)² / (n - 1)]
Here, xᵢ is each observation, x̄ is the sample mean, n is the number of observations, and n - 1 is the degrees of freedom. When you analyze student survey data in SPSS, the software usually reports Std. Deviation using the sample calculation. You can also see the article on standard deviation if you need to compare the concept and notation.
For example, take four scores: 2, 3, 4, and 5. The Mean is 3,5. The deviations from the Mean are -1,5, -0,5, 0,5, and 1,5. After squaring and adding them, divide by n - 1 and take the square root to obtain the sample SD. The final number describes the dispersion of the four scores. It does not tell you whether the average score is high or low.
The meaning of standard deviation in quantitative research
In quantitative research, you use SD to describe how different the responses or observations are. The Mean shows the central tendency, while the SD shows whether respondents cluster around that tendency. Two variables can have the same Mean of 3,8 and still have very different SD values.
For example, the variable Satisfaction has a Mean of 3,8 and an SD of 0,45. This shows that the responses are fairly concentrated around 3,8. Another variable can also have a Mean of 3,8 but an SD of 1,20, showing that the responses are more dispersed, possibly including both a low-rating group and a high-rating group.
SD helps you read descriptive statistics tables, check the data before regression, identify variables with little dispersion, and compare the relative consistency of variables measured on the same scale. SD by itself does not establish that a scale is good or bad. You need Cronbach's Alpha to assess reliability, while the model requires tests that match your research objective.
If SD equals 0, every observation has the same value. This usually means that the variable has no variation, such as when every respondent selects the same option. A variable with no variation usually does not provide information for correlation or regression analyses.
You also need to examine SD together with the number of observations, Mean, Min, and Max. A large SD may appear because the sample is genuinely diverse, because of an outlier, or because the data were entered incorrectly. When you suspect an unusual value, check the outlier instead of immediately deleting the data row.
What standard deviation is acceptable
This is where many students look for one cutoff, but SD does not have a single acceptable level for every variable. SD depends on the scale, coding, value range, and characteristics of the sample. With a Likert scale from 1 to 5, an SD of 0,30 and an SD of 1,10 describe two different levels of dispersion, but you cannot use those numbers alone to conclude that one result is acceptable and the other is not.
The table below helps you use SD for the right purpose. The thresholds in the table are rules or criteria from the relevant sources. They are not a universal cutoff for grading SD.
| What you need to assess | Level or interpretation | Source | Use in your thesis |
|---|---|---|---|
| SD equals 0 | No variation among observations | Direct interpretation from the formula | Check the coding and whether the variable can be used in the analysis |
| Large or small SD | No universal acceptable cutoff | Interpret it according to the scale and sample | Compare variables measured on the same scale, without judging from SD alone |
| 5 to 10 observations per item | A reference rule for sample size | (Hair et al., 2010) | Use when planning sample size, not as an SD cutoff |
| Corrected Item-Total Correlation of 0,3 or above | A criterion related to items and the scale | (Nunnally and Bernstein, 1994) | Do not replace SD with this criterion |
| Cronbach's Alpha of 0,7 or above | A commonly used reliability criterion | (Nunnally, 1978) | Report it separately in the reliability analysis |
Therefore, the question of what standard deviation is acceptable must be answered in context. If all variables are measured on a Likert scale from 1 to 5, you can describe which variable has greater dispersion than the others. You should not write that an SD below a certain number is always acceptable without a theoretical basis or specific guidance for the study.
When reporting the result, state the measurement unit clearly. SD for income measured in currency will have a completely different scale from SD for a Likert score. Comparing SD values for two variables with different units or ranges can also be misleading.
How to read standard deviation in the output
In SPSS, to calculate standard deviation, open Analyze > Descriptive Statistics > Descriptives. Move the variables you want to describe into Variable(s), select Options, tick Mean, Std. deviation, Minimum, and Maximum, then select Continue and OK. If you need a more detailed table by group, you can use Analyze > Descriptive Statistics > Explore.
The table below is illustrative output, not the result of a real study. The column names remain exactly as they appear in SPSS output so you can compare them directly with your file.
| Descriptive Statistics | N | Minimum | Maximum | Mean | Std. Deviation |
|---|---|---|---|---|---|
| HL1 | 185 | 1 | 5 | 3,76 | 0,82 |
| HL2 | 185 | 1 | 5 | 3,69 | 0,91 |
| HL3 | 185 | 2 | 5 | 3,88 | 0,67 |
| Valid N (listwise) | 185 |
In the illustrative table, HL1 has a Mean of 3,76 and an SD of 0,82. You could write that the average rating for HL1 is 3,76, with a dispersion of 0,82. HL2 has a slightly lower Mean and a higher SD, meaning that the sample's responses are more dispersed than those for HL1 on the same scale.
N is the number of valid observations used for that variable. If N differs between variables, check the missing values. Minimum and Maximum help you detect values outside the permitted range. For example, the questionnaire uses a scale from 1 to 5, but the output contains a value of 7.
If you calculate standard deviation in Excel, the functions you will commonly see are STDEV.S for a sample and STDEV.P for a population. Do not choose the function out of habit. Your survey data are usually a sample from the research population, so you need to define the reporting convention before calculating the result.
You can compare SD with variance, because variance is the square of standard deviation. When the output table contains Variance, remember that variance has squared units, while SD retains the original unit of the variable and is therefore usually easier to interpret.
What to do when standard deviation is not acceptable
First, do not delete data simply because the SD is large. SD has no universal cutoff, so your first step is to check the original data and the measurement context.
Check the range and coding
Review Minimum, Maximum, the frequency of each response option, and the value labels. If the scale runs only from 1 to 5 but contains 0, 6, or 99, determine whether the value is a missing-value code, an entry error, or a valid response. Correct the coding in the data file and record the change before running the analysis again.
Check unusual values
A few observations that differ sharply from the rest can increase SD. Examine the boxplot, frequency table, and data record for that row. Remove an observation only when you have a clear methodological reason, such as an entry error or failure to meet a screening criterion. Do not delete an outlier simply to make the SD look better.
Check how the variable was constructed
If you calculate the Mean of several items but one item was reverse-coded and has not been recoded, the SD of the composite variable may be difficult to interpret. For a reverse-worded item, recode it correctly before calculating the scale score. You also need to make sure that the variables combined into the same construct have the same measurement direction.
Keep the result and explain it transparently
If the data are valid and the large SD accurately reflects differences in the sample, keep the result. In your thesis, describe the level of dispersion and explain it in the context of the study instead of setting your own cutoff and deleting data. If your supervisor asks you to do additional processing, save both the pre-processing and post-processing files so you can trace every step.
Distinguishing standard deviation from variance and the mean
Standard deviation and variance use the same information about the distance between each observation and the Mean. Variance takes the average of the squared deviations. Standard deviation is the square root of variance, so the two measures are related as SD = √Variance.
For example, if Variance is 0,64, SD is 0,8. Variance is suitable for calculations and statistical models, while SD is usually easier to present in a descriptive table because it uses the same unit as the original variable. See the variance formula when you need to check each calculation step.
Mean is the average value, not a measure of dispersion. A sample can have a high Mean and a high SD if the responses are widely spread. The median is the middle value after the data are arranged in order, and it is often useful when the distribution is skewed or contains an outlier. You can read more about the median when choosing an appropriate measure of central tendency.
You also need to distinguish SD from Standard Error. SD describes the dispersion of observations in the sample. Standard Error describes the uncertainty of the estimated Mean and usually decreases as sample size increases. Do not substitute one for the other when writing the results chapter.
Common mistakes
Reading SD as a rating score. An SD of 0,7 does not mean that respondents gave a rating of 0,7. The rating is represented by the Mean, while SD shows the dispersion around the Mean.
Setting your own SD cutoff to remove a variable. There is no universal rule stating that every SD below 0,5 or above 1,0 is unacceptable. Review the scale range, distribution, and analysis objective.
Using the wrong sample and population formula. If your data come from a survey sample, state the calculation convention clearly. Do not combine results from STDEV.S and STDEV.P and then describe them as the same measure.
Ignoring N and missing values. An SD calculated from 80 observations should not automatically be interpreted in the same way as an SD calculated from 185 observations if the missing-data rates differ.
Rounding too early. Keep enough decimal places during the calculation and round only in the reporting table, usually to two decimal places if your department has no different requirement.
Deleting rows to reduce SD. This is difficult to explain if you have no evidence of a data error. Keep the original file, record the removal criteria, and report the number of observations before and after processing if anything changed.
Frequently asked questions
What standard deviation is acceptable?
There is no single number that applies to every variable. Examine SD together with the scale, Mean, Min, Max, N, and data distribution. For variables measured on the same Likert scale, you can compare which variable is more dispersed, but you should not treat a particular SD as acceptable without a basis.
How do the sample and population standard deviation formulas differ?
The population standard deviation divides the total squared deviation by N. The sample standard deviation divides by n - 1 to estimate population dispersion from a sample. Thesis survey data are usually treated as a sample, so state the calculation method or tool you used clearly.
Why are the standard deviations different in Excel and SPSS?
The usual reasons are that you used STDEV.S in one program and STDEV.P in the other, or that the two tools handled missing values differently. Check that both use the same observations, the same cell range, and the same sample or population convention. A small difference caused by rounding may also appear.
Does a standard deviation of 0 mean that the data are incorrect?
Not necessarily. An SD of 0 means that every valid value for the variable is identical. Check the frequencies and coding. If every respondent genuinely selected the same option, the variable has no variation. If one value was entered for every row, the data contain an error that needs to be corrected.
Should you delete a variable when its standard deviation is too large?
Do not delete it simply because the SD is large. Check the range, outliers, missing values, reverse coding, and survey context first. If the variable is valid but reflects genuine differences among respondents, keep it and report the result transparently.
How should you write standard deviation in a thesis?
How to write this in your thesis
You can write: “[Variable name] has a mean value of [Mean] and a standard deviation of [SD], indicating [an appropriate description] of the dispersion of observations around the mean.” Replace the placeholders with the actual values in the Descriptive Statistics table. Do not use the illustrative numbers in this article.
Open your file, run Analyze > Descriptive Statistics > Descriptives, check N, Mean, Std. Deviation, Minimum, and Maximum together, then record the data-processing steps before adding the table to Chapter 4. If you need to run the analysis on your own .sav or .csv file, see the M4 data analysis module.