Unveiling the One-Sample T-Test: A Comprehensive Guide

The one-sample t-test is a statistical tool used to determine whether a sample comes from a population with a specific mean. This population mean is not always known, but is sometimes hypothesized. It's a fundamental concept in inferential statistics, allowing researchers to draw conclusions about a population based on data from a single sample.

Introduction

The one-sample t-test is a statistical hypothesis test used to determine whether the mean calculated from sample data collected from a single group is different from a designated value specified by the researcher. This designated value does not come from the data itself, but is an external value chosen for scientific reasons. Often, this designated value is a mean previously established in a population, a standard value of interest, or a mean concluded from other studies. Like all hypothesis testing, the one sample t test determines if there is enough evidence reject the null hypothesis (H0) in favor of an alternative hypothesis (H1).

For example, imagine you want to show that a new teaching method for pupils struggling to learn English grammar can improve their grammar skills to the national average. Your sample would be pupils who received the new teaching method and your population mean would be the national average score. Alternately, you believe that doctors that work in Accident and Emergency (A & E) departments work 100 hour per week despite the dangers (e.g., tiredness) of working such long hours.

Core Concepts

At its heart, the one-sample t-test compares the mean of your sample data to a known value. For example, you might want to know how your sample mean compares to the population mean.

Hypotheses

There are two kinds of hypotheses for a one sample t-test, the null hypothesis and the alternative hypothesis. The alternative hypothesis assumes that some difference exists between the true mean (μ) and the comparison value (m0), whereas the null hypothesis assumes that no difference exists. The purpose of the one sample t-test is to determine if the null hypothesis should be rejected, given the sample data.

Null Hypothesis (H0): Assumes no difference exists between the true mean (μ) and the comparison value (m0).
Alternative Hypothesis (H1): Assumes a difference exists between the true mean (μ) and the comparison value (m0).

The alternative hypothesis can assume one of three forms depending on the question being asked. If the goal is to measure any difference, regardless of direction, a two-tailed hypothesis is used. If the direction of the difference between the sample mean and the comparison value matters, either an upper-tailed or lower-tailed hypothesis is used. The null hypothesis remains the same for each type of one sample t-test.

Note. It is important to remember that hypotheses are never about data, they are about the processes which produce the data.

Test Statistic

Prism calculates the t ratio by dividing the difference between the actual and hypothetical means by the standard error of the actual mean.

P-value

A p value is computed based on the calculated t ratio and the numbers of degrees of freedom present (which equals sample size minus 1).

Statistical significance is determined by looking at the p-value. The p-value gives the probability of observing the test results under the null hypothesis. The lower the p-value, the lower the probability of obtaining a result like the one that was observed if the null hypothesis was true. Thus, a low p-value indicates decreased support for the null hypothesis. However, the possibility that the null hypothesis is true and that we simply obtained a very rare result can never be ruled out completely. The cutoff value for determining statistical significance is ultimately decided on by the researcher, but usually a value of .05 or less is chosen.

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If the p value is large (usually defined to mean greater than 0.05), the data do not give you any reason to conclude that the population mean differs from the designated value to which it has been compared. This is not the same as saying that the true mean equals the hypothetical value, but rather states that there is no evidence of a difference.
If the p value is small (usually defined to mean less than or equal to 0.05), then it is unlikely that the discrepancy observed between the sample mean and hypothetical mean is due to a coincidence arising from random sampling. There is evidence to reject the idea that the difference is coincidental and conclude instead that the population has a mean that is different from the hypothetical value to which it has been compared.

Degrees of Freedom

The degrees of freedom used in this test are n − 1, where n is the sample size.

Assumptions of the One-Sample T-Test

When you choose to analyse your data using a one-sample t-test, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a one-sample t-test. You need to do this because it is only appropriate to use a one-sample t-test if your data "passes" four assumptions that are required for a one-sample t-test to give you a valid result.

As a parametric procedure (a procedure which estimates unknown parameters), the one sample t-test makes several assumptions. Though t-tests are robust, it’s good practice to check the degree of assumption deviation to assess result quality.

Before we introduce you to these four assumptions, do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated (i.e., is not met). This is not uncommon when working with real-world data rather than textbook examples, which often only show you how to carry out a one-sample t-test when everything goes well! However, don’t worry. Even when your data fails certain assumptions, there is often a solution to overcome this.

Continuous Data: The one sample t-test requires the sample data to be numeric and continuous, as it is based on the normal distribution. Continuous data can take on any value within a range (income, height, weight, etc.). The opposite of continuous data is discrete data, which can only take on a few values (Low, Medium, High, etc.). You can use the test for continuous data.

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Independence of Observations: The data are independent (i.e., not correlated/related), which means that there is no relationship between the observations. Independence of observations is usually not testable, but can be reasonably assumed if the data collection process was random without replacement. In our example, we would want to select laptop computers at random, compared to using any systematic pattern. Data is collected randomly.
No Significant Outliers: There should be no significant outliers. Outliers are data points within your data that do not follow the usual pattern (e.g., in a study of 100 students' IQ scores, where the mean score was 108 with only a small variation between students, one student had a score of 156, which is very unusual, and may even put her in the top 1% of IQ scores globally). The problem with outliers is that they can have a negative effect on the one-sample t-test, reducing the accuracy of your results. Fortunately, when using SPSS Statistics to run a one-sample t-test on your data, you can easily detect possible outliers. An outlier is a data value which is too extreme to belong in the distribution of interest. Let’s suppose in our example that the assembly machine ran out of a particular component, resulting in a laptop that was assembled at a much lower weight. This is a condition that is outside of our question of interest, and therefore we can remove that observation prior to conducting the analysis. Let’s suppose that our laptop assembly machine occasionally produces laptops which weigh significantly more or less than five pounds, our target value. In this case, these extreme values are absolutely essential to the question we are asking and should not be removed. Box-plots are useful for visualizing the variability in a sample, as well as locating any outliers.
Approximate Normal Distribution: Your dependent variable should be approximately normally distributed. We talk about the one-sample t-test only requiring approximately normal data because it is quite "robust" to violations of normality, meaning that the assumption can be a little violated and still provide valid results. You can test for normality using the Shapiro-Wilk test of normality, which is easily tested for using SPSS Statistics. To test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-world data are rarely perfectly normal, so you can consider the assumption met if the shape is roughly symmetric and bell-shaped. Normal distributions are symmetric, which means they are “even” on both sides of the center. Normal distributions do not have extreme values, or outliers. You can check these two features of a normal distribution with graphs.

You can check assumptions #3 and #4 using SPSS Statistics. Before doing this, you should make sure that your data meets assumptions #1 and #2, although you don't need SPSS Statistics to do this. When moving on to assumptions #3 and #4, we suggest testing them in this order because it represents an order where, if a violation to the assumption is not correctable, you will no longer be able to use a one-sample t-test. Just remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a one-sample t-test might not be valid.

Addressing Non-Normal Data

What if you know the underlying measurements are not normally distributed? Or what if your sample size is large and the test for normality is rejected? In this situation, you can use a nonparametric test. Nonparametric analyses do not depend on an assumption that the data values are from a specific distribution.

When the normality assumption does not hold, a non-parametric alternative to the t-test may have better statistical power. However, when data are non-normal with differing variances between groups, a t-test may have better type-1 error control than some non-parametric alternatives. Furthermore, non-parametric methods, such as the Mann-Whitney U test discussed below, typically do not test for a difference of means, so should be used carefully if a difference of means is of primary scientific interest.

Conducting a One-Sample T-Test

The procedure for a one sample t-test can be summed up in four steps.

State the null and alternative hypotheses.
Calculate the test statistic.
Calculate the probability of observing the test statistic under the null hypothesis. This value is obtained by comparing t to a t-distribution with ((n\ -\ 1)) degrees of freedom.
Compare the p-value to an acceptable significance level (alpha) and make a decision about whether to reject the null hypothesis.

Using SPSS Statistics

A researcher is planning a psychological intervention study, but before he proceeds he wants to characterise his participants' depression levels. He tests each participant on a particular depression index, where anyone who achieves a score of 4.0 is deemed to have 'normal' levels of depression. Lower scores indicate less depression and higher scores indicate greater depression. He has recruited 40 participants to take part in the study. Depression scores are recorded in the variable dep_score.

The 5-step One-Sample T Test… procedure below shows you how to analyse your data using a one-sample t-test in SPSS Statistics when the four assumptions in the previous section, Assumptions, have not been violated. At the end of these five steps, we show you how to interpret the results from this test.

Since some of the options in the One-Sample T Test… procedure changed in SPSS Statistics version 27, we show how to carry out a one-sample t-test depending on whether you have SPSS Statistics versions 27 to 30 (or the subscription version of SPSS Statistics) or version 26 or an earlier version of SPSS Statistics. The latest versions of SPSS Statistics are version 30 and the subscription version.

SPSS Statistics versions 27 to 30 (or the subscription version):

Click on Analyze > Compare Means and Proportions > One-Sample T Test… Note: If you have SPSS Statistics versions 27 or 28, click on Analyze > Compare Means > One-Sample T Test…
Transfer the dependent variable, dep_score, into the Test Variable(s): box by selecting it (by clicking on it) and then clicking on the button.
Enter the population mean you are comparing the sample against in the Test Value: box, by changing the current value of "0" to "4".
Keep Estimate effect sizes selected.
Click on the button.

Note 1: By default, SPSS Statistics uses 95% confidence intervals (labelled as the Confidence Interval Percentage in SPSS Statistics). This equates to declaring statistical significance at the p < .05 level. If you wish to change this you can enter any value from 1 to 99. For example, entering "99" into this box would result in a 99% confidence interval and equate to declaring statistical significance at the p < .01 level.

Note 2: If you are testing more than one dependent variable and you have any missing values in your data, you need to think carefully about whether to select Exclude cases analysis by analysis or Exclude cases listwise) in the -Missing Values- area. Selecting the incorrect option could mean that SPSS Statistics removes data from your analysis that you wanted to include.

SPSS Statistics version 26 or an earlier version:

Click on Analyze > Compare Means > One-Sample T Test…
Transfer the dependent variable, dep_score, into the Test Variable(s): box by selecting it (by clicking on it) and then clicking on the button.
Enter the population mean you are comparing the sample against in the Test Value: box, by changing the current value of "0" to "4".
Click on the button.

Interpreting the Results

If your data passed assumption #3 (i.e., there were no significant outliers) and assumption #4 (i.e., your dependent variable was approximately normally distributed for each category of the independent variable), which we explained earlier in the Assumptions section, you will only need to interpret these two main tables. However, since you should have tested your data for these assumptions, you will also need to interpret the SPSS Statistics output that was produced when you tested for them (i.e., you will have to interpret: (a) the boxplots you used to check if there were any significant outliers; and (b) the output SPSS Statistics produces for your Shapiro-Wilk test of normality to determine normality).

It is more common than not to present your descriptive statistics using the mean and standard deviation ("Std. Deviation" column) rather than the standard error of the mean ("Std. Error Mean" column), although both are acceptable. However, by running a one-sample t-test, you are really interested in knowing whether the sample you have (dep_score) comes from a 'normal' population (which has a mean of 4.0).

The One-Sample Test table reports the result of the one-sample t-test. In this example, you can see the 'normal' depression score value of "4" that you entered in earlier. Moving from left-to-right, you are presented with the observed t-value ("t" column), the degrees of freedom ("df"), and the statistical significance (p-value) ("Sig. (2-tailed)") of the one-sample t-test. In this example, p < .05 (it is p = .022). Therefore, it can be concluded that the population means are statistically significantly different.

Note: If you see SPSS Statistics state that the "Sig. (2-tailed)" value is ".000", this actually means that p < .0005. SPSS Statistics also reports that t = -2.381 ("t" column) and that there are 39 degrees of freedom ("df" column).

You can also include measures of the difference between the two population means in your written report. This section of the table shows that the mean difference in the population means is -0.28 ("Mean Difference" column) and the 95% confidence intervals (95% CI) of the difference are -0.51 to -0.04 ("Lower" to "Upper" columns). For the measures used, it will be sufficient to report the values to 2 decimal places.

After reporting the unstandardised effect size, we might also report a standardised effect size such as Cohen's d (Cohen, 1988) or Hedges' g (Hedges, 1981). There are many different types of standardised effect size, with different types often trying to "capture" the importance of your results in different ways. In SPSS Statistics versions 18 to 26, SPSS Statistics did not automatically produce a standardised effect size as part of a one-sample t-test analysis. However, it is easy to calculate a standardised effect size such as Cohen's d (Cohen, 1988) using the results from the one-sample t-test analysis.

Practical Significance vs. Statistical Significance

Although a statistically significant difference was found between the depression scores in the recruited subjects vs. the normal depression score, it does not necessarily mean that the difference encountered, 0.28 (95% CI, 0.04 to 0.51), is enough to be practically significant.

Reporting the Results

A one-sample t-test was run to determine whether depression score in recruited subjects was different to normal, defined as a depression score of 4.0. Depression scores were normally distributed, as assessed by Shapiro-Wilk's test (p > .05) and there were no outliers in the data, as assessed by inspection of a boxplot. There was a statistically significant difference between means (p < .05).

In our enhanced one-sample t-test guide, we show you how to write up the results from your assumptions tests and one-sample t-test procedure if you need to report this in a dissertation/thesis, assignment or research report. We do this using the Harvard and APA styles. We also explain how to interpret the results from the One-Sample Effect Sizes table, which include the two standardised effect sizes: Cohen's d and Hedges' g.

Real-World Example: Energy Bar Protein Content

Imagine we have collected a random sample of 31 energy bars from a number of different stores to represent the population of energy bars available to the general consumer. The label claims that the bars have 20 grams of protein.

Let’s start by answering: Is the t-test an appropriate method to test that the energy bars have 20 grams of protein?

The data values are independent. The grams of protein in one energy bar do not depend on the grams in any other energy bar. An example of dependent values would be if you collected energy bars from a single production lot.
The data values are grams of protein.

From a quick look at the histogram, we see that there are no unusual points, or outliers. From a quick look at the statistics, we see that the average is 21.40, above 20. Does this average from our sample of 31 bars invalidate the label's claim of 20 grams of protein for the unknown entire population mean?

For the t-test calculations we need the mean, standard deviation and sample size. We round the statistics to two decimal places. Software will show more decimal places, and use them in calculations. Next, we calculate the standard error for the mean. We now have the pieces for our test statistic.

To make our decision, we compare the test statistic to a value from the t- distribution. We calculate a test statistic. We decide on the risk we are willing to take for declaring a difference when there is not a difference. For the energy bar data, we decide that we are willing to take a 5% risk of saying that the unknown population mean is different from 20 when in fact it is not. In statistics-speak, we set α = 0.05. We find the value from the t- distribution based on our decision. For a t-test, we need the degrees of freedom to find this value. The degrees of freedom are based on the sample size.

The critical value of t with α = 0.05 and 30 degrees of freedom is +/- 2.043. Most statistics books have look-up tables for the distribution. You can also find tables online.

We compare the value of our statistic (3.07) to the t value. Since 3.07 > 2.043, we reject the null hypothesis that the mean grams of protein is equal to 20. Our null hypothesis is that the underlying population mean is equal to 20. The alternative hypothesis is that the underlying population mean is not equal to 20. The labels claiming 20 grams of protein would be incorrect. This is a two-sided test. We are testing if the population mean is different from 20 grams in either direction.

If we can reject the null hypothesis that the mean is equal to 20 grams, then we make a practical conclusion that the labels for the bars are incorrect.

Interpreting Software Output

You are likely to use software to perform a t-test. The software shows the null hypothesis value of 20 and the average and standard deviation from the data. The test statistic is 3.07. The software shows results for a two-sided test and for one-sided tests. We want the two-sided test. Our null hypothesis is that the mean grams of protein is equal to 20. Our alternative hypothesis is that the mean grams of protein is not equal to 20.

The software shows a p- value of 0.0046 for the two-sided test. This p- value describes the likelihood of seeing a sample average as extreme as 21.4, or more extreme, when the underlying population mean is actually 20; in other words, the probability of observing a sample mean as different, or even more different from 20, than the mean we observed in our sample. A p-value of 0.0046 means there is about 46 chances out of 10,000.

Alternatives to the One-Sample T-Test

As described, a one sample t test should be used only when data has been collected on one variable for a single population and there is no comparison being made between groups. In most cases involving data analysis, however, there are multiple groups of data either representing different populations being compared, or the same population being compared at different times or conditions. For these situations, it is not appropriate to use a one sample t test.

Independent Samples T-Test: The independent sample t test, also referred to as the unpaired t test, is used to compare the means of two different samples. the Welch's t test, which is less restrictive compared to the original Student's test. An example research question for an independent sample t-test is “Do boys and girls differ in their SAT scores?
Paired Samples T-Test: The paired sample t test is used to compare the means of two related groups of samples. Put into other words, it is used in a situation where you have two values (i.e., a pair of values) for the same group of samples. A dependent sample t-test compares two matched scores or measurements (e.g., before vs. after).

A Brief History of the T-Test

In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. The t-distribution also appeared in a more general form as Pearson type IV distribution in Karl Pearson's 1895 paper. However, the t-distribution, also known as Student's t-distribution, gets its name from William Sealy Gosset, who first published it in English in 1908 in the scientific journal Biometrika using the pseudonym "Student" because his employer preferred staff to use pen names when publishing scientific papers. Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples - for example, the chemical properties of barley with small sample sizes. Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material. Gosset devised the t-test as an economical way to monitor the quality of stout.

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