If you want to determine whether there are any statistically significant differences between the means of two or more independent groups, you can use a one-way analysis of variance (ANOVA). For example, you could use a one-way ANOVA to determine whether exam performance differed based on test anxiety levels amongst students (i.e., your dependent variable would be “exam performance”, measured from 0-100, and your independent variable would be “test anxiety level”, which has three groups: “low-stressed students”, “moderately-stressed students” and “highly-stressed students”). As another example, a one-way ANOVA could be used to understand whether there is a difference in salary based on degree subject (i.e., your dependent variable would be “salary” and your independent variable would be “degree subject”, which has five groups: “business studies”, “psychology”, “biological sciences”, “engineering” and “law”).

**Note:** The one-way ANOVA is also referred to as a between-subjects ANOVA or one-factor ANOVA. Although it can be used with an independent variable with only two groups, the independent-samples t-test is typically used in this situation instead. For this reason, you will come across the one-way ANOVA being described as a test to use when you have three or more groups (rather than two or more groups).

It is important to realize that the one-way ANOVA is an *omnibus* test statistic and cannot tell you which specific groups were significantly different from each other; it only tells you that at least two groups were different. Since you may have three, four, five or more groups in your study design, determining which of these groups differ from each other is important. You can do this using follow-up tests, which will be either a post hoc test or custom contrasts. If you are unfamiliar with the one-way ANOVA, you are most likely to benefit from reading about the basic requirements that must be met to run a one-way ANOVA, the null and alternative hypotheses being tested, the importance of choosing between running post hoc tests and custom contrasts, as well as issues relating to effect sizes and sample sizes and (un)balanced designs. However, if you already understand these characteristics, requirements and choices, we would suggest starting with the example we use throughout this guide. This also includes an SPSS Statistics data file you can download so that you can work through the example used in this guide (e.g., if you want to practice before carrying out the one-way ANOVA on your own data).

## Assumptions

In order to run a one-way ANOVA, there are six assumptions that need to be considered. The first three assumptions relate to your choice of study design and the measurements you chose to make, whilst the second three assumptions relate to how your data fits the one-way ANOVA model. These assumptions are:

- Assumption #1: You have
**one dependent variable**that is measured at the**continuous**level. Examples of**continuous variables**include height (measured in metres and centimetres), temperature (measured in °C), salary (measured in US dollars), revision time (measured in hours), intelligence (measured using IQ score), firm size (measured in terms of the number of employees), age (measured in years), reaction time (measured in milliseconds), grip strength (measured in kg), power output (measured in watts), test performance (measured from 0 to 100), sales (measured in number of transactions per month), academic achievement (measured in terms of GMAT score), and so forth.

- Assumption #2: You have
**one independent variable**that consists of**two or more categorical**,**independent groups**. Typically, a one-way ANOVA is used when you have three or more categorical, independent groups, but it can be used for just two groups (although an independent-samples t-test is more commonly used for two groups). Example independent variables that meet this criterion include ethnicity (e.g., three groups: Caucasian, African American and Hispanic), physical activity level (e.g., four groups: sedentary, low, moderate and high), profession (e.g., five groups: surgeon, doctor, nurse, dentist, therapist), and so forth. If you are unfamiliar with any of the terms above, you might want to read our Types of Variables guide. Also, if you have two independent variables rather than just one, you should consider a two-way ANOVA instead of a one-way ANOVA.

**Note:** The “groups” of the independent variable are also referred to as “categories” or “levels”, but the term “levels” is usually reserved for groups that have an order (e.g., fitness level, with three levels: “low”, “moderate” and “high”).

- Assumption #3: You should have
**independence of observations**, which means that there is no relationship between the observations in each group of the independent variable or between the groups themselves. Indeed, an important distinction is made in statistics when comparing values from either different individuals or from the same individuals. Independent groups (in a one-way ANOVA) are groups where there is no relationship between the participants in any of the groups. Most often, this occurs simply by having different participants in each group. For example, if you split a group of individuals into four groups based on their physical activity level (e.g., a “sedentary” group, “low” group, “moderate” group and “high” group), no one in the sedentary group can also be in the high group, no one in the moderate group can also be in the high group, and so forth. As another example, you might randomly assign participants to either a control trial or one of two interventions. Again, no participant can be in more than one group (e.g., a participant in the the control group cannot be in either of the intervention groups). This will be true of any independent groups you form (i.e., a participant cannot be a member of more than one group). In actual fact, the ‘no relationship’ part extends a little further and requires that participants in different groups are considered unrelated, not just different people (e.g., participants might be considered related if they are husband and wife, or twins). Furthermore, participants in one group cannot influence any of the participants in any other group. If you are using the same participants in each group or they are otherwise related, a one-way repeated measures ANOVA is a more appropriate test. It is also fairly common to hear this type of study design, with two or more independent groups, being referred to as “between-subjects” because you are concerned with the differences in the dependent variable between different subjects. An example of where related observations might be a problem is if all the participants in your study (or the participants within each group) were assessed together, such that a participant’s performance affects another participant’s performance (e.g., participants encourage each other to lose more weight in a ‘weight loss intervention’ when assessed as a group compared to being assessed individually; or athletic participants being asked to complete ‘100m sprint tests’ together rather than individually, with the added competition amongst participants resulting in faster times, etc.). Independence of observations is largely a study design issue rather than something you can test for, but it is an important assumption of the one-way ANOVA. If your study fails this assumption, you will need to use another statistical test instead of the one-way ANOVA (you can use our Statistical Test Selector to find the appropriate statistical test).

- Assumption #4: There should be no significant outliers in the groups of your independent variable in terms of the dependent variable. If there are any scores that are unusual in any group of the independent variable, in that their value is extremely small or large compared to the other scores, these scores are called outliers (e.g., 8 participants in a group scored between 60-75 out of 100 in a difficult maths test, but one participant scored 98 out of 100). Outliers can have a large negative effect on your results because they can exert a large influence (i.e., change) on the mean and standard deviation for that group, which can affect the statistical test results. Outliers are more important to consider when you have smaller sample sizes, as the effect of the outlier will be greater. Therefore, in this example, you need to investigate whether the dependent variable, coping_stress, has any outliers for each group of the independent variable, group (i.e., you are testing whether the coping stress score is outlier free for the “sedentary”, “low”, “moderate” and “high” groups).

- Assumption #5: Your dependent variable should be approximately normally distributed for each group of the independent variable. The assumption of normality is necessary for statistical significance testing using a one-way ANOVA. However, the one-way ANOVA is considered “robust” to violations of normality. This means that some violation of this assumption can be tolerated and the test will still provide valid results. Therefore, you will often hear of this test only requiring
*approximately*normal data. Furthermore, as sample size increases, the distribution can be very non-normal and, thanks to the Central Limit Theorem, the one-way ANOVA can still provide valid results. Also, it should be noted that if the distributions are all skewed in a similar manner (e.g., all moderately negatively skewed), this is not as troublesome as when compared to the situation where you have groups that have differently-shaped distributions (e.g., the distributions of Group A and Group D are moderately ‘positively’ skewed, whilst the distributions of Group B and Group C are moderately ‘negatively’ skewed). Therefore, in this example, you need to investigate whether coping_stress is normally distributed.

**Note:** Technically, it is the residuals (errors) that need to be normally distributed. However, for a one-way ANOVA the distribution of the scores (observations) in each group will be the same as the distribution of the residuals in each group. However, for a one-way ANOVA, this is equivalent to assuming normality of the actual scores (observations) in each group (Kirk, 2013).

- Assumption #6: You have homogeneity of variances (i.e., the variance is equal in each group of your independent variable)The assumption of homogeneity of variances states that the population variance for each group of your independent variable is the same. If the sample size in each group is similar, violation of this assumption is not often too serious. However, if sample sizes are quite different, the one-way ANOVA is sensitive to the violation of this assumption. Either way, SPSS Statistics uses
**Levene’s test of equality of variances**and two differently-calculated one-way ANOVAs, which will give you a valid result irrespective of whether you met or violated this assumption; that is, SPSS Statistics provides a one-way ANOVA that is calculated normally (with pooled variances) and another for when the assumption is violated that uses separate variances (i.e., non-pooled variances) and the Welch-Satterthwaite correction to the degrees of freedom. In addition, if your data fails this assumption, we show you how to run a Games-Howell post hoc test for your multiple comparisons instead of the Tukey post hoc test. Please contact us. We will show you how to carry out Levene’s test for equality of variances.

## Interpreting Results

We show you how to interpret and report the results of the one-way ANOVA, as well as post hoc tests and custom contrasts. If you are interested in running custom contrasts instead of post hoc tests, we also show you the procedures to carry out this analysis in SPSS Statistics.

- One-way ANOVA with post hoc testing: More specifically, if your data met the assumption of homogeneity of variances, you simply need to interpret the ‘standard’ one-way ANOVA output in SPSS Statistics. However, if your data violated the assumption of homogeneity of variances, you need to interpret the Welch ANOVA output in SPSS Statistics instead. We can 1) interpret the SPSS Statistics output for the one-way ANOVA, including the means, standard deviations,
*F*-value, degrees of freedom and*p*-value; 2) determine which group means are statistically significantly different; 3) determine if you can reject, or fail to reject, the null hypothesis; and (d) determine how you can bring all of this together into a single paragraph that explains your results. - One-way ANOVA with contrasts: If you are interested in the results of the one-way ANOVA and custom contrasts, you should follow up page 16 with pages 17 through 20. More specifically, if your data met the assumption of homogeneity of variances, you simply need to interpret the ‘standard’ one-way ANOVA output in SPSS Statistics. However, if your data violated the assumption of homogeneity of variances, you need to interpret the Welch ANOVA output in SPSS Statistics instead. Alternately, if you are only interested in custom contrasts and not the results from the one-way ANOVA, you should follow up page 16 with pages 21 through 23, where we set out the SPSS Statistics procedures required to run custom contrasts. We can accurately (1) interpret the SPSS Statistics output in order to determine which group means are statistically significantly different based on the simple or complex contrasts that you ran; (2) determine if you can reject, or fail to reject, the null hypothesis; and 3) determine how you can bring all of this together into a single paragraph that explains your results.

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