Question

The following ANOVA table is only partially completed. $$ \begin{array}{|lrrr|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS } \\ \hline \text { Between groups } & 3 & & 45 \\ \text { Within groups } & 12 & 337 & \\ \hline \text { Total } & & 472 & \\ \hline \end{array} $$ (a) Complete the table. (b) How many groups were there in the study? (c) How many total observations were there in the study?

Short answer

SSB = 135, MSW = 28.083, Number of groups = 4, Total observations = 16.

Step-by-step solution

Calculate SS for Between groups

To find the Sum of Squares Between (SSB), subtract the Sum of Squares Within (SSW) from the Total Sum of Squares (SST). Given SSW (Within groups) is 337 and SST (Total) is 472: SSB = SST - SSW = 472 - 337.

Calculate MS for Within groups

To calculate Mean Square Within (MSW), divide the Sum of Squares Within (SSW) by the degrees of freedom (df) within groups. MSW = SSW / df within = 337 / 12.

Count the number of groups

The number of groups in the study is equal to the degrees of freedom (df) plus one for the Between groups variability. Number of groups = df Between + 1 = 3 + 1.

Calculate the total number of observations

The total number of observations in the study is the sum of all degrees of freedom (df) plus one. Total observations = df Between + df Within + 1 = 3 + 12 + 1.

Key concepts

Sum of Squares Between (SSB)

Understanding the Sum of Squares Between (SSB) is crucial when completing an ANOVA table. This value helps us determine the amount of variance explained by the differences between group means. In essence, SSB measures how each group mean differs from the overall mean, highlighting whether group membership affects the dependent variable.

For example, when you are given the Total Sum of Squares (SST) and the Sum of Squares Within (SSW), calculating SSB is straightforward. You subtract the within-group variance (SSW) from the total variance (SST) to isolate the between-group variance. Mathematically, this is shown as: \[ SSB = SST - SSW \].

In the given exercise, the SST is known to be 472 and SSW is 337. Therefore, the calculation would be: \[ SSB = 472 - 337 = 135 \]. This resulting value represents the variance observed between the different groups in the study.

Mean Square Within (MSW)

Delving into the Mean Square Within (MSW), this statistic tells us about the average variation within the groups of a dataset. It's an essential component for determining whether the variation between groups is significant when compared to the variation within groups.

To calculate MSW, you take the Sum of Squares Within (SSW) and divide it by the appropriate degrees of freedom (df within). The equation looks like this: \[ MSW = \frac{SSW}{df_{within}} \].

The degrees of freedom for within groups typically equals the total number of observations in those groups minus the number of groups. In the exercise scenario, the SSW is 337 and the df within is 12. Hence, the calculation for MSW would be: \[ MSW = \frac{337}{12} \]. This computed MSW provides insight into the average variation within the groups, setting the stage for determining if the variation between groups is more than what would be expected by chance.

Degrees of Freedom (df)

The concept of degrees of freedom (df) might seem nebulous at first, but it is a critical component in statistical analysis. In an ANOVA context, degrees of freedom are used to calculate both the Mean Square Within (MSW) and Mean Square Between (MSB). They provide an insight into the number of independent values that can vary in the data set.

There are two types of degrees of freedom in ANOVA: df between (associated with the variance between groups) and df within (associated with the variance within groups). To calculate df between, you subtract one from the number of groups. For df within, it is the total number of observations minus the number of groups. The total df (df total) is the sum of df between and df within.

The exercise provided states that df between is 3, and df within is 12. This helps uncover that the study had 4 groups (df between + 1) and 16 total observations (df total + 1, where df total is df between + df within). Understanding these df values is pivotal for executing proper variance analyses.

Total Sum of Squares (SST)

The Total Sum of Squares (SST) plays a fundamental role in ANOVA as a measure of the total variability within the data. To put it simply, it quantifies how much the data points vary from the overall mean – considering all groups combined.

SST is a starting point from which we can partition total variability into its components: variability due to differences between group means (SSB), and variability within the groups themselves (SSW). By definition, \[ SST = SSB + SSW \], which means that knowing any two of these values will allow you to calculate the third.

In the given exercise, we are told that the total SST is 472. With this number, along with SSB or SSW, we can derive insights about the proportion of variance due to the grouping of data versus individual differences within those groups. A key aspect of understanding SST is realizing it captures all the variance present in the dataset, making it indispensable for the ANOVA calculation process.