Total Sum of Squares Around the Mean

Why Measure Variation Around the Mean?

A list of outcomes is easier to summarize when we know both its center and how far its values tend to lie from that center. In regression, the outcome’s sample mean provides a simple reference point: it is one value shared by the whole sample. The total sum of squares measures the observed outcomes’ squared distances from that reference point.

This quantity is useful to define precisely before using it in model-fit comparisons. It describes the outcome values in the sample; it is not a measure of predictor variation, and its calculation does not require fitting a regression model. Its numerical value also depends on the units used for the outcome and on which observations are in the sample.

The subscript \(i\) identifies an observation, not a predictor or a model coefficient. Each term \(y_i-\bar{y}\) is measured in the same units as the outcome; after squaring, TSS has squared outcome units. For example, if outcomes are measured in centimetres, TSS is measured in square centimetres.

TSS is a descriptive quantity for the particular observed sample. There are no regression-model assumptions needed to calculate it. A sample of only one outcome has TSS equal to zero, because that one outcome is also its sample mean. The usual sample variance is defined for \(n\geq 2\), so its connection to TSS below is stated for that case.

Calculating TSS and Connecting It to Sample Variance

A reliable calculation follows a simple sequence: find the sample mean, subtract that mean from each outcome, square each deviation, and add the squared deviations. Keeping the deviations visible helps prevent errors, especially when observations lie on both sides of the mean.

For \(n\geq 2\), the sample variance \(s^2\) is the sum of squared deviations from the sample mean divided by \(n-1\). Therefore, TSS is exactly \((n-1)s^2\). This relationship is a way to check a TSS calculation; it does not change the definition of TSS.

The connection to sample variance follows directly from its definition:

The factor \(n-1\) matters. Dividing TSS by \(n\) does not give the usual sample variance \(s^2\). TSS itself is the sum, while \(s^2\) is that sum divided by \(n-1\). Because variance has squared outcome units, this relationship also confirms that TSS has squared outcome units.

How Changing Outcome Units Changes TSS

TSS is affected by a change in the numerical scale of the outcome. Suppose every outcome is transformed using \(z_i=a+c y_i\), where \(a\) is a constant shift and \(c\) is a constant multiplier. The transformed sample mean is \(\bar{z}=a+c\bar{y}\). Each transformed deviation is therefore

The shift \(a\) cancels when deviations from the mean are taken. The multiplier \(c\), however, remains in every deviation and is squared in TSS. Thus a constant shift does not change TSS, while multiplying all outcomes by \(c\) multiplies TSS by \(c^2\).

How Changing the Sample Changes TSS

TSS is calculated using the mean of the particular sample being analyzed. If observations are added or removed, the mean may move, so the deviations of the original observations may change as well. TSS is therefore sample-dependent: it is not a fixed number that belongs to the outcome variable regardless of which observations are included.

The effect of adding one observation can be derived exactly. Suppose a sample of \(n\) observations has mean \(\bar{y}\) and TSS \(T\). Add one observation \(z\). The new mean is \(\bar{y}_{\mathrm{new}}=(n\bar{y}+z)/(n+1)\). For each original observation, its deviation from the new mean is its old deviation minus \((z-\bar{y})/(n+1)\). The new observation’s deviation from the new mean is \(n(z-\bar{y})/(n+1)\). Squaring these deviations and adding yields the update formula below.

For completeness, let \(d=z-\bar{y}\). The shift in the mean is \(\bar{y}_{\mathrm{new}}-\bar{y}=d/(n+1)\). The sum of the original deviations \(y_i-\bar{y}\) is zero, so when the original squared deviations are recentered around the new mean, their total becomes \(T+n d^2/(n+1)^2\). The new observation contributes \(n^2d^2/(n+1)^2\). Adding these contributions gives

which establishes the result. The key point is that the original observations must be recentered around the new mean; it is not enough to add the new observation’s squared distance from the old mean to the old TSS.

Interpreting and Using TSS Carefully

TSS summarizes the total squared spread of the observed outcomes around their sample mean. A larger value can arise because the outcomes are more dispersed, because the outcome is expressed on a larger numerical scale, or because the sample contains different observations. Consequently, TSS values should not be compared casually across different units or samples.

When two samples have the same size and outcome units, TSS can help describe which sample’s outcomes are more spread out around its own mean. The sample variance provides a related summary on a per-degree-of-freedom scale. But TSS alone does not say why the outcomes vary, whether any predictor explains that variation, or whether a regression model is adequate. Those questions require additional quantities and reasoning.

Common Mistakes

• Using the wrong center. TSS in this definition is centered at the sample mean \(\bar{y}\), not at zero or at a predictor’s mean.
• Forgetting to square deviations. Positive and negative deviations would cancel if added directly. Squaring makes every contribution nonnegative.
• Reporting the wrong units. If \(y_i\) is measured in minutes, each squared deviation and TSS are in minutes squared, not minutes.
• Assuming a unit conversion leaves the numerical value unchanged. A shift alone leaves TSS unchanged, but multiplying outcomes by \(c\) multiplies TSS by \(c^2\).
• Adding an observation without updating the mean. Recompute the mean and all deviations, or use the one-observation update formula.
• Treating TSS as a model-based quantity. It is calculated from the observed outcomes alone and does not require regression assumptions.