# Standard Error of the Estimate

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# The standard error of the estimate

The standard error of the estimate

- is closely related to this quantity and is defined below:
- is a measure of the accuracy of predictions

s_{est}is the standard error of the estimate, Y - actual score Y' - predicted score Y-Y' - differences between the actual scores and the predicted scores. Σ(Y-Y')^{2}- SSE N - number of pairs of scores

## Simple Example

- The graphs below shows two regression examples.
- You can see that in graph A, the points are closer to the line then they are in graph B.
- Therefore, the predictions in Graph A are more accurate than in Graph B.

## Example

Assume the data below are the data from a population of five X-Y pairs

- The last column shows that the sum of the squared errors of prediction is 2.791.
- Therefore, the standard error of the estimate is:

## Formula for the Standard Error

There is a version of the formula for the standard error in terms of Pearson's correlation:

where ρ is the population value of Pearson's correlation

SSY is

Similar formulas are used when the standard error of the estimate is computed from a sample rather than a population.

- The only difference is that the denominator is N-2 rather than N, since two parameters (the slope and the intercept) were estimated in order to estimate the sum of squares
- Formulas comparable to the ones for the population are shown below.

### Example

For the example data,

- μ
_{y}= 2.06 - SSY = 4.597
- ρ= 0.6268.

Therefore,

which is the same value computed previously.

# Quiz