# Standard Error of the Estimate

Site: | Saylor Academy |

Course: | MA121: Introduction to Statistics |

Book: | Standard Error of the Estimate |

Printed by: | Guest user |

Date: | Tuesday, September 17, 2024, 10:44 AM |

## Description

This section discusses how to compute the standard error of the estimate based on errors of prediction as well as how to compute the standard error of the estimate based on a sample.

## Standard Error of the Estimate

#### Learning Objectives

- Make judgments about the size of the standard error of the estimate from a scatter plot
- Compute the standard error of the estimate based on errors of prediction
- Compute the standard error using Pearson's correlation
- Estimate the standard error of the estimate based on a sample

where σest is the standard error of the estimate, is an actual score, is a predicted score, and is the number of pairs of scores. The numerator is the sum of squared differences between the actual scores and the predicted scores.

Note the similarity of the formula for σest to the formula for . It turns out that σest is the standard deviation of the errors of prediction (each - is an error of prediction).

Assume the data in Table 1 are the data from a population of five , pairs.

**Table 1.**Example data.

1.00 | 1.00 | 1.210 | -0.210 | 0.044 | |

2.00 | 2.00 | 1.635 | 0.365 | 0.133 | |

3.00 | 1.30 | 2.060 | -0.760 | 0.578 | |

4.00 | 3.75 | 2.485 | 1.265 | 1.600 | |

5.00 | 2.25 | 2.910 | -0.660 | 0.436 | |

Sum | 15.00 | 10.30 | 10.30 | 0.000 | 2.791 |

which is the same value computed previously.

##### R code

x=c(1,2,3,4,5)

y= c(1,2,1.3,3.75,2.25)

summary(lm(y~x))

Call:

lm(formula = y ~ x)

Residuals:

1 2 3 4 5

-0.210 0.365 -0.760 1.265 -0.660

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.785 1.012 0.776 0.494

x 0.425 0.305 1.393 0.258

Residual standard error: 0.9645 on 3 degrees of freedom

Multiple R-squared: 0.3929, Adjusted R-squared: 0.1906

F-statistic: 1.942 on 1 and 3 DF, p-value: 0.2578

Source: David M. Lane , https://onlinestatbook.com/2/regression/accuracy.html

This work is in the Public Domain.

## Video

## Questions

**Question 1 out of 4.**

In a regression line, the ________ the standard error of the estimate is, the more accurate the predictions are.

larger

smaller

The standard error of the estimate is not related to the accuracy of the predictions.

**Question 2 out of 4.**

Linear regression was used to predict Y from X in a certain population.
In this population, SSY is 50, the correlation between X and Y is .5,
and N is 100. What is the standard error of the estimate?

**Question 3 out of 4.**

You sample 10 people in a high school to try to predict GPA in 10th
grade from GPA in 9th grade. You determine that SSE = 5.8. What is the
standard error of the estimate?

**Question 4 out of 4.**

The graph below represents a regression line predicting Y from X. This
graph shows the error of prediction for each of the actual Y values.
Use this information to compute the standard error of the estimate in
this sample.

## Answers

- The standard error of the estimate is a measure of the accuracy of predictions. The regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error), and the standard error of the estimate is the square root of the average squared deviation.
- The standard error of the estimate for a population = sqrt[(1-rho
^{2})*SSY/N] = sqrt[(1-.5^{2})*50/100] = .61 - The standard error of the estimate for a sample = sqrt[SSE/(N-2)] = sqrt[5.8/8] = .85
- The standard error of the estimate for a sample = sqrt[SSE/(N-2)]. SSE
is the sum of the squared errors of prediction, so SSE = (-.2)
^{2}+ (.4)^{2}+ (-.8)^{2}+ (1.3)^{2}+ (-.7)^{2}= 3.02; sqrt(3.02/3) = 1.0