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## Statistics and probability

### Course: Statistics and probability > Unit 5

Lesson 4: Least-squares regression equations- Introduction to residuals and least squares regression
- Introduction to residuals
- Calculating residual example
- Calculating and interpreting residuals
- Calculating the equation of a regression line
- Calculating the equation of the least-squares line
- Interpreting slope of regression line
- Interpreting y-intercept in regression model
- Interpreting a trend line
- Interpreting slope and y-intercept for linear models

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# Introduction to residuals

Build a basic understanding of what a residual is.

We run into a problem in stats when we're trying to fit a line to data points in a scatter plot. The problem is this: It's hard to say for sure which line fits the data best.

For example, imagine three scientists, start text, start color #ca337c, A, n, d, r, e, a, end color #ca337c, end text, start text, start color #01a995, J, e, r, e, m, y, end color #01a995, end text, and start text, start color #aa87ff, B, r, o, o, k, e, end color #aa87ff, end text, are working with the same data set. If each scientist draws a different line of fit, how do they decide which line is best?

If only we had some way to measure how well each line fit each data point...

## Residuals to the rescue!

A residual is a measure of how well a line fits an individual data point.

Consider this simple data set with a line of fit drawn through it

and notice how point left parenthesis, 2, comma, 8, right parenthesis is start color #1fab54, 4, end color #1fab54 units above the line:

This vertical distance is known as a

**residual**. For data points above the line, the residual is positive, and for data points below the line, the residual is negative.For example, the residual for the point left parenthesis, 4, comma, 3, right parenthesis is start color #e84d39, minus, 2, end color #e84d39:

The closer a data point's residual is to 0, the better the fit. In this case, the line fits the point left parenthesis, 4, comma, 3, right parenthesis better than it fits the point left parenthesis, 2, comma, 8, right parenthesis.

## Try to find the remaining residuals yourself

## Want to join the conversation?

- what is the difference between error and residual?(50 votes)
- I think ysun means that:

An error is**a deviation from the**population mean.

A residual is**a deviation from the**sample mean.

Errors, like other population parameters (e.g. a population mean), are*usually*theoretical.

Residuals, like other sample statistics (e.g. a sample mean), are measured values from a sample. Sample statistics are often used to estimate population parameters, so in this case the residuals can be used to estimate the error.(49 votes)

- How do you do this On a calculator(9 votes)
- the explanation on how to do this using a calculator is confusing(3 votes)

- This article does not explain what to do with the residuals after calculating them. Are you supposed to sum them? When are you supposed to use them?(11 votes)
- The article is incomplete. It didn't circle back around to answer the question it posed at the beginning: "If each scientist draws a different line of fit, how do they decide which line is best?" Calculating the residuals for each line helps you decide which line best fits the data.(12 votes)

- If you have a really positive residual point that is quite far form the LSRL is that good or bad ? Like what can you say about the residual?(3 votes)
- That would be what is called an "outlier".

It could suggest that the measurement that led to that point was wrong — e.g. The value was 3000, but 30000 got entered by mistake.

Another possibility, especially if there aren't a lot of data points, is that the relationship between the variables is not linear — e.g. an exponential curve might be a better fit....

ADDENDUM: It is also possible that the data is actually very "noisy" (highly variable).(8 votes)

- Really dumb question: Why is it called least squares regression? What does least squares mean?(3 votes)
- The "squares" refers to the squares (that is, the 2nd power) of the residuals, and the "least" just means that we're trying to find the smallest total sum of those squares.

You may ask: why squares? The best answer I could find is that it's easy (minimizing a quadratic formula is easy) and still gives good results.(4 votes)

- how can you summarize a residual plot?(3 votes)
- What are estimates ? How are they different from residuals ?(3 votes)
- An estimate would be the y-value predicted by the regression line whereas a residual is the signed difference between the actual y-value and the estimate.(1 vote)

- If there are many points on a graph then how can you draw a line that is best for all of them?(3 votes)
- The line you make is a compromise that minimizes some function of the residuals.

The most commonly used function is the sum of squares of the residuals. You cannot just do the sum of the values of the residuals, since there are likely to be many lines for which that will be zero.(2 votes)

- in residuals how do you determine which one is best? do you mean it or do you do something else this article did not tell me how to.(4 votes)
- we sum the square of the distances from the mean..though just summing the residuals look intuitively appealing, but it does not take into consideration the "magnitude" of the distance.. e.g, suppose 10 and -10 are two residuals, they are too far from the mean, but they add to 0.(1 vote)

- In the article, it says that the closer the the data point's residual is to zero, it fits the line best. There's (4,3) and (2,8). The residuals are 4, and -2. It says 4 is closer ( aka (4,3) ) but isn't -2 closer to zero than 4? How is this possible?(2 votes)
- The point (4,3) is two units below the line. It has a residual of -2. However, the point (2,8) is four units above the line. It has a residual of 4. Since -2 is closer to zero than 4, the point (4,3) fits the line better than the point (2,8). I think you misunderstood that the residual of four is closer to the line. The article really meant that the point (4,3) is closer to the line. Hope this helped!(2 votes)