---
title: "Sum of Squared Errors (SSE) | Intro to Statistics"
description: "Sum of Squared Errors (SSE) is the total of squared residuals from a regression line in Intro to Statistics, used to judge how well predictions fit data."
canonical: "https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse"
type: "key-term"
subject: "Intro to Statistics"
unit: "Unit 12"
---

# Sum of Squared Errors (SSE) | Intro to Statistics

## Definition

Sum of Squared Errors (SSE) is the total of the squared residuals from a regression model. In Intro to Statistics, it tells you how far the predicted values are from the observed data overall.

## What It Is

Sum of Squared Errors (SSE) is the total squared distance between the observed y-values and the values predicted by a regression line in Intro to Statistics. In symbols, it is the sum of (y - y-hat)^2 for every data point.

This is not just a random formula. It is the score the least-squares regression line is trying to minimize. The line of best fit is chosen because it makes those vertical prediction errors, called residuals, as small as possible once they are squared and added up.

The squaring matters. A positive residual and a negative residual would otherwise cancel each other out, which would hide prediction error. Squaring turns every error into a nonnegative number, so big misses count more than small ones and the total reflects overall model fit.

SSE is always zero or larger. A value of 0 would mean every predicted point lands exactly on every observed point, which basically never happens with real data unless the model fits perfectly. In normal regression problems, a smaller SSE means the line is tracking the data more closely.

A quick example makes the idea clearer. If a model predicts y-values of 10, 12, and 14, but the actual values are 11, 9, and 15, the residuals are 1, -3, and 1. Squaring them gives 1, 9, and 1, so the SSE is 11. That number summarizes the total mismatch between the line and the data.

In Intro to Statistics, you usually see SSE when comparing regression models, finding the least-squares line, or moving on to statistics built from SSE, like mean squared error and R-squared. It is one of the main ways statistics turns a scatterplot into a measurable fit.

## Why It Matters

SSE shows how regression turns a scatterplot into a number you can work with. Instead of just saying one line looks better than another, you can compare how much error each line leaves behind. That makes regression less about eyeballing and more about measuring fit.

It also explains why the least-squares line is special. The best-fit line is not chosen because it looks nice, but because it produces the smallest total squared prediction error. Once you know that, the logic of regression makes a lot more sense, especially when you are asked why one line is the least-squares regression line.

SSE connects directly to other parts of intro stats. Mean Squared Error comes from SSE, and R-squared is built from how much variation the model explains compared with the total variation in the data. If you can read SSE well, those later formulas feel less abstract.

You will also see the same idea in problem sets where you calculate residuals from a table or a scatterplot, then decide which model fits better. The concept is simple, but it shows up again and again whenever the course asks you to judge a prediction model instead of just compute one.

## Connections

### Residuals

Residuals are the individual prediction errors that get squared and added up in SSE. Each residual is observed minus predicted, so SSE is really a summary of all those leftover distances. If you can find residuals on a graph or in a table, you are already halfway to finding SSE.

### [Sum of Squared Vertical Distances](/college-intro-stats/key-terms/sum-squared-vertical-distances)

This is the geometric way to think about SSE on a scatterplot. The errors are vertical because regression predicts y from x, so you measure straight up or down from each point to the line. Squaring those vertical distances gives the total that the least-squares line tries to minimize.

### Mean Squared Error (MSE)

MSE is SSE divided by the number of data points, so it changes the total squared error into an average squared error per point. SSE grows with sample size, while MSE helps you compare error on a per-observation basis. They are closely related, but they answer slightly different questions.

### R-squared

R-squared is built from SSE and total variation in the data. It tells you what fraction of the variation in y is explained by the regression model. A smaller SSE usually leads to a larger R-squared, but R-squared gives the fit in relative terms instead of raw error units.

## On the AP Exam

A regression problem set might give you predicted values and observed values and ask for SSE, or it might ask which of two lines fits better based on the residuals. Your job is to calculate each residual, square it, and add the squares. If a question gives you a scatterplot, you may need to identify the vertical errors that make up SSE.

You also use SSE when a quiz asks why the least-squares line was chosen. The correct move is to connect it to minimizing the total squared residuals, not to say it simply has the highest correlation. In interpretation questions, a smaller SSE means the model leaves less unexplained error, so the predictions are closer to the actual data overall.

## Sum of Squared Errors (SSE) vs Residuals

Residuals are the individual errors for each data point, while SSE is the sum of those residuals after each one is squared. So residuals are the pieces, and SSE is the total. If you mix them up, you may calculate one point's error when the question wants the full model error.

## Key Takeaways

- SSE stands for Sum of Squared Errors, and in regression it measures the total squared difference between observed and predicted y-values.
- A smaller SSE means the regression line fits the data better, because the predictions are closer to the actual points overall.
- The least-squares regression line is chosen because it minimizes SSE, not because it makes every residual zero.
- SSE uses squared residuals so that positive and negative errors do not cancel out and larger mistakes count more.
- SSE is the starting point for other statistics in regression, including Mean Squared Error and R-squared.

## FAQs

### What is Sum of Squared Errors (SSE) in Intro to Statistics?

SSE is the total of the squared residuals from a regression model. It measures how far the predicted values are from the observed values overall, using vertical errors on the scatterplot.

### How do you calculate SSE?

First find each residual by subtracting the predicted value from the observed value. Then square each residual and add all the squared results. The final sum is SSE.

### Is a smaller SSE better?

Yes. A smaller SSE means the regression model leaves less unexplained error, so the line fits the data more closely. SSE is only useful for comparison when the models are using the same response variable and data scale.

### What is the difference between SSE and residuals?

A residual is the error for one data point, while SSE is the sum of all the squared residuals. Residuals show the individual misses, and SSE gives you one number for the whole model.

## Related Study Guides

- [12.3 The Regression Equation](/college-intro-stats/unit-12/3-regression-equation/study-guide/tIAj1HV4y1upDCdh)

## About This Document

Canonical Fiveable pages are available as Markdown at the same path plus `.md`.

- [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns
- [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing
- [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`)
- [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`)

## Structured Data

```json
{"@context":"https://schema.org","@graph":[{"@type":"LearningResource","@id":"https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse#resource","name":"Sum of Squared Errors (SSE) | Intro to Statistics","url":"https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse","learningResourceType":"Concept explainer","educationalLevel":"AP® / High School","about":{"@id":"https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse#term"},"audience":{"@type":"EducationalAudience","educationalRole":"student"},"dateModified":"2026-07-03T02:21:06.614Z","isPartOf":{"@type":"Collection","name":"Intro to Statistics Key Terms","url":"https://fiveable.me/college-intro-stats/key-terms"},"author":{"@type":"Organization","@id":"https://fiveable.me/editorial#content-team","name":"Fiveable Content Team","url":"https://fiveable.me/editorial","parentOrganization":{"@id":"https://fiveable.me/#organization"},"member":[{"@type":"Person","name":"Amanda DoAmaral","sameAs":["https://www.linkedin.com/in/adoamaral"]},{"@type":"Person","name":"Becky Bahr","sameAs":["https://www.linkedin.com/in/becky-bahr-b52743132"]}]},"publisher":{"@type":"Organization","name":"Fiveable","url":"https://fiveable.me"}},{"@type":"DefinedTerm","@id":"https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse#term","name":"Sum of Squared Errors (SSE)","description":"Sum of Squared Errors (SSE) is the total of the squared residuals from a regression model. In Intro to Statistics, it tells you how far the predicted values are from the observed data overall.","url":"https://fiveable.me/college-intro-stats/key-terms/sum-of-squared-errors-sse","inDefinedTermSet":{"@type":"DefinedTermSet","name":"Intro to Statistics Key Terms","url":"https://fiveable.me/college-intro-stats/key-terms"}},{"@type":"FAQPage","mainEntity":[{"@type":"Question","name":"What is Sum of Squared Errors (SSE) in Intro to Statistics?","acceptedAnswer":{"@type":"Answer","text":"SSE is the total of the squared residuals from a regression model. It measures how far the predicted values are from the observed values overall, using vertical errors on the scatterplot."}},{"@type":"Question","name":"How do you calculate SSE?","acceptedAnswer":{"@type":"Answer","text":"First find each residual by subtracting the predicted value from the observed value. Then square each residual and add all the squared results. The final sum is SSE."}},{"@type":"Question","name":"Is a smaller SSE better?","acceptedAnswer":{"@type":"Answer","text":"Yes. A smaller SSE means the regression model leaves less unexplained error, so the line fits the data more closely. SSE is only useful for comparison when the models are using the same response variable and data scale."}},{"@type":"Question","name":"What is the difference between SSE and residuals?","acceptedAnswer":{"@type":"Answer","text":"A residual is the error for one data point, while SSE is the sum of all the squared residuals. Residuals show the individual misses, and SSE gives you one number for the whole model."}}]},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Intro to Statistics","item":"https://fiveable.me/college-intro-stats"},{"@type":"ListItem","position":2,"name":"Key Terms","item":"https://fiveable.me/college-intro-stats/key-terms"},{"@type":"ListItem","position":3,"name":"Unit 12","item":"https://fiveable.me/college-intro-stats/unit-12"},{"@type":"ListItem","position":4,"name":"Sum of Squared Errors (SSE)"}]}]}
```