You’ve probably seen slope and intercept in algebra. These concepts can also be used to predict and understand information in statistics. Take a look at this lesson!

## Interpreting Slope and Intercept

Lauren is collecting information for her auto mechanics class. She surveys six different auto mechanic shops in her town and collects information to see if there is a relationship between the number of times the oil is changed in a vehicle and the longevity in the engine of the vehicle. Once she gathers her information, Lauren puts all of it into a scatterplot with a regression line. Now that she has collected all of her data, how can she interpret this data into usable information?

In this lesson, you will learn how to interpret the meaning of slope and *y*-intercept in different examples of linear models.

## Identifying Slope and Intercept

A **linear model** is a comparison of two values, usually *x* and *y*, and the consistent change between those values. The easiest way to understand and interpret slope and intercept in linear models is to first understand the slope-intercept formula: *y* = *mx* + *b*. *M* is the slope or the consistent change between *x* and *y*, and *b* is the *y*-intercept. Often, the *y*-intercept represents the starting point of the equation.

Take a look at this graph:

The line in the center is known as a **regression line**, a straight line that attempts to predict the relationship between two points. This relationship is the same thing as the slope, and you may also hear the terms consistent change or interval. These three words are used interchangeably and mean the same thing in this case. The points around this line represent the data that is collected in this scenario. The equation for this line is *y* = .3136*x* + .2644.

## Interpreting Slope

Let’s take a look at our regression equation. For this scenario we have .3136 and .2644. .3136 is the slope in this equation, and .2644 is the intercept in this equation. First, let’s talk about slope and how we can interpret slope in this equation. Remember that the slope is the consistent change, or the relationship between two variables, in a linear model.

For example, let’s say you were getting paid eight dollars an hour at your job. The rate, eight dollars, would be multiplied by the number of hours that you worked to get how much you should be paid for the week. In this case, the two variables are the number of hours you worked and how much you get paid for the week. The relationship between the number of hours you work and how much you get paid is the amount you get paid per hour. In this case, you know the relationship between the two variables ahead of time, but sometimes you know the variables and not the relationship, also known as the slope.

Notice on our equation that slope is .3136. So what does this mean? Remember, our two variables are the number of times the oil is changed in the vehicle and the longevity of the engine. The slope is a positive number, which means that when the one variable increases, the other also increases. Just like the amount you get paid at the end of the week increases when the number of hours you work increases, so does the longevity of your engine increase as the number of times you change out the oil increases.

Since a positive slope tells us there is a positive relationship between the two variables, what does the number .3136 tell us? Remember, in the previous scenario, the eight told us how much you were being paid per hour. In this example, .3136 shows us how much the longevity of your engine increases.

Let’s look at it like this. You have your vehicle sitting in your garage. Maybe you’ve had it for a couple of years. Each time you change the oil out in your vehicle increases the likelihood that the engine will last by .3136 years. That’s right! In this case, the slope represents the number of years that you increase your engine’s lifespan every time you change the oil. Remember, this is just an example, and statistics doesn’t always show us the full picture. Obviously, if your vehicle’s engine is broken, changing the oil in it several times won’t fix the problem! Now that you understand slope in this scenario, let’s move on to the intercept.

## Interpreting Intercept

We know that the intercept is .2644 in our equation, but what does that mean? First, the intercept is also called the *y*-intercept. This is because it is the place in the equation where the line intercepts the *y*-axis. The ordered pair for the intercept is (0, .2644). This means that *x* = 0 and *y* = .2644. To understand the intercept, you need to understand the ordered pair.

The *x* variable represents the number of oil changes in the vehicle. Therefore, in this case we are saying that there have been zero oil changes in the vehicle. The *y* variable represents the number of years of longevity in the engine. Therefore, this ordered pair shows that the vehicle’s engine longevity is .2644 years. If we put this together, the intercept tells us that if there are zero oil changes in the vehicle the engine will last .2644 years. Obviously, in reality, this will differ depending on the maintenance on the vehicle, the age of the vehicle and the freshness of the oil that is currently in the vehicle. You will find that many data sets will have more variables that influence the slope and the intercept of the equation.

## Practice Problem

Lauren is now collecting data on the amount of gasoline used and how often a person sits idle when using their vehicle. After collecting the data and plotting the points, she has developed the following equation: *y* = 3.52*x* + 17.32, where *y* equals the number of gallons used in a month, and *x* equals the number of hours spent idle in a month. Can you interpret the slope and intercept of this equation?

First, let’s talk about the slope. In this example, the slope represents the gallons of gas that are used. Therefore, for every hour the vehicle spends idle, 3.52 extra gallons of gas are used per month according to this linear model. While this may or may not be true in real life, this is the interpretation of this particular model.

Second, it’s time to interpret the intercept. Remember that the ordered pair for the intercept is (0, 17.32). If *x* equals the number of hours spent idle in a month, and *y* equals the number of gallons of gas used in a month, then this means that a person will still use at least 17.32 gallons even if he or she never has the vehicle idle.

## Lesson Summary

Every linear model is different. However, you can identify some similarities that will help you interpret the slope and intercept of the model. Remember, a **linear model** is a comparison of two values, usually *x* and *y*, and the consistent change between the values. Linear models will have a **regression line**, a straight line that attempts to predict the relationship between two points.

You can use the slope-intercept formula, *y* = *mx* + *b*, to identify the slope and intercept of the regression line. In this equation, *m* is the slope, or the consistent change between *x* and *y*, and *b* is the *y*-intercept. Often, the *y*-intercept represents the starting point of the equation; in our examples, the *y*-intercept represented what would happen if the *x*-value did not exist, and this is true for all linear models. You can use these interpretations to predict information about a set of data. Learn more in our other lessons on regression and correlation.

## Learning Outcomes

After completing this lesson, take the next step and test your ability to:

- Recognize a linear model and regression line
- Use the slope-intercept formula to interpret real-world data