Line of Best Fit: A Line of best fit is a fundamental concept of statistics used to analyze the relationship between two variables. It helps predict the values of one variable based on the values of another variable(given).
Line of best fit is a straight line drawn through a scatter plot of data points that best represent their distribution by minimizing the distances between the line and these points. It results from regression analysis and serves to illustrate the relationship among the data. This line is also a predictive tool, useful for forecasting trends, such as market indicators and price movements.
In this article, we will learn about the Line of Best Fit, how to calculate the line of best fit, solved examples, and other in detail in this article.
What Is a Line of Best Fit?
Line of best fit also known as the trend line or regression line is a straight line that best represents the relationship between a set of data points. We can also say that “a straight line drawn on a scatter plot and lies near the majority of the data points is called the line of best fit.
Line of best fit is calculated using the statistical method of linear regression. The line of best fit minimizes the distance between the observed data points and the line itself indicating the overall data trends.
Meaning – Line of Best Fit
The line of best fit, also known as a trend line or linear regression line, is a straight line that is used to approximate the relationship between two variables in a set of data points on a scatter plot. This line attempts to show the pattern within the data by minimizing the total distance between itself and all the data points.
Line of Best Fit in Regression
Line of best fit is a straight line that best represents the relationship between two variables in a dataset. It is used in statistics to summarize and analyze the relationship between the variables. The line of best fit is often determined through regression analysis, a statistical technique used to model the relationship between variables. Regression analysis helps quantify the strength and direction of the relationship, allowing for predictions and further analysis.
Line of Best Fit in Statistics
In statistics, the line of best fit, also known as the trend line, is a straight line that best represents the data points on a scatter plot. This line attempts to show the relationship between two variables by minimizing the distances between the points and the line itself, specifically the vertical distances. The process of finding this line involves a method called linear regression, where the goal is to minimize the sum of the squares of these vertical distances, a technique often referred to as the “least squares” method.
The line of best fit is calculated using the least squares method, which minimizes the sum of the squares of the vertical distances between the observed data points and the line. The formula for the equation of the line of best fit is:
y = mx + b
where:
- m is Slope of Line
- b is Y-Intercept
Line of Best Fit Formula
How to Calculate the Line of Best Fit?
Calculating the line of best fit involves finding the slope and y-intercept of the line that minimizes the overall distance between the line and the data points. A regression with two independent is solved using a formula
y = c + b1(x1) + b2(x2)
where,
- y is Dependent Variable
- c is Constant
- b1 is First Regression Coefficient
- x1 is First Independent Variable
- b2 is Second Regression Coefficient
- x2 is Second Independent Variable
To find the line of best fit, you can use various statistical software or programming languages like Python or R which have built-in functions for regression analysis. Alternatively, you can manually calculate the line’s parameters using statistical formulas.
The line of best fit is a key concept in statistics, showcasing the relationship between two variables within a dataset. It serves as a foundational tool in regression analysis enabling the prediction of one variable’s value based on another. By determining the line of best fit, the aim is to minimize the vertical distances between the line and the data points providing the closest approximation of the overall trend.
Example:Consider a dataset representing the relationship between the number of hours studied and the score achieved on a test:
|
Hours Studied
|
Test Score
|
|
2
|
65
|
|
3
|
70
|
|
4
|
75
|
|
5
|
80
|
|
6
|
85
|
By calculating the line of best fit for this data, we can predict the test score based on the number of hours studied. Let’s assume the line of best fit equation given is:
y = mx +b
Test Score = 5 × Hours Studied + 60
where:
Is a Line of Best Fit Always Straight?
Line of best fit is typically assumed to be straight in linear regression analysis. However in more complex regression techniques like polynomial regression, the line of best fit can take on curved forms to better fit the data.
Thus, the line of best fit is not always required to be straight.
In fact, depending on the nature of the data the line of best fit can take various forms including linear, quadratic, exponential or logarithmic. For the example provided, let’s fit a quadratic curve to the dataset representing the growth of a bacterial colony over time:
|
Time (hours)
|
Bacterial Count
|
|
1
|
100
|
|
2
|
200
|
|
3
|
400
|
|
4
|
800
|
|
5
|
1600
|
General equation for a quadratic curve is:
y = ax2 + bx + c
We can use the given data points to solve for the coefficients a, b and c.
Using the first data point (1, 100):
100 = a(1)2 + b(1) + c
100 = a + b + c..(1)
Using the second data point (2, 200):
200 = a(2)2 + b(2) + c
200 = 4a + 2b + c …(2)
Using the third data point (3, 400):
400 = a(3)2 + b(3) + c
400 = 9a + 3b + c…(3)
Now, we have a system of three equations (1), (2) and (3) to solve for a, b and c.
Solving this system will give us the coefficients of the quadratic curve equation, which will represent the growth pattern of the bacterial colony over time.
Where Line of Best Fit is Used?
Line of best fit is essential for several reasons:
- Prediction: It enables us to predict the values of one variable based on the values of another variable.
- Trend Analysis: It helps in identifying trends and patterns in the data.
- Decision Making: Businesses use it to make informed decisions based on historical data trends.
Line of Best Fit Examples
Example 1: For the following data, equation of the line of best fit is y = -3.2x + 880. Find, how many people will attend the show if ticket price is $15?
|
Ticket price (x) (in $)
|
12
|
15
|
18
|
20
|
|
Number of people attending (y)
|
780
|
800
|
820
|
830
|
Solution:
Substitute ticket price as x = 15 into the equation of line of best fit
y = -3.2x + 880
y = -3.2 × 15 + 880 = 830
830 people will attend the show if the ticket price is $15.
Example 2: For the following data, equation of the line of best fit is y = -6.5x + 1350. Find, how many people will purchase chocolate if price of chocolate is $30?
|
Chocolate price (x)
|
$25
|
$30
|
$35
|
$40
|
|
Number of people purchasing chocolate (y)
|
480
|
600
|
720
|
840
|
Solution:
Substitute chocolate price as x = 30 into the equation of line of best fit
y = -6.5x + 1350
y = -6.5 × 30 + 1350 = 1155
1155 people will purchase chocolate if the price is $30.
Example 3: For the following data, equation of the line of best fit is y = -2.1x + 1220. Find, how many people will buy candy if price of candy is $10?
|
Candy price (x)
|
$8
|
$10
|
$12
|
$14
|
|
Number of people buying candy (y)
|
1180
|
1200
|
1220
|
1240
|
Solution:
Substitute x = 10 into the equation of line of best fit
y = -2.1x + 1220
y = -2.1 × 10 + 1220 = 1209
1209 people will buy candy if the price is $10.
Example 4: For the following data, equation of the line of best fit is y = -5.8x + 1320. Find, how many people will attend the concert show if ticket price is $25?
|
Concert ticket price (x)
|
$20
|
$25
|
$30
|
$35
|
|
Number of people attending concert (y)
|
660
|
500
|
340
|
180
|
Solution:
Substitute ticket price as x = 25 into the equation of line of best fit
y = -5.8x + 1320
y = -5.8 × 25 + 1320 = 1180
1180 people will attend the concert show if the ticket price is $25.
Example 5: For the following data, equation of the line of best fit is y = -3.7x + 1020. Find, how many people will buy cigarette if price is $25?
|
Cigarette price (x)
|
$15
|
$18
|
$21
|
$24
|
|
Number of people purchasing cigarette (y)
|
870
|
900
|
930
|
960
|
Solution:
Substitute ticket price as x = 18 into the equation of line of best fit
y = -3.7x + 1020
y = -3.7 × 18 + 1020 = 951.6
951.6 people will attend the show if the ticket price is $18.
Line of Best Fit – Practice Questions
Q1: For the following data, equation of the line of best fit is y = -3x + 100. Find, how many people will buy cigarette if price is $20?
|
Cigarette price (x)
|
$10
|
$12
|
$13
|
$14
|
|
Number of people purchasing cigarette (y)
|
850
|
920
|
900
|
960
|
Q2: For the following data, equation of the line of best fit is y = -2x + 120. Find, how many people will buy candy if price of candy is $10?
|
Candy price (x)
|
$8
|
$10
|
$14
|
|
Number of people buying candy (y)
|
180
|
120
|
140
|
Q3: For the following data, equation of the line of best fit is y = -3x + 800. Find, howmany people will attend the show if ticket price is $14?
|
Ticket price (x)
|
$2
|
$5
|
$8
|
$10
|
|
Number of people attending (y)
|
80
|
90
|
110
|
140
|
Q4: For the following data, equation of the line of best fit is y = -5x + 1200. Find, how many people will attend the concert show if ticket price is $27?
|
Concert ticket price (x)
|
$20
|
$25
|
$30
|
$35
|
|
Number of people attending concert (y)
|
600
|
500
|
300
|
200
|
Q5: For the following data, equation of the line of best fit is y = -4.0x + 1060. Find, how many many people will attend the concert show if ticket price is $25?
|
Ticket price (x)
|
$17
|
$20
|
$23
|
$26
|
|
Number of people attending (y)
|
740
|
700
|
660
|
620
|
FAQs on Line of Best Fit
What is the meaning of line of best fit?
Line of best fit is a straight line that represents the relationship between two variables in a dataset. It is used to summarize and analyze the relationship making predictions or inferences about future data points.
How is the line of best fit calculated?
Line of best fit is typically calculated using statistical techniques such as linear regression. It involves finding the line that minimizes the overall distance between the line and the data points, often using methods like the least squares regression.
What does the slope of the line of best fit represent?
Slope of the line of best fit represents the rate of change between the two variables. It indicates how much the dependent variable (y) changes for a one-unit increase in the independent variable (x).
What does the y-intercept of the line of best fit signify?
Y-intercept of the line of best fit represents the predicted value of the dependent variable when the independent variable is zero. It provides a baseline reference point for the relationship between the variables.
Is the line of best fit always straight?
In linear regression analysis, the line of best fit is assumed to be straight. However, in more complex regression techniques like polynomial regression, the line of best fit can take on curved forms to better fit the data.
How reliable is the line of best fit for making predictions?
Reliability of the line of best fit depends on the strength of the relationship between the variables and the variability of the data. Generally, if the data points closely follow the trend of the line, predictions tend to be more accurate.
How can I interpret the goodness-of-fit of the line of best fit?
Goodness-of-fit of the line of best fit is often assessed using metrics like the coefficient of determination (R-squared). A higher R-squared value indicates a better fit, meaning the line explains more of the variability in the data.
What are some common applications of the line of best fit?
Line of best fit is used in various fields such as economics, engineering, biology and social sciences. Common applications include forecasting sales based on historical data analyzing trends in stock prices and predicting crop yields.
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