What criterion is used to establish the best fit of a straight line through the points on a scatter diagram?

A line of best fit is a straight line drawn through the maximum number of points on a scatter plot balancing about an equal number of points above and below the line.

It is used to study the nature of relation between two variables.

The line of best fit in the scatter plot above rises from left to right; so, the variables have a positive correlation .

Here, the line of best fit drops from left to right, so the variables have a negative correlation.

Example 1:

Draw a line of best fit for the scatter plot given.

Solution:

Draw a line through the maximum number of points, balancing about an equal number of points above and below the line.

Example 2:

Draw a line of best fit for the scatter plot given.

Solution:

Plot the age in the x -axis and the income in the y -axis and mark the points.

Draw a line through the maximum number of points balancing about an equal number of points above and below the line.

A line of best fit is a straight line that is the best approximation of the given set of data.

It is used to study the nature of the relation between two variables. (We're only considering the two-dimensional case, here.)

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

A more accurate way of finding the line of best fit is the least square method .

Use the following steps to find the equation of line of best fit for a set of ordered pairs ( x 1 , y 1 ) , ( x 2 , y 2 ) , ... ( x n , y n ) .

Step 1: Calculate the mean of the x -values and the mean of the y -values.

X ¯ = ∑ i = 1 n x i n Y ¯ = ∑ i = 1 n y i n

Step 2: The following formula gives the slope of the line of best fit:

   m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2

Step 3: Compute the y -intercept of the line by using the formula:

b = Y ¯ − m X ¯

Step 4: Use the slope m and the y -intercept b to form the equation of the line.

Example:

Use the least square method to determine the equation of line of best fit for the data. Then plot the line.

Solution:

Plot the points on a coordinate plane .

Calculate the means of the x -values and the y -values.

X ¯ = 8   +   2   +   11   +   6   +   5   +   4   +   12   +   9   +   6   +   1 10 = 6.4 Y ¯ = 3   +   10   +   3   +   6   +   8   +   12   +   1   +   4   +   9   +   14 10 = 7

Now calculate x i − X ¯ , y i − Y ¯ , ( x i − X ¯ ) ( y i − Y ¯ ) , and ( x i − X ¯ ) 2 for each i .

Calculate the slope.

m = ∑ i = 1 n ( x i − X ¯ ) ( y i − Y ¯ ) ∑ i = 1 n ( x i − X ¯ ) 2 = − 131 118.4 ≈ − 1.1

Calculate the y -intercept.

Use the formula to compute the y -intercept.

b = Y ¯ − m X ¯       = 7 − ( − 1.1 × 6.4 )         = 7 + 7.04         ≈ 14.0

Use the slope and y -intercept to form the equation of the line of best fit.

The slope of the line is − 1.1 and the y -intercept is 14.0 .

Therefore, the equation is y = − 1.1 x + 14.0 .

Draw the line on the scatter plot.