A heat of best fit is a right line that is the ideal approximation that the given set of data.

that is offered to research the nature of the relation in between two variables. (We"re just considering the two-dimensional case, here.)

A heat of ideal fit have the right to be roughly determined making use of an eyeball an approach by illustration a right line top top a scatter plot so that the variety of points over the line and also below the line is about equal (and the line passes through as countless points as possible).

A more accurate way of detect the line of ideal fit is the least square technique .

usage the adhering to steps to find the equation of heat of best fit for a collection of ordered pairs ( x 1 , y 1 ) , ( x 2 , y 2 ) , ... ( x n , y n ) .

action 1: calculation the average of the x -values and the mean of the y -values.

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

step 2: The complying with formula provides the slope of the line of ideal fit:

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

action 3: Compute the y -intercept the the line by using the formula:

b = Y ¯ − m X ¯

step 4: use the slope m and the y -intercept b to type the equation the the line.




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Example:

usage the least square technique to recognize the equation of heat of best fit because that the data. Climate plot the line.
x 8 2 11 6 5 4 12 9 6 1
y 3 10 3 6 8 12 1 4 9 14


Solution: Plot the clues on a coordinate plane .

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calculate the way of the x -values and also 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

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

ns x ns y ns x i − X ¯ y i − Y ¯ ( x i − X ¯ ) ( y i − Y ¯ ) ( x i − X ¯ ) 2
1 8 3 1.6 − 4 − 6.4 2.56
2 2 10 − 4.4 3 − 13.2 19.36
3 11 3 4.6 − 4 − 18.4 21.16
4 6 6 − 0.4 − 1 0.4 0.16
5 5 8 − 1.4 1 − 1.4 1.96
6 4 12 − 2.4 5 − 12 5.76
7 12 1 5.6 − 6 − 33.6 31.36
8 9 4 2.6 − 3 − 7.8 6.76
9 6 9 − 0.4 2 − 0.8 0.16
10 1 14 − 5.4 7 − 37.8 29.16
∑ ns = 1 n ( x ns − X ¯ ) ( y ns − Y ¯ ) = − 131 ∑ ns = 1 n ( x i − X ¯ ) 2 = 118.4

calculation the slope.

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

calculate the y -intercept.

usage the formula come compute the y -intercept. b = Y ¯ − m X ¯       = 7 − ( − 1.1 × 6.4 )         = 7 + 7.04         ≈ 14.0

usage the slope and also y -intercept to type the equation that the line of finest fit.

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