Line of Best Fit

Line of Best Fit

The line of best fit is a line that runs across a scatter plot of data points and best reflects the connection between them. To arrive at the geometric equation for the line, statisticians often utilise the least squares approach, which may be done manually or with regression analysis software. A simple linear regression study of two or more independent variables will provide a straight line. In rare instances, a regression with numerous linked variables might result in a curved line. A line of best fit is a straight line drawn across the most points on a scatter plot, with an equal number of points along each line.

 

Basis of line of Best Fit

One of the most significant outcomes of regression analysis is the line of best fit. A quantitative assessment of the connection between one or more independent factors and a consequent dependent variable is referred to as regression. Experts in a wide range of professions, from research and public administration to financial analysis, employ regression.

A statistician gathers a series of data points, each with a full set of dependent as well as independent variables, to conduct a regression analysis. If the stock is not included in the S&P 500, the dependent variable might be the stock price of the company, as well as the independent variables could be the Standards as well as Poor's 500 index and the national unemployment rate. For the previous 20 years, the sample set might be either of these three data sets. These measured values would show as a scatter plot on a chart, a collection of points which might or might not be grouped along any lines. If a linear pattern emerges, a line of best fit that minimises the distance between those points and the line may be drawn. If no organising axis is visible, regression analysis using the least squares approach can construct line. This approach creates a line that minimises each point's squared distances from the line of best fit. The statistician inserts these three outcomes over the previous 20 years into a regression software tool to get the formula for this line. The programme generates a linear formula that expresses the causal link between the S& P 500, the unemployment rate, and the company's stock price. The formula for the line of best fit is this equation. It's a forecasting technique that allows analysts and traders to forecast the firm's future stock price using those two independent factors.

The Line of Best Fit Equation and Its Components

A regression using two independent variables, such as the one described above, yields a formula that looks like this:

y= c(x1) + b2(x2) (x2)

The dependent variable is y, the constants is c, the first regression coefficient is b1, and the very first independent variable is x1. b2 and x2 are the second coefficient and second independent variable, respectively. The shareprice would be y, the S&P 500 would be x1, as well as the unemployment rate would be x2 in the case above. Each independent variable's coefficient reflects the amount of change in y for each extra unit in that variable. If the S&P 500 rises by one, the corresponding y, or share price, will rise by the same amount. The unemployment rate, the second independent variable, is the same. The gradient of the best fit line is the coefficient in a simple regression with one independent variable. The slope in this case, like in any regression with two independent variables, is a combination of the two coefficients. The y-intercept of the line of best fit is represented by the constant .

How to Find the Line of Best Fit

A best-fit line is a straight line that best approximates the given collection of data. It's a tool for determining the nature of a relationship amongst two variables. (We're only looking at the two-dimensional example right now.) Creating a straight line on a scatter plot so that the number of points above and below the line is roughly equal can be used to calculate a line of best fit using an eyeball approach.

The least square approach is a more precise means of determining the line of best fit.

For a collection of ordered pairs (x1,y1), (x2,y2),...(xn,yn), use the procedures below to obtain the equation of line of best fit.

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

X¯¯¯=∑i=1nxinY¯¯¯=∑i=1nyin

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

   m=∑i=1n(xi−X¯¯¯)(yi−Y¯¯¯)∑i=1n(xi−X¯¯¯)2

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

b=Y¯¯¯−mX¯¯¯

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.

xx	88	22	1111	66	55	44	1212	99	66	11
yy	33	1010	33	66	88	1212	11	44	99	1414

Plot the points on a coordinate plane .

Calculate the means of the xx -values and the yy -values.

X¯¯¯=8 + 2 + 11 + 6 + 5 + 4 + 12 + 9 + 6 + 110=6.4Y¯¯¯=3 + 10 + 3 + 6 + 8 + 12 + 1 + 4 + 9 + 1410=7X¯=8 + 2 + 11 + 6 + 5 + 4 + 12 + 9 + 6 + 110=6.4Y¯=3 + 10 + 3 + 6 + 8 + 12 + 1 + 4 + 9 + 1410=7

Now calculate xi−X¯¯¯xi−X¯ , yi−Y¯¯¯yi−Y¯ , (xi−X¯¯¯)(yi−Y¯¯¯)(xi−X¯)(yi−Y¯) , and (xi−X¯¯¯)2(xi−X¯)2 for each ii .

ii	xixi	yiyi	xi−X¯¯¯xi−X¯	yi−Y¯¯¯yi−Y¯	(xi−X¯¯¯)(yi−Y¯¯¯)(xi−X¯)(yi−Y¯)	(xi−X¯¯¯)2(xi−X¯)2
11	88	33	1.61.6	−4−4	−6.4−6.4	2.562.56
22	22	1010	−4.4−4.4	33	−13.2−13.2	19.3619.36
33	1111	33	4.64.6	−4−4	−18.4−18.4	21.1621.16
44	66	66	−0.4−0.4	−1−1	0.40.4	0.160.16
55	55	88	−1.4−1.4	11	−1.4−1.4	1.961.96
66	44	1212	−2.4−2.4	55	−12−12	5.765.76
77	1212	11	5.65.6	−6−6	−33.6−33.6	31.3631.36
88	99	44	2.62.6	−3−3	−7.8−7.8	6.766.76
99	66	99	−0.4−0.4	22	−0.8−0.8	0.160.16
1010	11	1414	−5.4−5.4	77	−37.8−37.8	29.1629.16
					∑i=1n(xi−X¯¯¯)(yi−Y¯¯¯)=−131∑i=1n(xi−X¯)(yi−Y¯)=−131	∑i=1n(xi−X¯¯¯)2=118.4∑i=1n(xi−X¯)2=118.4

Calculate the slope.

m=∑i=1n(xi−X¯¯¯)(yi−Y¯¯¯)∑i=1n(xi−X¯¯¯)2=−131118.4≈−1.1m=∑i=1n(xi−X¯)(yi−Y¯)∑i=1n(xi−X¯)2=−131118.4≈−1.1

Calculate the yy -intercept.

Use the formula to compute the yy -intercept.

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

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

The slope of the line is −1.1−1.1 and the yy -intercept is 14.014.0 .

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

Draw the line on the scatter plot.