Least Square Method: Notes, Definition, Formula & Solved Examples

We typically deal with a significant volume of numerical data in statistics. We collect, coordinate, handle, and measure while we calculate the statistical data in this branch. This knowledge may be full of noise or variance. Often, it is important to know the data pattern that moves in whatever direction it goes, increases or declines, etc. The way of doing so is called the least-square method.

Also Read: Class 12 Maths NCERT Solutions

Least-square Example

An analyst who wants to measure the relationship between the stock returns of a business and the returns of the index for which the stock is a part is an example of the least-square approach. 

The Line of Best Fit Equation

The best fit line calculated by the system of the least-squares has an equation that tells the tale of the correlation between the data points. 

Regression Line with the least-squares

The line that ideally suits this linear equation is defined as a least-square regression line, which minimizes the vertical distance from the data points to the regression line if the data shows a slimmer relationship between two variables. 

Limitations for Least-Square Method

The least-square method is a rather beneficial curve-fitting approach. It also has a few shortcomings, including several advantages. Here, one of the main drawbacks is addressed.

It is invariably presumed that the errors in the independent variable are negligible or zero in the regression analysis procedure, which utilises the least-square approach for curve fitting. In such situations, simulations are susceptible to estimation errors where independent variable errors are not insignificant. The least square approach will also lead to hypothesis testing here, where parameter ranges and confidence intervals are taken into account due to the presence of errors found in the independent variables.

The least-square method says that a curve with a minimum total of the squared residuals (or variations or errors) from the specified data points is said to be the curve that best matches a given collection of observations. Let us assume that (x1,y1), (x2,y2), (x3,y3),..., (xn,yn) are the data points provided, in which all x's are independent variables, while all y's are dependent variables. Also, assume that the appropriate curve is f(x) and that d is a mistake or deviation from each given value.

In chapter statistics and probability, you will get to learn about some of the basic topics of least-squares. The weightage is 5-6 marks.

In chapter statistics, you will get to learn about it along with other properties and methods. The weightage of this chapter is 4-5 marks.

1. Fit the straight-line curve with the help of the least-square method.

Solution.

The normal equation is:
Σy = aΣx + nb and Σxy = aΣx² + bΣx

Substituting the values, we get,

819 = 798a + 10b

66045 = 64422a + 798b

Solving, we get

a = 0.9288 and b = 7.78155

Therefore, the straight line equation is:

y = 0.9288x + 7.78155.

2. If the equation y=ax^b can be written in linear form Y=A+BX, what is Y, X, A, B?

Solution.

The curve given is y=ax^b.
Taking log on both ends, we're going to have,
Log y = log a + b log x.
It can be written as Y=A+BX,
Where 
Y=log y, A=log a, B=b and X=logx.

3. If the equation y = ae^bx is written in linear form Y=A + BX, then what is Y, X, A, B?

Solution.

The equation is y = ae^bx
Take log to base e on both ends,
we get log y = log a + bx.
This can be substituted as Y=A+BX,
Where Y = log y, A = log a, B = b and X = x.

Q: Why do we use the least-square method?

Q: What is the least-squares theory?

Q: What is the least-square Line of Regression?

Q: What does the approach of minimal squares minimize?

Q: What is the principle that the least-square method is based on?