How do you calculate r squared by hand?

Hint: To solve the given question, first we should know what is r squared value and its significance. r squared is a statistical measure of fit that indicates how much variation of a dependent variable is explained by the independent variable in a regression model. R-squared values range from 0 to 1 and are commonly stated as percentages from 0% to 100%. An R-squared of 100% means that all movements of the dependent variable are completely explained by movements in the independent variable.

Complete step-by-step solution:
Let’s say that we are given a data set of independent variables as \[{{x}_{i}}\] and dependent variables as \[{{y}_{i}}\]. The r squared value is calculated as \[{{r}^{2}}=1-\dfrac{S{{S}_{err}}}{S{{S}_{tot}}}\].
By substituting the value of numerator and denominator in RHS, we can find the value of r squared. Here the numerator is \[S{{S}_{err}}\] which is also called the sum of residuals. It is calculated as \[S{{S}_{err}}=\sum{{{\left( {{y}_{i}} \right)}^{2}}}-{{B}_{0}}\sum{{{y}_{i}}}-{{B}_{1}}\sum{\left( {{x}_{i}}{{y}_{i}} \right)}\]. For a system with 1 unknown parameter, the variable y is the sum of \[{{B}_{0}}\And {{B}_{1}}x\].
Now, the denominator is calculated as \[S{{S}_{tot}}=\sum{{{\left( {{y}_{i}} \right)}^{2}}}-\dfrac{{{\left( \sum{{{y}_{i}}} \right)}^{2}}}{n}\]. n is the total number of independent/ dependent variables.

Note: Here, we assumed that r squared has relation with regression. Assuming it a general question, having no relation with regression. We can find the r square value by multiplying r with itself.
In statistics, the closer r is to +1 or -1, the more closely the two variables are related. If r is close to 0, it means there is no relationship between the variables.