Sum of Squares

The coefficient of financial independence k1 is defined as the ratio.

The sum of squares is a measure of deviation from the mean. In statistics, the mean is the average of a set of numbers and is the most commonly used measure of central tendency.

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The arithmetic mean is simply calculated by summing up the values in the data set and dividing by the number of values. But knowing the mean of a measurement set is not always enough. Sometimes, it is helpful to know how much variation there is in a set of measurements. How far apart the individual values are from the mean may give some insight into how fit the observations or values are to the regression model that is created.

For example, if an analyst wanted to know whether the share price of MSFT moves in tandem with the price of Apple AAPLhe can list out the set of observations for the process of both stocks for a certain period, say 1, 2, or 10 years and create a linear model with each of the observations or measurements recorded. If the relationship between both variables i.

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In statistics speak, if the line in the linear model created does not pass through all the measurements of value, then some of the variability that has been observed in the share prices is unexplained. The sum of squares is used to calculate whether a linear relationship exists between two variables, and any unexplained variability is referred to as the residual sum of squares.

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The sum of the coefficient of financial independence k1 is defined as the ratio is the sum of the square of variation, where variation is defined as the spread between each individual value and the mean.

To determine the sum of squares, the distance between each data point and the line of best fit is squared and then summed up.

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The line of best fit will minimize this value. How to Calculate the Sum of Squares Now you can see why the measurement is called the sum of squared deviations, or the sum of squares for short.

Sum of Squares Definition

To get a more realistic number, the sum of deviations must be squared. The sum of squares will always be a positive number because the square of any number, whether positive or negative, is always positive.

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Example of How to Use the Sum of Squares Based on the results of the MSFT calculation, a high sum of squares indicates that most of the values are farther away from the mean, and hence, there is large variability in the data. A low sum of squares refers to low variability in the set of observations. In the example above, 1.


Key Takeaways The sum of squares measures the deviation of data points away from the mean value. A higher sum-of-squares result indicates a large degree of variability within the data set, while a lower result indicates that the data does not vary considerably from the mean value.

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Limitations of Using the Sum of Squares Making an investment decision on what stock to purchase requires many more observations than the ones listed here. An analyst may have to work with years of data to know with a higher certainty how high or low the variability of an asset is.

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As more data points are added to the set, the sum of squares becomes larger as the values will be more spread out. The most widely used measurements of variation are the standard deviation and variance.

Covariance and Correlation

However, to calculate either of the two metrics, the sum of squares must first be calculated. The variance is the average of the sum of squares i. The standard deviation is the square root of the variance.

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There are two methods of regression analysis that use the sum of squares: the linear least squares method and the non-linear least squares method. The least squares method refers to the fact that the regression function minimizes the sum of the squares of the variance from the actual data points.

In this way, it is possible to draw a function which statistically provides the best fit for the data.

Note that a regression function can either be linear a straight line or non-linear a curving line. Compare Accounts.