What is Mean Square Error?
A mean-squared error is an essential part of the estimation of the statistics. It helps in measuring the discrepancies between the estimated value and the actual value giving insight through its unique formula to calculate the square of the errors and their average. The square and the average gives the value of the MSE.
The calculation of errors is based on the difference between the value being estimated and the estimated value. As it forecasts the value of the loss, it is a risk function. This quantification helps in analyzing the reason for the loss that can be the estimator as well. MSE is mostly positive.
There are other ways to calculate what is MSE. For instance, if the estimator is not biased, then the variance and the mean-squared error is the same. The unit varies according to the primary measure of the quantity.
Mean Square Error Formula
The MSE formula is pretty easy to understand.
MSE = (1/n)\[\sum_{i=1}^{n}\] (Xobs,i -Xmodel,i)²
The summation sign denotes the sum of all the points that are being considered to estimate the error. Starting from the first point to the nth point or the last point of estimation, the sum has to be calculated. Xi represents all the true values meaning that the obs and the model value represent the actual and predicted values, respectively.
The MSE Formula can also be seen in the light of a specific value, which can be estimated with the same formula replacing the obs value with it and the actual value with the model value.
When the mean square formula is used concerning a regression line on the graph, the Xobs, i value is replaced by the y coordinate of the points from which the errors are to be estimated. The Xmodel, i is taken from the regression line, and in this way, the formula is solved.
What is the Root Mean Square Error Formula?
The confusion of root mean square error vs. mean square error can make the task of finding the estimate difficult. So it is better to analyze what root mean square error formula is for better understanding. The RMSE formula, in simple words, would be the root of the MSE formula.
The primary use of the formula is to understand the magnitude of errors between the predicted and actual values. One of the major differences between MSE and RMSE is that in RMSE, the formula is used for only specific variables but not a range of variables. This is not true for the mean square error formula as root mean square errors are scale-dependent. The formula for
RMSE = \[\sqrt{1/n}\] \[\sum_{i=1}^{n}\] (Xobs,i -Xmodel,i)²
Root Mean Square Error:
This formula is responsible for understanding the accuracy of the model better. Understanding root mean square error vs. mean square error will clear the conjectures relating to what is MSE.
Steps to Use the Formula (with Graphs)
On a graph, the MSE can be seen as the distance between the regression line and the set of points given. In this case, understanding the formula is easier. Therefore, let’s understand how to look at the MSE equation from this perspective.
The reference point for the calculation is the regression line. It has to be calculated to go ahead with the other steps using the coordinates mentioned and the line equation.
Find the new value of the Y’s using the X values to find the exact position of the regression line.
The new Y values are to be subtracted from the actual values to find the error.
Use the formula to calculate the sum and squares of the errors and find the mean.
The example given below would help to understand better.
Solved Example
Q. Find the Mean Squared Error or mse Equation for the Following Set of Values: (43,41),(44,45),(45,49),(46,47),(47,44)
A. On calculating the regression line using an online computation tool, it is found to be y= 9.2 + 0.8x.
The new Y’ values are as follows:
9.2 + 0.8(43) = 43.6
9.2 + 0.8(44) = 44.4
9.2 + 0.8(45) = 45.2
9.2 + 0.8(46) = 46
9.2 + 0.8(47) = 46.8
The error can be calculated as (Y-Y’):
41 – 43.6 = -2.6
45 – 44.4 = 0.6
49 – 45.2 = 3.8
47 – 46 = 1
44 – 46.8 = -2.8
Adding the squares of the errors: 6.76 + 0.36 + 14.44 + 1 + 7.84 = 30.4. The mean of the squares of the errors are: 30.4 / 5 = 6.08
Regression line =
y=9.2+0.8x
Facts about Mean Squared Error
The efficacy depends on the proximity of the value to zero.
The ideal value for MSE is zero. However, that is not very probable.
Score functions like Brier score are used in forecasting, and further prediction is based on what is mean square error.
FAQs on Mean Squared Error
1. What is Mean Squared Error (MSE) and what does it tell you?
Mean Squared Error (MSE) is a standard metric used to measure the average squared difference between the actual values and the predicted values in a dataset. It essentially provides a measure of the average magnitude of error for a set of predictions. A lower MSE value indicates that a model's predictions are, on average, closer to the true values, signifying a better model fit.
2. What is the formula for calculating Mean Squared Error?
The formula for calculating Mean Squared Error (MSE) is given by:
MSE = (1/n) * Σ(yᵢ - ŷᵢ)²
Where:
- n represents the total number of data points or observations.
- Σ is the summation symbol, indicating the sum of all terms.
- yᵢ is the actual or observed value for the i-th data point.
- ŷᵢ (pronounced y-hat) is the value predicted by the model for the i-th data point.
3. Is a higher or lower MSE value considered better?
A lower Mean Squared Error (MSE) is always better. An MSE value of 0 represents a perfect model, where every prediction exactly matches the actual value. As the MSE increases, it signifies that the model's predictions are further from the actual values, indicating a poorer performance. Therefore, in statistics and machine learning, the primary goal is often to find a model that minimises the MSE.
4. Why is the Mean Squared Error never a negative number?
The Mean Squared Error (MSE) can never be negative due to the squaring operation in its formula. The error for each data point is calculated as the difference (yᵢ - ŷᵢ), which can be positive, negative, or zero. However, this difference is then squared, (yᵢ - ŷᵢ)², which ensures the result is always non-negative (positive or zero). Since MSE is the average of these non-negative values, it will always be a positive number or, in the case of a perfect model, zero.
5. How is Mean Squared Error used in linear regression?
In linear regression, Mean Squared Error serves as the primary cost function. The goal of the regression algorithm is to find the best-fit line that most accurately represents the data. It achieves this by systematically adjusting the line's slope and intercept to find the values that minimise the MSE. A lower MSE indicates that the average squared distance from the data points to the regression line is smaller, meaning the line is a better fit for the data.
6. What is considered a “good” MSE value?
There is no single “good” value for Mean Squared Error, as it is highly dependent on the context and the scale of the target variable. For instance, an MSE of 50 might be very small when predicting house prices in thousands of dollars but extremely large when predicting test scores out of 100. A more practical approach is to compare the MSE of one model to another for the same dataset. A “good” MSE is one that is low relative to the range of your data and is better than the MSE from a baseline or alternative model.
7. What is the difference between Mean Squared Error (MSE) and Root Mean Squared Error (RMSE)?
The key difference between MSE and RMSE is their scale and interpretability.
- Mean Squared Error (MSE): Measures the average of the squared errors. Its units are the square of the original data's units (e.g., if predicting height in cm, MSE is in cm²).
- Root Mean Squared Error (RMSE): Is the square root of the MSE. Its main advantage is that its units are the same as the original variable (e.g., cm), making it easier to interpret the typical error size.
While both penalise large errors heavily, RMSE is often preferred for reporting model accuracy because its value is more intuitive.
8. How do outliers affect the Mean Squared Error?
Outliers can have a significant and disproportionate impact on the Mean Squared Error. Because the error terms in the MSE formula are squared, a single large error from an outlier becomes a very large squared error value. This can dramatically inflate the overall MSE, making a model appear less accurate than it actually is for the non-outlier data points. This sensitivity to outliers is a crucial characteristic to consider when using MSE as an evaluation metric.
