# How to Calculate MSE in Python (4 Examples)

MSE stands for Mean Squared Error. MSE is used to compare our estimated Y (DV) and observed Y in a model.

This tutorial shows how you can calcuate biased and unbiased MSE in Python using 4 examples.

## Biased MSE and unbiased MSE

The following is the formulas for biased MSE and unbiased MSE.

Biased MSE

$MSE=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n}$

Unbiased MSE

$MSE=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n-p-1}$

## How to Calculate MSE in Python

Method 1: Use Python Numpy

• Biased MSE: np.square(np.subtract(Y_Observed,Y_Estimated)).mean()
• Unbiased MSE: sum(np.square(np.subtract(Y_Observed,Y_Estimated)))/(n-p-1)

Method 2: Use sklearn.metrics

• Biased MSE: mean_squared_error(Y_Observed,Y_Estimated)
• Unbiased MSE: (n/(n-p-1))*mean_squared_error(Y_Observed,Y_Estimated)

## Example 1: Use Numpy for biased MSE

The following Python code calculate biased MSE using Numpy.

import numpy as np
# Obseved values
Y_Observed = [5,4,3,5,1,4,5]

# Estimated values
Y_Estimated = [4.4,5.2,2.5,4.5,2,4,4.5]

# Use Numpy to calculate biased Mean Squared Error (MSE)
np.square(np.subtract(Y_Observed,Y_Estimated)).mean()

Output:

0.5071428571428571

## Example 2: Use Numpy for unbiased MSE

Suppose we only estimate 1 parameter. Thus, the degree of freedom is 7-1-1=5. The following Python code calculate unbiased MSE using Numpy.

import numpy as np
# Obseved values
Y_Observed = [5,4,3,5,1,4,5]

# Estimated values
Y_Estimated = [4.4,5.2,2.5,4.5,2,4,4.5]

# Use Numpy to calculate unbiased Mean Squared Error (MSE)
sum(np.square(np.subtract(Y_Observed,Y_Estimated)))/(7-1-1)

Output:

0.71

## Example 3: Use sklearn.metrics for biased MSE

The folllowing Python codes uses sklearn.metrics mean_squared_error to calculate biased MSE.

from sklearn.metrics import mean_squared_error

import numpy as np
# Obseved values
Y_Observed = [5,4,3,5,1,4,5]

# Estimated values
Y_Estimated = [4.4,5.2,2.5,4.5,2,4,4.5]

#Use sklearn.metrics mean_squared_error to calculate biased MSE
mean_squared_error(Y_Observed,Y_Estimated)

Output:

0.5071428571428571

## Example 4: Use sklearn.metrics for unbiased MSE

Suppose we only estimate 1 parameter. Thus, the degree of freedom is 7-1-1=5. The folllowing Python codes uses mean_squared_error to calculate unbiased MSE.

from sklearn.metrics import mean_squared_error

import numpy as np
# Obseved values
Y_Observed = [5,4,3,5,1,4,5]

# Estimated values
Y_Estimated = [4.4,5.2,2.5,4.5,2,4,4.5]

#Use sklearn.metrics mean_squared_error to calculate unbiased MSE
(7/5)*mean_squared_error(Y_Observed,Y_Estimated)

Output:

0.71