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How to Find Variance in MATLAB?

Published in MATLAB Statistics 3 mins read

To find the variance in MATLAB, you use the built-in function var().

The primary way to calculate the variance of a dataset A in MATLAB is by using the command:

V = var(A)

According to the reference, this function call V = var( A ) returns the variance of the elements of A.

Understanding var(A)

Let's break down how the var(A) function works based on the provided information:

  • Input A: This is the data (a vector, matrix, or multi-dimensional array) for which you want to calculate the variance.
  • Output V: This is the calculated variance.
  • Dimension: The function calculates the variance along the first array dimension whose size does not equal 1.
    • If A is a vector (either a row or column vector), the variance is calculated for all elements in the vector, and the result V is a scalar.
    • If A is a matrix, var(A) calculates the variance for each column (the first dimension whose size is typically greater than 1 in a matrix), and V will be a row vector containing the variances of each column.
  • Normalization: By default, the variance is normalized by N-1, where N is the number of observations along the dimension the variance is calculated. This is the standard sample variance often used as an unbiased estimator of the population variance.

Basic Example

Let's see a simple example using a vector.

% Define a vector of observations
data_vector = [1, 2, 3, 4, 5];

% Calculate the variance
variance_value = var(data_vector);

% Display the result
disp(variance_value);

In this case, data_vector is a vector of observations. As stated in the reference, if A is a vector, V is a scalar. So, variance_value will be a single number representing the variance of [1, 2, 3, 4, 5].

Variance for Arrays (Matrices)

When A is a matrix, var(A) computes the variance for each column.

% Define a matrix
data_matrix = [1, 5;
               2, 6;
               3, 7;
               4, 8];

% Calculate the variance of each column
column_variances = var(data_matrix);

% Display the result
disp(column_variances);

Here, data_matrix has two columns. var(data_matrix) will return a row vector where the first element is the variance of the first column [1; 2; 3; 4] and the second element is the variance of the second column [5; 6; 7; 8].

Normalization Explained

The reference mentions that the variance is normalized by N-1 by default. This is crucial for understanding the result.

  • N-1 normalization: This is the sample variance formula, often denoted as $s^2 = \frac{\sum (x_i - \bar{x})^2}{N-1}$. It's used when your data is a sample taken from a larger population, and you want to estimate the population variance unbiasedly.
  • MATLAB's var() function allows specifying other normalization methods (like dividing by N for population variance), but the default is N-1.

In summary, the var(A) function in MATLAB is your primary tool for calculating variance. It automatically handles vectors and arrays, calculating along the appropriate dimension and using the standard N-1 normalization by default.

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