How can the standard deviation of each column in a matrix be determined?

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The standard deviation of each column in a matrix can be determined using the following steps:

1. Calculate the mean of each column in the matrix.
2. For each element in a column, subtract the mean of that column from the element to get the deviation.
3. Square each deviation.
4. Calculate the average of the squared deviations for each column by dividing the sum of the squared deviations by the total number of elements in that column minus 1.
5. Take the square root of the result obtained in step 4 to get the standard deviation of each column.

Mathematically, the formula to calculate the standard deviation of each column in a matrix can be represented as follows:

𝜎 = √(∑(xᵢ - 𝑥̄)²/(n-1))

Where:

• 𝜎: Standard deviation of a column
• xᵢ: Each element of the column
• 𝑥̄: Mean of the column
• n: Total number of elements in the column

Note that the process and formula are the same whether the matrix is a population or a sample. The only difference is the value of n used in the formula (n for a population and n-1 for a sample).