Standard Deviation Notes

Published on: September 29, 2026

This Standard Deviation Calculator finds how spread out a list of numbers is and shows the working line by line. The mean is found first, then the distance from it to each value, and those distances are squared so they cannot cancel out. Their average is the variance, and its square root brings the answer back to the units of the original values. Following the same pattern every time makes the spread easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Count the values.
  3. Add the values together.
  4. Divide to get the mean.
  5. Subtract the mean from each value.
  6. Square each difference.
  7. Add the squared differences.
  8. Write the formula.
  9. Substitute the total and the count into the formula.
  10. Divide to get the variance.
  11. Take the square root to get the standard deviation.

Formula:

\(\sigma = \sqrt{\frac{\Sigma (x - \bar{x})^{2}}{n}}\)
See Example 1 Hide Example 1

Problem

\(6,\; 6,\; 7,\; 10\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Count the values.
  3. Add the values together.
  4. Divide to get the mean.
  5. Subtract the mean from each value.
  6. Square each difference.
  7. Add the squared differences.
  8. Write the formula.
  9. Substitute the total and the count into the formula.
  10. Divide to get the variance.
  11. Take the square root to get the standard deviation.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding how far the values \(6,\; 6,\; 7,\; 10\) sit from their own mean, on average.

\(6,\; 6,\; 7,\; 10\)

Step 2

Step 2 - Count the values.

In this problem: Counting along the list reaches \(4\) at the last value, so \(n = 4\).

\(n = 4\)

Step 3

Step 3 - Add the values together.

In this problem: Adding the values in order gives \(6 + 6 + 7 + 10 = 29\).

\(\Sigma x = 6 + 6 + 7 + 10 = 29\)

Step 4

Step 4 - Divide to get the mean.

In this problem: Dividing the total by the count gives \(29 \div 4 = 7.25\).

\(\bar{x} = \frac{29}{4} = 7.25\)

Step 5

Step 5 - Subtract the mean from each value.

In this problem: Taking the mean away from each value in turn: \(6 - 7.25 = -1.25\), \(6 - 7.25 = -1.25\), \(7 - 7.25 = -0.25\), \(10 - 7.25 = 2.75\).

\(\begin{array}{c|c} x & x - \bar{x} \\ \hline 6 & -1.25 \\ 6 & -1.25 \\ 7 & -0.25 \\ 10 & 2.75 \end{array}\)

Step 6

Step 6 - Square each difference.

In this problem: Squaring each of those differences removes the minus signs: \((-1.25)^{2} = 1.56\), \((-1.25)^{2} = 1.56\), \((-0.25)^{2} = 0.06\), \((2.75)^{2} = 7.56\).

\(\begin{array}{c|c} x - \bar{x} & (x - \bar{x})^{2} \\ \hline -1.25 & 1.56 \\ -1.25 & 1.56 \\ -0.25 & 0.06 \\ 2.75 & 7.56 \end{array}\)

Step 7

Step 7 - Add the squared differences.

In this problem: Adding the squared differences gives \(1.56 + 1.56 + 0.06 + 7.56 = 10.75\).

\(\Sigma (x - \bar{x})^{2} = 1.56 + 1.56 + 0.06 + 7.56 = 10.75\)

Step 8

Step 8 - Write the formula.

In this problem: The variance is the average of those squared differences, so their total is divided by how many values there are.

\(\sigma^{2} = \frac{\Sigma (x - \bar{x})^{2}}{n}\)

Step 9

Step 9 - Substitute the total and the count into the formula.

In this problem: Replacing \(\Sigma (x - \bar{x})^{2}\) with \(10.75\) and \(n\) with \(4\) gives \(\sigma^{2} = \frac{10.75}{4}\).

\(\sigma^{2} = \frac{10.75}{4}\)

Step 10

Step 10 - Divide to get the variance.

In this problem: Dividing gives \(10.75 \div 4 \approx 2.69\).

\(\sigma^{2} = \frac{10.75}{4} \approx 2.69\)

Step 11

Step 11 - Take the square root to get the standard deviation.

In this problem: Squaring made the units too big, and the square root undoes that: \(\sqrt{2.69} \approx 1.64\).

\(\sigma = \sqrt{2.69} \approx 1.64\)

Final Answer

\(\sigma \approx 1.64\)
See Example 2 Hide Example 2

Problem

\(4,\; 9,\; 11\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Count the values.
  3. Add the values together.
  4. Divide to get the mean.
  5. Subtract the mean from each value.
  6. Square each difference.
  7. Add the squared differences.
  8. Write the formula.
  9. Substitute the total and the count into the formula.
  10. Divide to get the variance.
  11. Take the square root to get the standard deviation.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding how far the values \(4,\; 9,\; 11\) sit from their own mean, on average.

\(4,\; 9,\; 11\)

Step 2

Step 2 - Count the values.

In this problem: Counting along the list reaches \(3\) at the last value, so \(n = 3\).

\(n = 3\)

Step 3

Step 3 - Add the values together.

In this problem: Adding the values in order gives \(4 + 9 + 11 = 24\).

\(\Sigma x = 4 + 9 + 11 = 24\)

Step 4

Step 4 - Divide to get the mean.

In this problem: Dividing the total by the count gives \(24 \div 3 = 8\).

\(\bar{x} = \frac{24}{3} = 8\)

Step 5

Step 5 - Subtract the mean from each value.

In this problem: Taking the mean away from each value in turn: \(4 - 8 = -4\), \(9 - 8 = 1\), \(11 - 8 = 3\).

\(\begin{array}{c|c} x & x - \bar{x} \\ \hline 4 & -4 \\ 9 & 1 \\ 11 & 3 \end{array}\)

Step 6

Step 6 - Square each difference.

In this problem: Squaring each of those differences removes the minus signs: \((-4)^{2} = 16\), \((1)^{2} = 1\), \((3)^{2} = 9\).

\(\begin{array}{c|c} x - \bar{x} & (x - \bar{x})^{2} \\ \hline -4 & 16 \\ 1 & 1 \\ 3 & 9 \end{array}\)

Step 7

Step 7 - Add the squared differences.

In this problem: Adding the squared differences gives \(16 + 1 + 9 = 26\).

\(\Sigma (x - \bar{x})^{2} = 16 + 1 + 9 = 26\)

Step 8

Step 8 - Write the formula.

In this problem: The variance is the average of those squared differences, so their total is divided by how many values there are.

\(\sigma^{2} = \frac{\Sigma (x - \bar{x})^{2}}{n}\)

Step 9

Step 9 - Substitute the total and the count into the formula.

In this problem: Replacing \(\Sigma (x - \bar{x})^{2}\) with \(26\) and \(n\) with \(3\) gives \(\sigma^{2} = \frac{26}{3}\).

\(\sigma^{2} = \frac{26}{3}\)

Step 10

Step 10 - Divide to get the variance.

In this problem: Dividing gives \(26 \div 3 \approx 8.67\).

\(\sigma^{2} = \frac{26}{3} \approx 8.67\)

Step 11

Step 11 - Take the square root to get the standard deviation.

In this problem: Squaring made the units too big, and the square root undoes that: \(\sqrt{8.67} \approx 2.94\).

\(\sigma = \sqrt{8.67} \approx 2.94\)

Final Answer

\(\sigma \approx 2.94\)
See Example 3 Hide Example 3

Problem

\(2,\; 4,\; 4,\; 6\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Count the values.
  3. Add the values together.
  4. Divide to get the mean.
  5. Subtract the mean from each value.
  6. Square each difference.
  7. Add the squared differences.
  8. Write the formula.
  9. Substitute the total and the count into the formula.
  10. Divide to get the variance.
  11. Take the square root to get the standard deviation.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding how far the values \(2,\; 4,\; 4,\; 6\) sit from their own mean, on average.

\(2,\; 4,\; 4,\; 6\)

Step 2

Step 2 - Count the values.

In this problem: Counting along the list reaches \(4\) at the last value, so \(n = 4\).

\(n = 4\)

Step 3

Step 3 - Add the values together.

In this problem: Adding the values in order gives \(2 + 4 + 4 + 6 = 16\).

\(\Sigma x = 2 + 4 + 4 + 6 = 16\)

Step 4

Step 4 - Divide to get the mean.

In this problem: Dividing the total by the count gives \(16 \div 4 = 4\).

\(\bar{x} = \frac{16}{4} = 4\)

Step 5

Step 5 - Subtract the mean from each value.

In this problem: Taking the mean away from each value in turn: \(2 - 4 = -2\), \(4 - 4 = 0\), \(4 - 4 = 0\), \(6 - 4 = 2\).

\(\begin{array}{c|c} x & x - \bar{x} \\ \hline 2 & -2 \\ 4 & 0 \\ 4 & 0 \\ 6 & 2 \end{array}\)

Step 6

Step 6 - Square each difference.

In this problem: Squaring each of those differences removes the minus signs: \((-2)^{2} = 4\), \((0)^{2} = 0\), \((0)^{2} = 0\), \((2)^{2} = 4\).

\(\begin{array}{c|c} x - \bar{x} & (x - \bar{x})^{2} \\ \hline -2 & 4 \\ 0 & 0 \\ 0 & 0 \\ 2 & 4 \end{array}\)

Step 7

Step 7 - Add the squared differences.

In this problem: Adding the squared differences gives \(4 + 0 + 0 + 4 = 8\).

\(\Sigma (x - \bar{x})^{2} = 4 + 0 + 0 + 4 = 8\)

Step 8

Step 8 - Write the formula.

In this problem: The variance is the average of those squared differences, so their total is divided by how many values there are.

\(\sigma^{2} = \frac{\Sigma (x - \bar{x})^{2}}{n}\)

Step 9

Step 9 - Substitute the total and the count into the formula.

In this problem: Replacing \(\Sigma (x - \bar{x})^{2}\) with \(8\) and \(n\) with \(4\) gives \(\sigma^{2} = \frac{8}{4}\).

\(\sigma^{2} = \frac{8}{4}\)

Step 10

Step 10 - Divide to get the variance.

In this problem: Dividing gives \(8 \div 4 = 2\).

\(\sigma^{2} = \frac{8}{4} = 2\)

Step 11

Step 11 - Take the square root to get the standard deviation.

In this problem: Squaring made the units too big, and the square root undoes that: \(\sqrt{2} \approx 1.41\).

\(\sigma = \sqrt{2} \approx 1.41\)

Final Answer

\(\sigma \approx 1.41\)
See Example 4 Hide Example 4

Problem

\(3,\; 5,\; 7\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Count the values.
  3. Add the values together.
  4. Divide to get the mean.
  5. Subtract the mean from each value.
  6. Square each difference.
  7. Add the squared differences.
  8. Write the formula.
  9. Substitute the total and the count into the formula.
  10. Divide to get the variance.
  11. Take the square root to get the standard deviation.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding how far the values \(3,\; 5,\; 7\) sit from their own mean, on average.

\(3,\; 5,\; 7\)

Step 2

Step 2 - Count the values.

In this problem: Counting along the list reaches \(3\) at the last value, so \(n = 3\).

\(n = 3\)

Step 3

Step 3 - Add the values together.

In this problem: Adding the values in order gives \(3 + 5 + 7 = 15\).

\(\Sigma x = 3 + 5 + 7 = 15\)

Step 4

Step 4 - Divide to get the mean.

In this problem: Dividing the total by the count gives \(15 \div 3 = 5\).

\(\bar{x} = \frac{15}{3} = 5\)

Step 5

Step 5 - Subtract the mean from each value.

In this problem: Taking the mean away from each value in turn: \(3 - 5 = -2\), \(5 - 5 = 0\), \(7 - 5 = 2\).

\(\begin{array}{c|c} x & x - \bar{x} \\ \hline 3 & -2 \\ 5 & 0 \\ 7 & 2 \end{array}\)

Step 6

Step 6 - Square each difference.

In this problem: Squaring each of those differences removes the minus signs: \((-2)^{2} = 4\), \((0)^{2} = 0\), \((2)^{2} = 4\).

\(\begin{array}{c|c} x - \bar{x} & (x - \bar{x})^{2} \\ \hline -2 & 4 \\ 0 & 0 \\ 2 & 4 \end{array}\)

Step 7

Step 7 - Add the squared differences.

In this problem: Adding the squared differences gives \(4 + 0 + 4 = 8\).

\(\Sigma (x - \bar{x})^{2} = 4 + 0 + 4 = 8\)

Step 8

Step 8 - Write the formula.

In this problem: The variance is the average of those squared differences, so their total is divided by how many values there are.

\(\sigma^{2} = \frac{\Sigma (x - \bar{x})^{2}}{n}\)

Step 9

Step 9 - Substitute the total and the count into the formula.

In this problem: Replacing \(\Sigma (x - \bar{x})^{2}\) with \(8\) and \(n\) with \(3\) gives \(\sigma^{2} = \frac{8}{3}\).

\(\sigma^{2} = \frac{8}{3}\)

Step 10

Step 10 - Divide to get the variance.

In this problem: Dividing gives \(8 \div 3 \approx 2.67\).

\(\sigma^{2} = \frac{8}{3} \approx 2.67\)

Step 11

Step 11 - Take the square root to get the standard deviation.

In this problem: Squaring made the units too big, and the square root undoes that: \(\sqrt{2.67} \approx 1.63\).

\(\sigma = \sqrt{2.67} \approx 1.63\)

Final Answer

\(\sigma \approx 1.63\)