# How to Calculate the Standard Deviation of the Mean

The standard deviation is a widely used statistical measure that provides valuable information about the variability within a set of data. It allows you to assess the dispersion of data points around the mean, which can be crucial in various disciplines such as finance, science, and engineering. In this article, we will explain how to calculate the standard deviation of the mean in a simple step-by-step manner.

**Step 1: Calculate the Mean**

The first step in calculating standard deviation is determining the mean (average) of your dataset. Add up all the data points in your dataset and divide the sum by the total number of data points (n).

Mean = (Sum of all data points) / n

For example, if you have five test scores (50, 60, 70, 80, 90), you can calculate their mean as follows:

Mean = (50 + 60 + 70 + 80 + 90) / 5

Mean = 350 / 5

Mean = 70

**Step 2: Calculate Deviation from Mean for Each Data Point**

Now that you have calculated the mean, you need to determine the deviation of each data point from the mean. To do this, subtract the mean from each individual data point. You are essentially finding how much each value differs from your calculated mean.

Deviation[i] = Data point[i] – Mean

Using our test score example:

Deviation[1] = 50 – 70 = -20

Deviation[2] = 60 – 70 = -10

Deviation[3] = 70 – 70 = 0

Deviation[4] = 80 – 70 = 10

Deviation[5] = 90 – 70 =20

**Step3: Square Each Deviation**

To eliminate negative values that can interfere with our calculations, we need to square each deviation value.

Squared Deviation[i] = Deviation[i]²

For our example:

Squared Deviation[1] = (-20)² = 400

Squared Deviation[2] = (-10)² = 100

Squared Deviation[3] = (0)² = 0

Squared Deviation[4] = (10)² = 100

Squared Deviation[5] = (20)² = 400

**Step 4: Calculate the Mean of the Squared Deviations**

Next, add up all the squared deviations and divide by the total number of data points (n).

Mean of Squared Deviations =(Sum of all squared deviations)/ n

For our example:

Mean of Squared Deviations=(400+100+0+100+400)/5

Mean of Squared Deviations= (1000)/5

Mean of Squared Deviations=200

**Step 5: Find the Square Root of the Mean of Squared Deviations**

Finally, calculate the square root of the mean of squared deviations to determine the standard deviation.

Standard Deviation= √(Mean of Squared Deviations)

Using our example:

Standard Deviation= √(200)

Standard Deviation ≈ 14.14

**Conclusion**

In this article, we have outlined a step-by-step process for calculating standard deviation. By understanding and applying this method, you can assess the variability in your data and make more informed decisions based on your findings.