How to Calculate Standard Deviation: A Step-by-Step Guide with Example

standard deviation

Standard deviation is a statistical measure that represents the variation or spread of data from the mean or average value

Standard deviation is a statistical measure that represents the variation or spread of data from the mean or average value. It gives us an idea of how spread out the data is from the average. In other words, it tells us how much the individual data points deviate from the mean.

To calculate the standard deviation, follow these steps:

1. Calculate the mean of the data set. Add up all the values and divide the sum by the total number of values. Let’s call this value “x̄”.

2. Subtract the mean (x̄) from each data point in your set.

3. Square each of the differences obtained in the previous step.

4. Find the mean of the squared differences obtained in step 3. This is known as the variance.

5. Finally, take the square root of the variance to get the standard deviation.

The formula for standard deviation is:

σ = √(Σ(x – x̄)^2 / N)

Where:
σ = standard deviation
Σ = sum of
x = individual data point
x̄ = mean
N = total number of data points

Here’s an example to illustrate the calculation of standard deviation:

Let’s say we have the following data set: 2, 4, 6, 8, 10.

1. Calculate the mean:
x̄ = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

2. Subtract the mean from each data point:
2 – 6 = -4
4 – 6 = -2
6 – 6 = 0
8 – 6 = 2
10 – 6 = 4

3. Square each of the differences:
(-4)^2 = 16
(-2)^2 = 4
0^2 = 0
2^2 = 4
4^2 = 16

4. Find the mean of the squared differences:
Variance = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8

5. Take the square root of the variance to get the standard deviation:
Standard deviation = √8 ≈ 2.83

Therefore, the standard deviation of the data set {2, 4, 6, 8, 10} is approximately 2.83. This indicates that the data points are relatively close to the mean since the standard deviation is relatively low.

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