Standard deviation is a measure used in statistics to determine the level of dispersion or variability within a set of data values. It is an important concept as it helps to identify how individual data points deviate from the average. In this article, we will walk you through the steps required to calculate the standard deviation of a sample.

Step 1: Gather your sample data

First, collect the data points for which you want to calculate the standard deviation. It is essential to have a representative and relevant sample for accurate results.

Step 2: Calculate the mean

The mean (or average) is simply the sum of all data points divided by the number of observations (n). Let’s denote each observation as x_i, where i ranges from 1 to n.

Mean (µ) = Σ(x_i) / n

Step 3: Determine deviations from the mean

For each data point in your sample, calculate its deviation from the mean. Subtract the mean from each observation.

(x_i – µ)

Step 4: Square these deviations

Square each deviation obtained in Step 3.

(xi – µ)^2

Step 5: Sum up squared deviations

Sum up all squared deviations obtained in Step 4.

Σ[(xi – µ)^2]

Step 6: Calculate variance

To find the sample variance, divide the sum of squared deviations (Step 5) by n -1:

Variance (σ^2) = Σ[(xi – µ)^2] / (n-1)

The denominator uses (n-1) instead of n which is known as Bessel’s correction aiming at accounting for bias in estimating sample standard deviation from a population.

Step 7: Compute Standard Deviation

Finally, take the square root of your calculated variance.

Standard Deviation (σ) = √σ^2

Example:

Let’s consider a small sample of data: 4, 6, 8, 10.

Step 1: Sample data = {4, 6, 8, 10}

Step 2: Mean (µ) = (4 + 6 + 8 + 10) / 4 = 7

Step 3: Deviations = {-3, -1, 1, 3}

Step 4: Squared deviations = {9, 1, 1, 9}

Step 5: Sum of squared deviations = (9 +1 +1 +9) =20

Step 6: Variance (σ^2) = Σ[(xi – µ)^2] / (n-1) =20 / (4-1)=6.67

Step 7: Standard Deviation (σ)= √σ^2 = √6.67 ≈2.58

Now you know how to calculate the standard deviation of a sample with ease! Remember that standard deviation is an essential tool in understanding the variability and distribution of your data and can be used in various fields such as science, finance, and social sciences.