Mastering Standard Deviation: A Simple Guide with Examples

It is fundamental in statistics as it involves the square root of the variance to determine the standard deviation. Often abbreviated as SD and symbolized by σ.  It reflects the extent of deviation of data points from the mean.
A reduced SD indicates that the values are more tightly clustered around the mean, while a high standard deviation indicates that the values are more scattered afar from the mean.

This article’s goal is to cover and explain the foundational terms linked to Standard Deviation, encompassing:

• Standard Deviation: What Does it Represent?
• Standard Deviation Calculation.
• Steps to Calculate SD.
• Grouped Data: SD.
• Problems linked to Standard deviation.

Let’s explain each term and gain an understanding of the Standard Deviation.

What is SD?

Standard deviation represents the dispersion or spread of data points about their mean. It defines how widely values are distributed across a sample and quantifies the deviation of data points from the mean.

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When we work with n observations denoted as x₁, x₂, …, xₙ, calculating the mean deviation from the mean value employs the formula Σᵢ₌₁ⁿ (xi – x̄) ². However, only using the total of squared differences from the average might not appear to be the best method to measure how spread out the values are.

A smaller average of squared differences indicates a closer of the observations x_i to the mean x̄, suggesting a lesser dispersion. Alternatively, a greater sum suggests a wider scattering of observations from the mean x̄. Therefore, Σᵢ₌₁ⁿ (x_i – x̄) ² stands as a reasonable gauge of dispersion or spread.

We adopt 1/n * Σᵢ₌₁ⁿ (x_i – x̄) ² as a suitable measure of dispersion and this is known as variance (σ²). The positive Sqrt of variance produces the standard deviation.

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SD Calculation:

These formulas help to determine the dispersion or spread of data values around the mean. The formulas used to calculate SD are:

For Population: SD _Population = √ [∑ (xᵢ – x̅) ² / N]

For Sample: SD _sample = √ [∑ (xᵢ – µ) ² / (N – 1)]

Here, x_i​ represents individual data points, μ is the population mean, x̄ is the sample mean, N is the number of data points in the population, and n is the number of data points in the sample.

Calculating the Standard Deviation: Stepwise Procedure

The Computation of standard deviation involves several steps:

Step 1: Detect the x̅ (Mean) of the data set.

Step 2: Subtract the x̅ (Mean) from each data point to find out the differences.

Step 3: Calculate the square for every difference.

Step 4:  Compute the x̅ (mean) of the squared differences. (VAR = total Sum of the squared differences / Number of observations).

Step 5: Take the sqrt of this mean (x̅) to gain the SD.

Problems of Standard Deviation:

Let’s solve some examples and try to understand how we calculate the SD.

Problem 1:

Consider a set of test scores: 85, 90, 88, 92, 84. Find the standard deviation.

Solution:

Step 1: Detect the x̅:

Average (x̅) = 90 + 85 + 88 + 92 + 84 / 5 = 87.8

Step 2: Subtract the mean (x̅) from each data point to determine the differences.

Step 3: Square each difference.

Step 4: Find the mean (x̅) of the squared differences.

Var = 44.8 / 4 = 11.2

Step 5: Take the sqrt of the step 4 answer to gain the standard deviation.

SD = √ 11.2 = 3.347.

Problem 2:

For a population of numbers: 10, 15, 20, 25, 30, 35. Determine the population standard deviation.

Solution:

Step 1: Detect the x̅

µ = 10 + 15 + 20 + 25 + 30 +35 / 6 = 22.5.

For Population: SD _Population = √ [∑ (xᵢ – µ) ² / N]

Step 2: Subtract the mean (x̅) from each data point to determine the differences.

Step 3: Square Each difference

Step 4:  Use Formula: (But consider ‘N’ as 6 for the population standard deviation calculation).

For Population: SD _Population = √ [∑ (xᵢ – µ) ² / N (Submit Values)

SD _Population (σ) = √ [∑1 / 6(437.5)​]

SD _Population (σ) = √ (0.1666) (437.5)​ = √72.92

SD _Population (σ) = 8.54

Var = 72.917

Conclusion:

In this article, we explored Standard Deviation, a crucial measure of data spread around the mean. Understanding its calculation helps grasp data variability. Through examples like test scores and population sets, we learned step-by-step calculations, identifying the mean, finding differences, and squaring them. Using these, we found the Standard Deviation for the given data sets.