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Exploring Standard Deviation: Definition, Formulas, & Calculations

by Usman CB
Exploring Standard Deviation: Definition, Formulas, & Calculations

A statistical technique used to quantify the degree of variability or dispersion within a dataset is the standard deviation. It plays a fundamental role in statistics, providing insights into how data points cluster around the mean.  In standard deviation, the concept if data points are closely grouped this is called smaller standard deviation, and if data points are spread this type is called larger standard deviation.

It is a crucial tool for assessing the spread of data points around the mean (average) and plays a fundamental role in various fields, from finance and science to social sciences and quality control. We’ll explore the idea of standard deviation in this article, along with its definition, formulas, calculations, and solved examples..

Standard Deviation

A statistical technique for calculating how far a dataset’s data points depart from the mean is known as standard deviation. It is a statistical measure that quantifies the spread or variability of data points in a dataset, indicating how closely they cluster around the mean.

Mathematically standard deviation is a different sign representing population and sample standard deviation, the symbol sigma (σ) denotes the population standard, and the letter s represents the sample of standard deviation.

Now we discuss below the formula we used for standard deviation.

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Population Standard Deviation (σ):

Sample Standard Deviation (s):

Steps for Finding Standard Deviation

Follow the below steps for finding the standard deviation:

Step 1: Find the average

Calculate the mean (average) of the data points by summing all the values and dividing by the total count (N for the population, n for the sample).

Step 2: Determine deviation

For each data point, subtract the mean (μ for the population, x̄ for the sample). This is the deviation score set of yield. The answer to this is a yield a set of deviation scores.

Step 3: Deviation square

Square each deviation score to eliminate negative values and emphasize the magnitude of the deviation.

Step 4: Take the sum of the answer value of step 3.

Sum all the squared deviation scores to obtain the total sum of squares.

Step 5: Divide by the Total Count (Population or Sample)

Divide the total sum (n-1) for the sample and N for the population.

Step 6: Using step 5’s square root, get the solution.

In the last step, we take the square root of standard deviation (s for the sample and σ for the population)

Uses of standard deviation

Standard deviation serves multiple important purposes in statistics and data analysis:

Measure of Variability:

A dataset’s variability or dispersion is measured by its standard deviation. It helps analysts understand how data points are spread out, providing critical information about the data’s distribution.

Quality Control:

In manufacturing and quality control, standard deviation helps monitor and improve the consistency and quality of products. A lower standard deviation implies more consistent products, while a higher standard deviation indicates inconsistency.

Scientific Research:

 In scientific experiments, standard deviation is used to assess the reliability and consistency of results. It aids in determining the precision of measurements and the significance of observed differences.


Standard deviation is crucial for decision-making in various fields, such as healthcare, where it is used to evaluate the variability of patient data, or education, where it helps assess student performance.

Statistical Inference: In statistical hypothesis testing, standard deviation plays a role in determining the significance of results. It helps establish confidence intervals and assess the validity of statistical tests.

How to calculate standard deviation?

Below are a few solved examples to learn how to calculate standard deviation for sample and population data values.

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Example number 1:

Suppose you are a teacher, and you have collected exam scores from a class of 30 students. You want to understand the variability in the scores to evaluate the difficulty of the exam and plan future teaching strategies.

 Here are the scores:

87, 92, 78, 85, 88, 90, 95, 84, 79, 75, 93, 91, 89, 86, 82, 96, 88, 94, 81, 83, 89, 90, 86, 77, 87, 94, 79, 81, 92, 84


Step# 1:

 Calculate the Mean

Sum all the scores and divide by the total count:

   μ = (87 + 92 + 78 + … + 84) / 30 = 87.6

Step# 2:

 Calculate the Deviations

For deviations subtract each score from mean

  Deviation: (87-87.6), (92-87.6), (78-87.6), …

Step# 3:

 Square the Deviations

Square each deviation:

S.D: (-0.6) ², (4.4) ², (-9.6) ², …

Step# 4:

Sum the Squared Deviations

Sum all the squared deviations:

   Σ (xi – μ) ² = (0.6) ² + (4.4) ² + (9.6) ² + … = 655.4

Step# 5:

 Divide by the Total Count

Divide the sum of squared deviations by 30 (population standard deviation) to find the variance:

   Variance = 655.4 / 30 = 21.847

Step# 6:

Take the Square Root

Finally, take the square root of the variance to obtain the standard deviation:

   σ = √21.847 ≈ 4.674

Example number 2:

 If sample data is 9, 2, 23, 38, 23, and 11 given determine the standard deviation of sample space.


Step# 1:

Input data

9, 2, 23, 38, 23, 11

Step# 2:

Mean = sum of all input data / total number of term

x̄ = (9 +2 + 23 + 38 + 23 + 11)/7

x̄ = 144/7

x̄  = 20.5714

Step# 3:

The formula used for sample space

s=√(1/ (n-1) ∑ni=1 (Xi – x̄)2)

Step# 4:

s = √ (1/ (7-1) ((9-20.5714)2 + (2- 20.5714)2+ (38 – 20.5714)2 + (23-20.5714)2 +(38 – 20.5714)2 +(11 – 20.5714)2)

s= √ (0.1667) (1189.7143)

s = 14.08

Wrap Up

In this article, we have discussed the standard deviation with the help of basic concepts, calculating steps, significance, and understanding this article with the help of detailed examples. After studying this article anyone can defend this article anywhere.


Question number 1:

What does a high standard deviation indicate?


A high standard deviation indicates that data points are widely spread out from the mean, suggesting greater variability or dispersion in the dataset.

Question number 2:

What does a low standard deviation mean?


A low standard deviation suggests that data points are closely clustered around the mean, indicating less variability or dispersion in the dataset.

Question number 3:

What are some real-life applications of standard deviation?


Standard deviation is applied in quality control to monitor product consistency, in healthcare to assess patient data variability, in scientific research for analyzing experimental results, and in education to evaluate student performance and more.

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