Standard deviation is the process that involves calculating the squared difference between each data point and the mean, summing those squared differences, dividing by the number of data points, and then taking the square root of the result.
Standard deviation is useful in various fields, such as statistics, finance, and science, because it provides an estimate of the amount of uncertainty or risk associated with a set of values.
This article explains the concept of standard deviation and a step-by-step guide of how to calculate it. Learn how to calculate the standard deviation of population data, and sample data, with some examples.
What is standard deviation?
The measuring process of dispersion or spreading of data is known as standard deviation. It exposes the average absolute distance of every point from the mean. The larger the standard deviation value, the more the data tends to move away from the mean.
Standard Deviation (STD) is the positive square root of the average squared deviation taken from the mean. It is denoted by the lowercase Greek letter σ read as sigma.
Population standard deviation (σ)
The population standard deviation tells you how much individual values in a population differ from the mean (mean) of the entire population. The formula for calculating population standard deviation is:
where N = the size of the population, xi = each value from the population and µ = the population mean
Note: ∑ means the sum of (this will make more sense with the examples later)
Step-by-step guide to calculating the population STD
Step 1
Calculate the sum of the given data (∑xi)
Step 2
Count the total number of observations (or population size) (N)
Step 3
Determine the population arithmetic mean (μ) by dividing the sum from step 1 (∑xi) with the population size from step 2 (N)
Step 4
For each value, calculate the squared difference from the arithmetic mean – (xi – μ)2
Step 5
Sum up the squared differences – ∑(xi – μ)2
Step 6
Determine the variance by dividing the sum of the squared deviations from step 5 by N
Step 7
Take the square root of the variance obtained in Step 6
Sample Standard Deviation (s)
The sample STD is used as an estimate of the population data when we have access only to a sample of data, not the entire population. The formula for calculating the sample standard deviation differs from that of the population standard deviation, primarily in the adjustment for degrees of freedom.
The degrees of freedom in the sample standard deviation formula account for this by dividing by N – 1 instead of N (where N is the number of observations in the sample).
The formula for finding sample STD is:
How to calculate sample STD
Step 1
Calculate the sum of the given data (∑xi)
Step 2
Count the total number of observations (or population size) (N)
Step 3
Determine the sample arithmetic mean (x̄) by dividing the sum from step 1 (∑xi) with the population size from step 2 (N)
Step 4
For each value, calculate the squared difference from the arithmetic mean – (xi – x̄)2
Step 5
Sum up the squared differences – ∑(xi – x̄)2
Step 6
Determine the variance by dividing the sum of the squared deviations from step 5 by (N-1)
Step 7
Take the square root of the variance obtained in Step 6
Examples of finding sample & population STD
Example 1
Find the standard deviation of the given sample data: 12, 15, 18, 24, 31
Σ(x) = 12 + 15 + 18 + 24 + 31 = 100
x̄ = Σ(x)/N = 100/5 = 20
xi – x̄ = (12 – 20), (15 – 20), (18 – 20), (24 – 20), (31 – 20) = (-8), (-5), (-2), (4), (11)
(xi – x̄)2 = (-8)2, (-5)2, (-2)2, (4)2, (11)2 = (64), (25), (4), (16), (121)
Σ(xi – x̄)2 = 64 + 25 + 4 + 16 + 121 = 230
Σ(xi – x̄)2 / (N-1) = 230/(5-1) = 230/4 = 57.5
s = √57.5 = 7.6 (to 1 decimal place)
Hence Sample STD of (12, 15, 18, 24, 31) = 7.6 (to 1 decimal place)
Wrap-up
In this article, we have learned about the idea of standard deviation, which measures the dispersion of data. It is most reliable and sufficient. We discussed a step-by-step guide to estimating standard deviation from population data and sample data with examples.
I hope you have found this explanation useful. This article was written by Talha Alvi. Talha Alvi, an experienced SEO specialist and skilled writer, passionately explores the new ideas of cutting-edge technology and business success. His articles and insights offer valuable perspectives on the latest advanced tech solutions for individuals, brands, and businesses, contributing to a thriving community engaged in tech innovation. LinkedIn: https://www.linkedin.com/in/talha-alvi-022702294/
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