Enter the data set values to calculate the standard deviation (σ).
Add this calculator to your site
Use this standard deviation calculator to calculate the standard deviation either for a sample or population data set. You can use this calculator to find:
Limitation: The calculator is not able to handle complex data structures such as matrices or time series data etc, it only focuses on general numerical data sets.
Standard deviation (σ) is a statistical measurement that informs how spread out the data points are in a set of data relative to the mean (average). A higher standard deviation points towards a higher variability in the data. A Greek symbol sigma “σ” is used to represent standard deviation. The following image shows the graphical representation of a normal distribution having a width of 1 band.
It can be used for:
The given formula is used for finding the standard deviation of a sample (subset of data drawn from the population):
\(s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}\)
Where
When all the members of the population can be sampled, then the following standard deviation formula is used:
\(σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2}\)
Where
The standard deviation calculator uses the following formulas to make statistical calculations of standard deviation:
\(\text{Sum} = \sum_{i=1}^{n}x_i\)
\(\text{Size} = n = \text{count}(x_i)_{i=1}^{n}\)
For Sample:
\(\overline{x} = \dfrac{\sum_{i=1}^{n}x_i}{n}\)
For Population:
\(\mu = \dfrac{\sum_{i=1}^{n}x_i}{n}\)
For Sample:
\(\ SS = \sum_{i=1}^{n}(x_i - \overline{x})^{2}\)
For Population:
\(\ SS = \sum_{i=1}^{n}(x_i - \mu)^{2}\)
For Sample:
\(\ s^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}\)
For Population:
\(\sigma^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n}\)
These are the steps:
Suppose you have a data set (3, 4, 9, 7, 2, 5 ), find its standard deviation.
Solution:
Step #1(Calculate Mean Value):
\(\bar{x} = {\dfrac{3 + 4 + 9 + 7 + 2 + 5}{6}}\)
\(\bar{x} = \dfrac{30}{6}\) \(\bar{x} = 5\)
Step #2(Calculate The Value Of \(\left(x_{i} – \bar{x}\right)\):
Data Values (xi) |
xi - x̅ | (xi - x̅ )2 |
3 | 3 - 5 = -2 |
(-2)2 = 4 |
4 |
4 - 5 = -1 | (-1)2 = 1 |
9 | 9 - 5 = 4 |
(4)2 = 16 |
7 |
7 - 5 = 2 | (2)2 = 4 |
2 | 2 - 5= -3 |
(-3)2 = 9 |
5 |
5 - 5 = 0 |
(0)2 = 0 |
Step # 3 (Calculate Sample Standard Deviation):
Now we know that:
\(\ s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}\)
\(\ s = \sqrt {\dfrac { 4+1+16+4+9+0}{ 6-1}}\)
\(\ s = \sqrt {\dfrac { 34 }{5}}\)
\(\ s = \sqrt {6.8}\)
Standard Deviation =\(\ s = 2.6076...\)
The standard deviation of data set {5 5 9 9 9 10 5 10 10} is approximately 2.2913. To check your answers, use the above standard deviation calculator.
It depends on what you are trying to understand about your data:
References:
From the source of Wikipedia: General understanding and basic examples
From the site of scribbr.com: Formulas for population and sample standard deviation
Support
Calculator Online Team Privacy Policy Terms of Service Content Disclaimer Advertise TestimonialsEmail us at
[email protected]© Copyrights 2024 by Calculator-Online.net