Standard deviation calculator :
The standard deviation calculator is used to find the standard deviation of the given data set also calculate mean, sum of square, variance whethere it is population or sample. This calculator provides step by step solution so that you can easily understand the calculation behind the answer.
How to Use the Standard Deviation Calculator
- Enter Data: Input your data values separated by commas or spaces in the text box provided.
- Select Data Type: Choose whether your data represents a sample or the entire population.
- Rounding Preferences: Specify the number of decimal places for rounding the mean, variance, and standard deviation.
What is Standard Deviation ?
The standard deviation is a statistical measure that used to measure the amount of variation in a set of data
values. It tells us how much the individual data points varies from the mean of the data set.
Understanding standard deviation is must for data analysis, as it helps in interpreting the spread of data
points.
When to use population vs sample standard deviation
Population standard deviation
The population standard deviation is used when we have access to data for the entire population.
Example:
Measuring the height of all students in a school. ( Since we are considering every student, this is the entire population )
Sample standard deviation
When we are working with sample of the population used sample standard deviation.
Example:
Measuring the height of a random sample of \(50\) students out of \(500\) in a school. As this is just a sample, we use the sample standard deviation.
Table which shows difference between formula of sample and population.
| Measure | Sample | Population |
|---|---|---|
| Mean | \( \mathrm{ \bar{x} = \frac{\sum {x} }{n} } \) | \( \mathrm{ \mu = \frac{\sum {x} }{n} } \) |
| Variance | \( \mathrm{ s^{2} =\frac{\sum_{i=1}^{n}(x_{i}-\bar{x} )^{2}}{n-1}} \) | \( \mathrm{ \sigma^{2} =\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n}} \) |
| Standard Deviation | \( \mathrm{ s = \sqrt{ \frac{\sum_{i=1}^{n}(x_{i}- \bar{x} )^{2}}{n-1}}} \) | \( \mathrm{ \sigma = \sqrt{ \frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n}}} \) |
How to calculate population and sample standard deviation
Lets take below Example to calculate standard deviation.
Solution :
Given data is \(25, 9, 21, 16, 5\)
Sample size, \(\mathrm{n = 5}\)
Steps to calculate sample standard deviation :
Sample mean :
\( \begin{align} \mathrm{\bar{x}} &= \mathrm{\frac{\sum x}{n}} \\ &=\frac{\text{25}+\text{9}+......+\text{5}}{\text{5}} \\ &=\frac{\text{76}}{\text{5}} \\ &=\text{15.2} \end{align} \)
Sample standard deviation :
\( \begin{align} \mathrm{s} &=\mathrm{ \sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n-1}}} \\ &=\sqrt{ \frac{(\text{25}-\text{15.2})^{2}+(\text{9}-\text{15.2})^{2}+.....+(\text{5}-\text{15.2})^{2}}{\text{5}-1} } \\ &=\sqrt{ \frac{ \text{272.8} } { \text{4} } } \\ &=\sqrt{\text{68.2}} \\ &= \text{8.3} \end{align} \)
The sample standard deviation is \(\text{8.3}\)
Steps to calculate population standard deviation :
Population mean :
\( \begin{align} \mathrm{\mu } &= \mathrm{\frac{\sum x}{n}} \\ &=\frac{\text{25}+\text{9}+......+\text{5}}{\text{5}} \\ &=\frac{\text{76}}{\text{5}} \\ &=\text{15.2} \end{align} \)
Population standard deviation :
\( \begin{align} \mathrm{\sigma} &=\mathrm{ \sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\mu )^{2}}{n}}} \\ &=\sqrt{ \frac{(\text{25}-\text{15.2})^{2}+(\text{9}-\text{15.2})^{2}+.....+(\text{5}-\text{15.2})^{2}}{\text{5}} } \\ &=\sqrt{ \frac{ \text{272.8} } { \text{5} } } \\ &=\sqrt{\text{54.56}} \\ &= \text{7.4} \end{align} \)
The population standard deviation is \(\text{7.4}\)
Excel standard deviation formula :
For sample standard deviation :
The Excel formula to calculate the sample standard deviation is
=STDEV.S( Numbers )
For population standard deviation :
The Excel formula to calculate the population standard deviation is
=STDEV.P( Numbers )
Note : Replace the numbers with your data set for e.g \(1,2,3,4,5\)
Why Use Our standard deviation Calculator?
- Accurate Results: Get precise calculations for mean, variance, and standard deviation.
- Easy to Use: User-friendly interface designed for quick and efficient calculations.
- Educational: Learn how the calculations are performed with our detailed step-by-step solutions.
- Versatile: Suitable for students, teachers, and professionals needing reliable statistical analysis.