Finding standard deviation was never so simple, thanks to the sample standard deviation calculator. The calculations offered by sample mean calculator are:
- Sample Variance
- Sample Standard Deviation
- Population Variance
- Population Standard Deviation
- Differences, and
- Square of Differences
Let’s explore a few key points such as standard deviation definition, how to find standard deviation without using relative standard deviation calculator, and standard deviation equation.
What is standard deviation?
A measure of extent to which numbers are spread out.
Standard Deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is denoted by the Greek symbol sigma σ.
Below, you can find the plot of normal distribution with width of 1 band.
Standard Deviation Formula
Standard deviation formula can be expressed by taking the square root of the variance.
Sample Standard Deviation Formula
- s refers to sample standard deviation
- N is the number of observations
- x_{i} is the observed values of sample item, and
- x̄ is the mean value of the sample
Population Standard Deviation Formula
- σ refers to population standard deviation
- N is the size of population
- x_{i} is the observed values from population, and
- μ is population mean
How To Calculate Standard Deviation?
Standard Deviation Calculation can be carried out using mean and standard deviation calculator above. However, we will explain the method to calculate SD with an example
Example:
Find the standard deviation of the given sample:
30, 20, 28, 24, 11, 17
Step 1: Calculate the mean value of sample data:
N = 6
$\frac{∑X}{N} = \frac{(30+20+28+24+11+17)}{6} = \frac{130}{6} = 21.67$
Step 2: Calculate (x_{i} - x̄) by subtracting mean value from each value of data set.
x_{1} - x̅ = 30 - 21.67 = 8.33
x_{2} - x̅ = 20 - 21.67 = -1.67
x_{3} - x̅ = 28 - 21.67 = 6.33
x_{4} - x̅ = 24 - 21.67 = 2.33
x_{5} - x̅ = 11 - 21.67 = -10.67
x_{6} - x̅ = 17 - 21.67 = -4.67
Step 3: Now, take the square of (x_{i} - x̄) for each value to find (x_{i} - x̄)^{2}.
(x_{1} - x̅)^{2} = 8.33^{2} = 69.4
(x_{2} - x̅)^{2} = -1.67^{2} = 2.78
(x_{3} - x̅)^{2} = 6.33^{2} = 40
(x_{4} - x̅)^{2} = 2.33^{2} = 5.43
(x_{5} - x̅)^{2} = -10.67^{2} = 113.85
(x_{6} - x̅)^{2} = -4.67^{2} = 21.80
Step 4: Get the sum of all values for (x_{i} - x̄)^{2}.
(x_{i} - x̅)^{2} |
69.4 |
2.78 |
40 |
5.43 |
113.85 |
21.80 |
∑ (x_{i} - x̅)^{2} = 253.26 |
Step 5: Divide ∑ (x_{i} - x̅)^{2} with (N-1).
$\frac{∑(x_i - \overline x)^2 }{N - 1} = \frac{253.26}{5} = 50.65$
Variance = 50.65
Step 6: Take the square root of $\frac{∑(x_i - \overline x)^2 }{N - 1}$ to get the standard deviation.
$\sqrt{\frac{∑(x_i - \overline x)^2 }{N}} = \frac{(30+20+28+24+11+17)}{6} = \frac{130}{6} = 21.67$
s = 7.11
Use the population standard deviation calculator above to cross-check the values for SD calculations.