Standard deviation calculator
This calculator computes standard deviation, variance, skewness and kurtosis of the given dataset. The calculator shows a step-by-step explanation on how to find these statistics.
What a is standard deviation?
Definition:
The standard deviation measures how close the set of data is to the mean value of the data set.
If the dataset has a high standard deviation, the values are widely distributed. If the dataset has a low standard deviation, the data points are quite near to the mean.
When the data is normally distributed, roughly 66 percent of the data points fall within one standard deviation of the mean.
How to calculate the standard deviation?
The Standard deviation formula is:
$$ \sigma = \sqrt{\dfrac{1}{N} \sum_{i=1}^N{\left(x_i - \overline{X} \right)} } $$Examples
Example 01:
We will show how to find the standard deviation using a simple dataset. Take, for example, 1, 4, 5, 6. The process can be separated into four steps.
Step 1: Find the mean of the dataset:
mean = (1 + 4 + 5 + 6)/4 = 16/4 = 4
Step 2: Find the square of the difference between the mean and each data point:
(1 - 4)2 = 9
(4 - 4)2 = 0
(5 - 4)2 = 1
(6 - 4)2 = 4
Step 3: Sum all squares
9 + 0 + 1 + 4 = 14
Step 4: Apply the standard deviation formula.
$$ \sigma = \sqrt{ \frac{ \sum{\left(x - X\right)} }{ n } } = \sqrt{ \frac{14}{4}} = 1.8708 $$Example 02:
Find the standard deviation of the following dataset: 2, 5, 10, 11, 12.
In this example:
x1=2, x2=5, x3=10, x4=11, x5=12,
N = 5
X = 8
After applying standard deviation formula we get:
σ = 4.3012
Sample vs whole population
It is important to note that the above formula only works if data is collected from the entire population, which is most often not the case.
If we have data from a subset of a population, then we can estimate the standard deviation for the whole population.
The estimation formula is
$$ \sigma = \sqrt{\dfrac{1}{N-1} \sum_{i=1}^N{\left(x_i - \overline{X} \right)} } $$Note that in estimation formula, N was replaced with N-1.