Best Sample and Population standard deviation calculator - With explanations

Standard deviation calculator

Enter all the numbers separated by comma ,

E.g. 23,55,77,89,100 (no comma at the end)

Statistics in not magic, click on the question mark before each result you get after pressing the calculate button. We explain how we did it

Total Numbers:

This is the total number of elements in your dataset. If you label your dataset from `x_1` to `x_n` such that `x_1` is the first element, `x_2` is the second element, `x_3` is the third element all the way to `x_n`, where `x_n` is the last element on the list, the total number of elements on your list is give by n. In this case,

Sample Mean:

This is the average of the numbers in your dataset. It is given by the formula

Mean,`barX=(sum_(i=1)^nx_i)/n`

This implies you sum all the elements of your data form `x_1` to `x_n` and divide by the total number of elements, n

In this case,

Sample Standard deviation:

This is the measure of the spread of squares within a set of data. It is given by the formula:

Sample Standard deviation = `S=root_((sum_(i=1)^n(x_i-barx)^2)/(n-1))`

This can be broken down to 4 steps

STEP 1: Calculate the mean, `barx`, as we did above.

STEP 2: Subtract the mean from each element in the data and square the difference

STEP 3: Add the squared differences and divide the total obtained by the total number of elements minus 1, that is, (n-1).

STEP 4: Obtain the square root of the result obtained in step 3. This will be your standard deviation.

For the data you provided above,

Sample Variance

Variance(S^{2}) is the average of the squared differences from the mean of the sample.

It is denoted as follows:

`S^2=(sum_(i=1)^n(x_i-barx)^2)/(n-1)`

From the data you have provided,

From this formula, you can see the Standard deviation is the square root of sample variance

Therefore, we actually obtained the sample variance when calculating the sample standard deviation above. The answer we obtained in STEP 3 above was the sample variance.

Population Variance:

Population variance (`sigma^2`) is the average of the squared differences from the mean of the data provided. It is similar to the sample variance but this time, we divided the cumulative difference of squares by n and not n-1. *If the question does not state whether it is sample variance or population variance we calculate population variance.

The formula is denoted as:

`sigma^2=(sum_(i=1)^n(x_i-barx)^2)/n`

From the data you have provided above,

Population Standard deviation:

Population standard deviation (`sigma`) is the squareroot of the population variance we obtained above