Standard Deviation Calculator
Get our easy to operate Standard Deviation Calculator to get the standard deviation, sum of squares, count, mean, and variance values of the given data set. Just you need to provide data set values separated by commas in the input of the calculator and hit on the calculate button to get the accurate output immediately.
Standard Deviation Calculator
Standard Deviation Calculator: Make use of this free calculator to check the multiple things at one place. Our Standard Deviation Calculator not only gives the exact result, it also provides the detailed steps engaged in solving those statistics parameters. This is the right place for the one who are looking for an equipment that computes standard deviation and other statistics parameters involved while solving them. For your convenience, we are also giving the step by step process in evaluating standard deviation, variance, sum of squares, mean and count of the data set.
Steps to Solve Standard Deviation
Find out the simple steps to calculate the standard deviation manually in the below segments. For the sake of your comfort and convenience, we listed the details below. Use them and make the calculations easily.
- Take sample or population data set.
- Find mean by calculating the sum of data values and dividing the result by number of data values.
- Calculate the difference between each data value in the set and mean.
- Square the result ad sum up all the values.
- Divide the obtained answer by total number of values to get the standard deviation.
Standard Deviation Formula
Sum:
Sum is defined as the result obtained after adding x1 + x2 + x3 + x4 . . xn.
Sum = ∑i = 1nxi
Size:
Size is defined as the number of values in the data set.
Count = n = count(xi)i = 1n
Mean:
Mean is the average of the data set. Its formula is
For population data set
μ = (∑i = 1nxi) / n = Sum / size
For sample data set
xmean = (∑i = 1nxi) / n
Sum of Squares:
Sum of squares is the sum of the squares of the difference between each value and mean of the data set.
For a Population
SS = ∑i = 1n (xi - μ)²
For a Sample
SS = ∑i = 1n (xi -xmean)²
Standard Deviation:
Standard deviation is square root of variance. The formula to calculate the standard deviation is listed here:
For a population
σ = √[(∑i = 1n (xi - μ)²) / n]
For a sample
s = √[(∑i = 1n (xi - xmean)²) / (n - 1)]
Variance:
Variance is the sum of squares per number of values in the data set or the square of standard deviation.
Variance formula for a a population is
σ² = [(∑i = 1n (xi - μ)²) / n]
For a sample is
s² = [(∑i = 1n (xi - xmean)²) / (n - 1)]
Example
Question: Find standard deviation, variance, sum of squares and mean of the following population data set. Data set is {9, 12, 3, 4, 6, 7, 8, 18}
Solution:
Given data set is {9, 12, 3, 4, 6, 7, 8, 18}
Sum = ∑i = 1nxi
= (9 + 12 + 3 + 4 + 6 + 7 + 8 + 18)
= 67
Size = count(xi)i = 1n
= 8
Mean = μ = (∑i = 1nxi) / n = Sum / size
= (9 + 12 + 3 + 4 + 6 + 7 + 8 + 18) / 8
= 67 / 8
= 8.375
Sum of Squares = SS = ∑i = 1n (xi - μ)²
= (9 - 8.375)² + (12 - 8.375)² + (3 - 8.375)² + (4 - 8.375)² + (6 - 8.375)² + (7 - 8.375)² + (8 - 8.375)² + (18 - 8.375)²
= 0.625² + 3.625² + (-5.375)² + (-4.375)² + (-2.375)² + (-1.375)² + (-0.375)² + 9.625²
= 0.39 + 13.14 + 28.89 + 19.14 + 5.64 + 0.14 + 92.64 + 1.89
= 161.87
Standard Deviation = σ = √[(∑i = 1n (xi - μ)²) / n]
= √[[(9 - 8.375)² + (12 - 8.375)² + (3 - 8.375)² + (4 - 8.375)² + (6 - 8.375)² + (7 - 8.375)² + (8 - 8.375)² + (18 - 8.375)²] / 8]
= √[(0.625² + 3.625² + (-5.375)² + (-4.375)² + (-2.375)² + (-1.375)² + (-0.375)² + 9.625²) / 8]
= √[[0.39 + 13.14 + 28.89 + 19.14 + 5.64 + 0.14 + 92.64 + 1.89 / 8]
= √[161.87 / 8]
= √[20.23]
= 4.49
Variance = σ² = [(∑i = 1n (xi - μ)²) / n]
= [(9 - 8.375)² + (12 - 8.375)² + (3 - 8.375)² + (4 - 8.375)² + (6 - 8.375)² + (7 - 8.375)² + (8 - 8.375)² + (18 - 8.375)²] / 8]
= (0.625² + 3.625² + (-5.375)² + (-4.375)² + (-2.375)² + (-1.375)² + (-0.375)² + 9.625²) / 8
= 0.39 + 13.14 + 28.89 + 19.14 + 5.64 + 0.14 + 92.64 + 1.89 / 8
= 161.87 / 8
= 20.23
∴ Mean μ = 8.375, Standard Deviation σ = 4.49, Variance σ² = 20.23, Sum of Squares SS = 161.87, Sum = 67.
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FAQs on Standard Deviation Calculator
1. What is meant by standard deviation?
Standard deviation is a measure of how spread out numbers are. It is represented using σ symbol. Square root of average of the squared differences from the mean is called standard deviation.
2. What is the difference between standard deviation and variance?
Standard deviation can be defined as the observations that get measured through dispersion within a data set. It is the perfect indicator of the observations in a data set. Variance can be defined as the numerical value, which describes how variable the observations are. It is the perfect indicator of the individuals spread out in a group. Standard deviation is nothing but the root of the mean square deviation. Variance is defined as the average taken out of the squared deviations.
3. What is standard deviation formula?
The formula to calculate standard deviation σ = √[(∑i = 1n (xi - xmean)²) / n].
4. Find the standard deviation of 4, 9, 11, 12?
Mean = (4 + 9 + 11 + 12) / 4 = 9
σ = √[((4-9)² + (9-9)² + (11-9)² + (12-9)²) / 4]
= √[(25 + 0 + 4 + 9) / 4]
= √[38/4] = √9.5
= 3.082