Standard Deviation Calculator

How to calculate the Standard Deviation

1. Enter a set of numeric values

   To get started, enter a comma-separated set of numerical data and also select the type of calculation (Population SD or Sample SD).

2. Learn a step-by-step solution

   Next, click the "Calculate" button and you will immediately get the standard deviation result with a step-by-step solution and a graphic chart.

3. Save the calculation result

   Now you can save the result as a .pdf document, .png image, print it, or just copy it to the clipboard.

Standard Deviation and Variance

Standard is a fundamental math concept with varied applications in fields from finance to science and technology. A clear understanding of the calculation of both standard deviation and variance is critical for the formulation of effective statistical strategies.

What is Standard Deviation?

The Standard Deviation is a proportion of how data is spread out from the mean and SD is symbolized by sigma σ. The equation is simple: SD is arrived at by calculating the square root of the variance

Variance calculations rely on squares since it gauges outliers on a greater scale as compared to data closer to the mean. This estimation likewise keeps differences above the mean from cancelling those beneath, which can bring about a variance of zero.

Population standard deviations differ slightly from sample standard deviations. The difference is both qualitative and quantitative. The population standard deviation is a fixed parameter while the sample standard deviation is a statistic. The parametric SD is calculated from each deviation while the statistical SD is calculated from only some deviations in the population.

You can arrive at population standard deviation by calculating the square root of variance. If your group of numbers is more spread out from mean, the result is a higher standard deviation. The reverse is true.

1. First, add up all the data from and get the main
2. Subtract the mean from each data value
3. Square each of these deviations and find their sum
4. Divide the result by the total number of data points, n
5. The SD is the square root of the quotient

The sample standard deviation still shows how distributed data is from the mean. Except, sample SD calculation is a little different from the population standard deviation.

1. Add up all the data and get the mean
2. Calculate the difference between the mean and each of the data values
3. Square each of the differences and add them up
4. From your original number of data points, subtract 1 (n - 1)
5. Divide the result in step 4 by (n - 1)
6. The SD is the square root of the quotient in Step 5

Example calculation of the Standard Deviation

1. We have a data set of 20, 50, 60, 100
2. The mean is (20 + 50 + 60 + 100) / 4 = 57.5
3. Variance = (37.52 + 7.52 + (-2.5)² + (-42.5)²) / 4 = (1406.25 + 56.25 + 6.25 + 1806.25) / 4 = 3275/4 = 818.75
4. Population SD = √818.75 = 28.6138
5. Sample SD = √1091.6667 = 33.0404

What is the variance?

Variance measures how far each value is from the mean. You can arrive at variance by:

1. Subtracting each value in your data set from the mean
2. Squaring each of the differences in step 1 and adding them up
3. The variance is the answer you get in step 2 divided by the number of values in your data set

What is the mean?

Mean is the average of a data set and is arrived by adding up all the numbers in the data set and dividing the value by the number of total items in the set.

1. Subtracting each value in your data set from the mean
2. Squaring each of the differences in step 1 and adding them up
3. The variance is the answer you get in step 2 divided by the number of values in your data set