Standard Deviation Calculator

Standard deviation definition

Standard deviation is a measure of the amount of variation or dispersion of a set of values. It tells us to what degree a set of numbers are dispersed around an average. The higher the standard deviation value, the more widely distributed the values are in the sample. Conversely, the lower the standard deviation value, the more tightly clustered these values are.

So, what is low standard deviation? And what’s the difference of low vs. high standard deviation? A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Population standard deviation and sample standard deviation

Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter sigma “σ”, for the population standard deviation, or the Latin letter “s”, for the sample standard deviation. Different formulas are used for calculating standard deviations depending on whether you have data from a whole population or a sample.

What is the population standard deviation?

The population standard deviation is the experimental standard deviation under infinitely many measurements, also known as the theoretical standard deviation. A small overall standard deviation “σ” value indicates that the measured values are relatively concentrated, and a large “σ” value indicates that the measured values are relatively scattered. Below is the population standard deviation formula:

\(σ\) = population standard deviation
\(N\) = the size of the population
\( x_i \)= each value from the population
\(μ\) = the population mean

What is the sample standard deviation?

The sample is a part of the individual observed or investigated, and the population is the whole of the research object. In a real scenario where standard deviation is applied, it is impractical to find the true standard deviation of an aggregate, except in some special cases. Therefore it is necessary to take a sample data from a large set of data, which is the sample standard deviation. Below is the sample standard deviation formula:

\( S \) = sample standard deviation
\( \sum \) = sum of…
\( X \) = each value
\( \bar{x} \)= sample mean
\( N \) = number of values in the sample

How to solve the standard deviation?

Standard deviation is usually calculated automatically by software used for statistical analysis, such as UpStudy standard deviation calculator. But you can also do the calculations by hand to better understand how the formula works.

Here are the steps to calculate the standard deviation:

Step 1: find the mean, add up all the scores, and divide them by the number of scores (click to learn how to calculate the mean).

Step 2: subtract the mean from each score to get the deviations from the mean, then square each deviation from the mean.

Step 3: find the sum of squares and the variance.

Step 4: the square root of the variance is the standard deviation.

Does it sound difficult? Actually not, now let's take a look at the following standard deviation examples to practice.