Standard Deviation Calculator

Please enter numbers separated by commas to compute standard deviation, variance, mean, total, and margin of error.

In statistics, the standard deviation (σ) is a measure of variance or dispersion between values in a set of data. A reduced standard deviation indicates that the data points are closer to the mean (or expected value) (μ). In contrast, a higher standard deviation indicates a broader range of values. Standard deviation, like other mathematical and statistical ideas, can be applied to a wide range of situations, resulting in a wide variety of equations. In addition to representing population variability, the standard deviation is commonly employed to calculate statistical outcomes like the margin of error. When used in this context, standard deviation is commonly referred to as the standard error of the mean or the standard error of an estimate with respect to a mean. The calculator above calculates population standard deviation, sample standard deviation, and confidence interval approximations.

Population Standard Deviation

The population standard deviation (σ) is the square root of a given data set’s variation and is used to measure a whole population. When every member of a population can be sampled, the following equation can be used to calculate the whole population’s standard deviation:

Where

xi is an individual value

μ is the mean/expected value

N is the total number of values

For those inexperienced with summation notation, the equation above may appear daunting, but when broken down into separate components, it is not extremely hard. The i=1 in the summing represents the initial index; for the data set 1, 3, 4, 7, 8, i=1 is 1, i=2 is 3, and so on. The summation notation simply means to perform the operation of (xi – μ)2 on each value through N, which in this example is 5 due to the number of values in the data collection.

EX:           μ = (1+3+4+7+8) / 5 = 4.6        

σ = √[(1 – 4.6)2 + (3 – 4.6)2 + … + (8 – 4.6)2)]/5

σ = √(12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577

Sample Standard Deviation

In many circumstances, it is not possible to sample every member of a community, hence the above equation must be changed so that the standard deviation can be computed using a random sample of the population under study. The sample standard deviation (s) is an often used estimator for σ.It is worth mentioning that there are numerous formulae for computing sample standard deviation since, unlike sample mean, there is no one estimator that is unbiased, efficient, and has the highest likelihood.  The equation shown below is the “corrected sample standard deviation.” It is a corrected version of the equation generated by altering the population standard deviation equation to use the sample size as the population size, so removing some of the equation’s bias. Unbiased estimation of standard deviation, on the other hand, is complex and varies with the distribution.  As a result, the “corrected sample standard deviation” is the most often used estimator for population standard deviation, and it is often simply referred to as the “sample standard deviation.” The adjusted estimate outperforms the uncorrected version, but has a considerable bias for small sample sizes (N<10).

Where

xi is one sample value

x̄ is the sample mean

N is the sample size

The “Population Standard Deviation” section provides an example of how to work with summations. The equation is nearly identical, with the exception of the N-1 term in the corrected sample deviation equation and the usage of sample values.

Applications of Standard Deviation

Standard deviation is commonly used in experimental and industrial contexts to compare models to real-world data. One example of this in industrial applications is quality control for certain items. Standard deviation can be used to compute the minimum and maximum values within which some feature of the product should fall a significant proportion of the time. If values fall beyond the calculated range, adjustments to the production process may be required to maintain quality control.

The standard deviation is also used in meteorology to identify regional climate variances.  Consider two cities, one on the coast and one deep inland, with the same average temperature of 75°F. While this may lead to the conclusion that the temperatures in these two cities are nearly identical, the reality may be obscured if only the mean is considered and the standard deviation neglected. Coastal cities have far more stable temperatures due to regulation by large bodies of water, as water has a higher heat capacity than land; essentially, this makes water far less susceptible to temperature changes, and coastal areas remain warmer in winter and cooler in summer due to the amount of energy required to change the temperature of the water. As a result, a coastal city may have temperature ranges of 60°F to 85°F during a particular period of time, resulting in a mean of 75°F, whereas an inland city may have temps ranging from 30°F to 110°F, resulting in the same mean.

Standard deviation is also widely used in finance to assess the risk associated with price changes of an asset or portfolio of assets.  In these circumstances, standard deviation is used to quantify the uncertainty of future investment returns. For example, when comparing stock A, which has an average return of 7% and a standard deviation of 10%, to stock B, which has the same average return but a standard deviation of 50%, the first stock is clearly the safer option because stock B’s standard deviation is significantly higher for the same return. That is not to conclude that stock A is clearly a better investment option in this case, because standard deviation might bias the mean in either direction. While Stock A is more likely to have an average return of around 7%, Stock B has the potential for a substantially bigger return (or loss).

These are just a few examples of how to use standard deviation; there are many more. In general, calculating standard deviation is useful when determining how distant from the mean a typical number from a distribution can be.