What is Standard deviation?
A quantity expressing the size of the spread of data or of a random variable, often written σ (sigma). The variance is the average of the squared differences between each value and the mean, and the standard deviation is its positive square root. It has the same unit as the original data, so the width of the spread can be expressed directly, as in the mean ±1σ or ±3σ. It is widely used for control limits in process control, process capability indices, evaluating measurement errors and calculations with the normal distribution.
Japanese name: 標準偏差
Standard deviation appears in 14 practice questions across 5 topics, including Probability, Statistics and Queueing, Yield and quality control, Ion implantation.
Field: Probability, Statistics and Queueing
How to calculate it
With mean μ, the variance is the average of (x − μ)², and the standard deviation is its square root. For 2, 4, 4, 4, 5, 5, 7, 9, for example, the mean is 5, the sum of the squared differences is 32, the variance is 32 ÷ 8 = 4 and the standard deviation is 2. The variance has the square of the unit of the data, so the standard deviation is used to show the width of the spread in the same unit as the data.
Dividing by n or by n − 1
To describe the spread of the data at hand, the sum of the squared differences is divided by the number of values n. To estimate the variance of a population from a sample, the unbiased variance, divided by n − 1, is generally used. Because the sample mean is the value closest to the sample data, dividing by n slightly underestimates the population variance. Spreadsheet software has functions for both, so check which one is being used.
Where it is used
In process control, the control limits of a control chart are drawn at the centerline ±3σ, and the process capability index Cp compares the specification width with 6σ. For normally distributed data, about 68% lies within the mean ±1σ, about 95% within ±2σ and about 99.7% within ±3σ. The standard error, which describes the spread of the sample mean, is σ/√n, the population standard deviation divided by the square root of the sample size.
FAQ
When is the variance used and when the standard deviation?
The standard deviation is used to express and compare the width of the spread in the same unit as the data. The variance is convenient during calculations: the variance of a sum of independent quantities is the sum of their variances. Results are usually reported after taking the square root to convert back to the standard deviation.
What happens to the standard deviation when two independent sources of variation are combined?
If they are independent, their variances add, so the combined standard deviation is √(σ₁² + σ₂²). For example, independent variations with standard deviations of 3 nm and 4 nm combine to a standard deviation of 5 nm, smaller than the simple sum of 7 nm. This idea is used for tolerance stack-up and combining errors; it does not hold when the two variations are correlated.