Title: Uncertainty, error and confidence in data
Headings: Uncertainty, error and confidence in data
Uncertainty, error and confidence in data
Accuracy, precision and error
Precision and Imprecision
Error
Uncertainty
Estimating the experimental uncertainty for a single measurement
Estimating the experimental uncertainty for a repeated measure (Standard deviation)
Standard deviation of the means (Standard error of mean (SEM))
Standard error of mean (SEM) versus standard deviation
Confidence levels
Glossary
Works Cited

Content: If measured values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number of measurements. Standard error is often used to test (in terms of null hypothesis testing) differences between means. For example two populations of salmon fed on two different diets may be considered significantly different if the 95% confidence intervals (2 std errors) around the estimated fish sizes under Diet A do not cross the estimated mean fish size under Diet B.
Note that the standard error of the mean depends on the sample size, the standard error of the mean shrink to 0 as sample size increases to infinity. Standard error of mean (SEM) versus standard deviation
In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases. Confidence levels
The confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value of the unknown population parameter. ...