1.7:
Standard Deviation of Calculated Results
A plot of relative deviation from the mean and its frequency of occurrence appears as a Gaussian curve. This probability distribution curve of a population can be described by the population mean and standard deviation.
The standard deviation measures the spread of clustered data about the mean.
When the standard deviation is lower, the distribution about the mean is narrower, leading to high precision.
For a finite number of measurements – the population subset – the mean and standard deviation are denoted as the sample mean and sample standard deviation, respectively.
A set of N measurements can have N deviations from a reference point – true mean, which means N degrees of freedom. However, if the estimated mean is set as a reference point, only N-1 independent deviations are required, as the last one is predetermined.
The standard deviation expressed as a fraction of the mean is called the relative standard deviation. When given as a percentage, it is the coefficient of variation. Variance is the square of standard deviation.
1.7:
Standard Deviation of Calculated Results
Standard deviation measures the spread of data around the mean value. Many large data sets follow a Gaussian distribution, also known as a normal distribution. This distribution is bell-shaped curved, with the most frequently observed value (mean or central value) in the middle. The farther away from the central value, the greater the deviation from the central value, and the lower the frequency.
A broad Gaussian distribution curve has a wider standard deviation, representing a data set with lower precision. For data with high precision, the magnitude of the standard deviation is small, and the width of the distribution curve is narrow.
The mean and standard deviation of the entire data or population are known as the population mean and population standard deviation, respectively. The standard deviation for a population subset is the sample standard deviation, and the estimated mean of the subset is known as the sample mean. A useful way to express the standard deviation is the 'relative standard deviation', which expresses the standard deviation as a fraction of the mean. This quantity is also known as the coefficient of variation and is often expressed as a percentage.