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standard deviation (dict)
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Standard DeviationIn probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. Simply put, it measures how spread out the values in a data set are. The standard deviation is defined as the square root of the variance. This means it is the root mean square (RMS) deviation from the average. It is defined this way in order to give us a measure of dispersion that is (1) a non-negative number, and (2) has the same units as the data. A distinction is made between the standard deviation σ (sigma) of a whole population or of a random variable, and the standard deviation s of a subset-population sample. The formulae are given below. The term standard deviation was introduced to statistics by Karl Pearson (On the dissection of asymmetrical frequency curves, 1894). Interpretation and application The standard deviation is a measure of the degree of dispersion of the data from the mean value. Stated another way, the standard deviation is simply the "average" or "expected" variation around an average (i.e., square all individual deviations around the average, divide by 'N', then take the square root. You then have the 'root' of the mean squared deviation RMS: in a very simple sense the "average" or expected variation around the average). A large standard deviation indicates that the data points are far from the mean and a small standard deviation indicates that they are clustered closely around the mean. For example, the sets {0, 5, 9, 14}, {0, 0, 14, 14}, and {5, 6, 8, 9} each have an average of 7, but the third set has a much smaller standard deviation than the other two, because its values are all close to 7. Standard deviation may be thought of as a measure of uncertainty. In physical science for example, the standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then we consider the measurements as contradicting the prediction. This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct. See prediction interval. Definition and shortcut calculation of standard deviation Suppose we are given a population x1, ..., xN of values (which are real numbers). The arithmetic mean of this population is defined as - .
(see summation notation) and the standard deviation of this population is defined as - .
A slightly faster way to compute the same number is given by the formula -
- This is the standard deviation.
Rules for normally distributed data In practice, one often assumes that the data are from an approximately normally distributed population. If that assumption is justified, then about 68% of the values are at within 1 standard deviation away from the mean, about 95% of the values are within two standard deviations and about 99.7% lie within 3 standard deviations. This is known as the "68-95-99.7 rule". Relationship between standard deviation and mean The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is the "natural" measure of statistical dispersion if the center of the data is measured by the mean. The precise statement is the following: suppose x1, ..., xn are real numbers and define the function -
Using calculus, it is not difficult to show that σ(r) has a unique minimum for -
Geometric interpretation To gain some geometric insights, we will start with a population of three values, x1, x2, x3. This defines a point P = (x1, x2, x3) in R3. Consider the line L = {(r, r, r) : r in R}. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L. And that is indeed the case. Moving orthogonally from P to the line L, one hits the point -
whose coordinates are the mean of the values we started out with. A little algebra shows that the distance between P and R (which is the same as the distance between P and the line L) is given by σ√3. An analogous formula (with 3 replaced by N) is also valid for a population of N values; we then have to work in RN. Related articles External links
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