STANDARD DEVIATION

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. The standard deviation is usually denoted with the letter σ (lower case sigma). It is defined as the square root of the variance.

To understand standard deviation, keep in mind that variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared. Standard deviation, being the square root of that quantity, therefore measures the spread of data about the mean, measured in the same units as the data.

Stated more formally, the standard deviation is the root mean square (RMS) deviation of values from their arithmetic mean.

For example, in the population {4, 8}, the mean is 6 and the deviations from mean are {−2, 2}. Those deviations squared are {4, 4} the average of which (the variance) is 4. Therefore, the standard deviation is 2. In this case 100% of the values in the population are at one standard deviation from the mean.

The standard deviation is the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If many data points are close to the mean, then the standard deviation is small; if many data points are far from the mean, then the standard deviation is large. If all the data values are equal, then the standard deviation is zero.

A simple example

Suppose we wished to find the standard deviation of the set of the numbers 4 and 8.

Step 1: find the arithmetic mean (or average) of 4 and 8,

(4 + 8) / 2 = 6.

Step 2: find the deviation of each number from the mean,

4 − 6 = − 2

8 − 6 = 2.

Step 3: square each of the deviations (amplifying larger deviations and making negative values positive),

( − 2)² = 4

2² = 4.

Step 4: sum the obtained squares (as a first step to obtaining an average),

4 + 4 = 8.

Step 5: divide the sum by the number of values, which here is 2 (giving an average),

8 / 2 = 4.

Step 6: take the non-negative square root of the quotient (converting squared units back to regular units),

So, the standard deviation is 2.

 

For Normal Distributions

The empirical rule states that approximately 68 percent of the data in a normally distributed dataset is contained within one standard deviation of the mean, approximately 95 percent of the data is contained within 2 standard deviations, and approximately 99.7 percent of the data falls within 3 standard deviations.

As an example, the verbal or math portion of the SAT has a mean of 500 and a standard deviation of 100. This means that 68% of test-takers scored between 400 and 600, 95% of test takers scored between 300 and 700, and 99.7% of test-takers scored between 200 and 800 assuming a completely normal distribution (which isn't quite the case, but it makes a good approximation).