Standard deviation is a quantity used to indicate the average deviation of a group from the mean. In other words, how close are scores to their mean (on average). If this seems confusing, consider the following example. Let there be a set of three test scores \({70, 75, 80}\). It can be easily calculated that this set has a mean of \(75\). The next question is, how close are the scores in this set to that mean? This is what standard deviation lets us know. Standard deviation is calculated using the following formula:
$$ \sigma = \sqrt\frac{\sum (x - \mu)^2}{N} $$
Where \(x\) is the value or score at issue, \(\mu\) is the average, and \(N\) is the size of the data set. The reason that standard deviation squares values and then takes the square root of the final result is that it gives more weightage to extreme values than absolute mean deviation.
The mean is
The standard deviation is
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