# What does it mean if the standard deviation is greater than the mean?

The standard deviation and the arithmetic mean measure two different characteristics of a set of data. The standard deviation measures how spread out the data is, whereas the arithmetic mean measures where the data is centered. Because of this, there is no particular relation that must be satisfied because the standard deviation is greater than the mean.

*Actually, there IS a relationship between the mean and
standard deviation. A high (large) standard deviation indicates a
wide range of scores = a great deal of variance. Generally
speaking, the greater the range of scores, the less representative
the mean becomes (if we are using "mean" to indicate "normal"). For
example, consider the following example:*

*10 students are given a test that is worth 100 points. Only 1
student gets a 100, 2 students receive a zero, and the remaining 7
students get a score of 50.*

*(Arithmetic mean) = 100 + 0(2) + 7(50) = 100 + 0 + 350 =
450/10 students*

**SCORE = 45**

*In statistics, the median refers to the value at the 50%
percentile. That means that half of the scores fall below the
median & the other half are above the median. Using the example
above, the scores are: 0, 0, 50, 50, (50, 50), 50, 50, 50, 100. The
median is the score that has the same number of occurrences above
it and below it. For an odd number of scores, there is exactly one
in the middle, and that would be the median. Using this example, we
have an even number of scores, so the "middle 2" scores are
averaged for the median value. These "middle" scores are bracketed
by parenthesis in the list, and in this case are both equal to 50
(which average to 50, so the median is 50). In this case, the
standard deviation of these scores is 26.9, which indicates a
fairly wide "spread" of the numbers. For a "normal" distribution,
most of the scores should center around the same value (in this
case 50, which is also known as the "mode" - or the score that
occurs most frequently)* *& as you move towards the
extremes (very high or very low values), there should be fewer
scores.*