Q: Is it possible for a standard deviation to be negative?

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No. Standard deviation is the square root of a non-negative number (the variance) and as such has to be at least zero. Please see the related links for a definition of standard deviation and some examples.

it is possible to distribute standard deviation and mean but you dont have to understand how the mouse runs up the clock hicorky dickory dock.

If all four numbers are the same, there is no standard deviation. The mean will be equal to all 4 numbers, resulting in a 0 standard deviation. Ex) 5,5,5,5

When you don't have the population standard deviation, but do have the sample standard deviation. The Z score will be better to do as long as it is possible to do it.

The standard deviation is the standard deviation! Its calculation requires no assumption.

Related questions

Standard deviation can never be negative.

No. Neither the standard deviation nor the variance can ever be negative.

The mean would be negative, but standard deviation is always positive.

No. Standard deviation is the square root of a non-negative number (the variance) and as such has to be at least zero. Please see the related links for a definition of standard deviation and some examples.

A negative Z-Score corresponds to a negative standard deviation, i.e. an observation that is less than the mean, when the standard deviation is normalized so that the standard deviation is zero when the mean is zero.

There is a calculation error.

No. The standard deviation is not exactly a value but rather how far a score deviates from the mean.

Yes, a standard deviation can be less than one.

Yes.

it is possible to distribute standard deviation and mean but you dont have to understand how the mouse runs up the clock hicorky dickory dock.

If all four numbers are the same, there is no standard deviation. The mean will be equal to all 4 numbers, resulting in a 0 standard deviation. Ex) 5,5,5,5

No, a standard deviation or variance does not have a negative sign. The reason for this is that the deviations from the mean are squared in the formula. Deviations are squared to get rid of signs. In Absolute mean deviation, sum of the deviations is taken ignoring the signs, but there is no justification for doing so. (deviations are not squared here)