Question

If the standard deviation is larger than the mean we know:

a) the distribution of scores will be normally shaped
b) te distrubution of scores cannot be normally shaped
c) the distribution of scores is positively skewed
d) the distribution of scores is negatively skewed

Is it c?

Answer 1

u sure the question is correct?

Answer 2

it should be :

If the \(\color{red}{median}\) is larger than the mean we know:

Answer 3

could u check once ?

Answer 4

Nope, the question asks about the standard deviation but Im unsure of the correct answer to it

Answer 5

me too, @amistre64

Answer 6

@kirbykirby

Answer 7

Honestly I am not sure about this, unless there are implicit assumptions about the distributions you see in class?

For example, if you have a \(\text{GAM}(\alpha,\beta)\)distribution (where \(\alpha>0, \beta>0\)) , then the mean \(E(X)=\alpha\beta\) and \(Var(X)=\alpha\beta^2\implies StDev(X)=\sqrt{a}\beta\). Now, if you had \(\alpha=0.2\) and \(\beta=1\), then clearly the standard deviation is larger than the mean.

You can check the skewness of this distribution with the coefficient of skewness:
\[ \frac{E[(X-\mu)^3]}{\sigma^3}\], where \(\mu=E(X)\) and \(\sigma=StDev(X)\).

It can be shown (but I won't do it in detail) that \(E[(X-\mu)^3]=2\alpha\beta^3\), hence after simplication, the coefficient of skewness reduces to \(2\alpha^{-1/2}\), which is \(>0\) since \(\alpha >0\), so the distribution would be positively skewed.
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Now, you can still have a situation in which the standard deviation is larger than the mean. For example, you can have anormal distribution \(\text{N}(0,1)\) where the standard deviation = 1 > 0 = mean.
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Also, you can think of some more empirical distribution like this ordered data (like a time series plot):
{0,0,0,1,20,100,5}, which will have a negative-skew but the standard deviation = 97.127 > 31.5 = mean.

So from the above, I do not now how any of these choices will be true o_O.

Answer 8

In summary:

The example with the gamma distribution illustrates that you can have st. dev > mean with b) non-normal data, and c) positively skewed data

The N(0,1) example shows that you can have st. dev > mean with a) normal data

The last example shows that you can have st. dev > mean with d) negatively skewed data

Answer 9

yeah ... a normal 'z' distribution has sd(1) > mean(0)