How can I prove that the standard deviation of an asymmetric function (mean>median>mode) is larger than the standard deviation of a symmetric one (mean=median=mode) of the same variable?

I was asked if the statement "the standard deviation of an asymmetric function (mean>median>mode) is larger than the standard deviation of a symmetric one (mean=median=mode) of the same variable?" true or false and asked to prove the answer.

I was asked if the statement "the standard deviation of an asymmetric function (mean>median>mode) is larger than the standard deviation of a symmetric one (mean=median=mode) of the same variable?" true or false and asked to prove the answer.

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