Suppose I have a set of numbers, for example - exam scores of 20 students.
Can there be more than half of students scoring above the mean ? similarly, can more than half the class score above the median and mode ?
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Sign up to join this communitySuppose I have a set of numbers, for example - exam scores of 20 students.
Can there be more than half of students scoring above the mean ? similarly, can more than half the class score above the median and mode ?
It is not possible for over half of the students to score above the median:
Notice that I said "at most" above: when multiple students score at the median, the number of students scoring over the median must be even less than the $\frac{n}{2}$ or $\frac{n-1}{2}$ figures I gave above.
But is is possible for more than half the students to score above the mean: this may happen when the distribution is negatively-skewed, like in image (a) below, and the mean is below the median.
And it is possible for more than half the students to score above the mode: this may happen when the distribution is positively-skewed, like in image (c) below, and the mode is below the median.
(For the avoidance of doubt, the perfectly symmetrical distribution in image (b) above is actually a bit of a special case, where the mean exists (unlike the Cauchy distribution) and there is a single mode. If those conditions do not hold, we cannot state that the mean, median & mode coincide, so being perfectly symmetrical is neither necessary nor sufficient to do so. One famous example of such a symmetric distribution where those conditions do hold is the Normal/Gaussian ("bell curve") distribution.)