Skewness

# Problems with skewness

Skewness is a counter-intuitive concept. This is so because what catches our attention is not so much the "tail" of the distribution but the bulk of it. For example, a skewed left distribution is that where the data is piled up towards the right of the distribution, while a skewed right distribution is one were the data is piled up towards the left of the distribution. In time, it is easy to forget any mnemonics (such as that skewness is the "tail" of the distribution), thus confusing skewness with the pile up of the data and misinterpreting skewness.

This counter-intuition grows bigger if skewness is provided in numerical form, as positive numbers, which increase as we move towards the right side of the 'X' axis, identify a distribution skewed left, while negative numbers, which increase as we move towards the left side of the 'X' axis, identify a distribution skewed right.

A way of helping resolve this counter-intuition is to gain an intuitive understanding of skewness, as follows:

• A perfectly symmetrical distribution will have no skewness whatsoever (skewness = 0). Such is the case of the normal distribution. And one thing that we know about a normal distribution is that its median and mean (as well as its mode) are the same. From this property we can derive a 'postulate': the ideal mean is in the middle when the distribution has no skewness.
• A distribution skewed right has a longer tail towards the right and the bulk of data towards the left. Because of the greater influence of the tail on the mean, what such distribution does is to "pull" the mean towards the right of where it should be were it normal, ie, towards the tail. In most circumstances, this also implies that the mean is pulled towards the right while the median stays within the bulk of the distribution; thus, we should observe a mean greater than the median. (However, bear in mind that this is not always the case - see, for example here.) Thus, we could say that the mean is "skewed" right of its ideal location were it normal (and, most of the time, the mean is "skewed" right of the median).
• Equally, a positive skewness (eg, skewness = +1.96) also benefits from this understanding, as a higher positive skewness moves towards the right of the 'X' axis, describing a mean that is also "separating" from its 'normal' location (and, most of the time, from the median) in that same direction, which happens when the tail grows longer towards the right.
• Meanwhile, a distribution skewed left has a longer tail towards the left and the bulk of data towards the right. Because of the greater influence of the tail on the mean, what such distribution does is to "pull" the mean towards the left of where it should be were it normal, ie, towards the tail. In most circumstances, this also implies that the mean is pulled towards the left while the median stays within the bulk of the distribution; thus, we should observe a mean smaller than the median. (However, bear in mind that this is not always the case.) Thus, the mean is "skewed" left of its ideal location were it normal (and, most of the time, the mean is "skewed" left of the median).
• Equally, a negative skewness (eg, skewness = -2.15) also benefits from this understanding, as a higher negative skewness moves towards the left of the 'X' axis, describing a mean that is also "separating" from its 'normal' location (and, most of the time, from the median) in that same direction, which happens when the tail grows longer towards the left.

The intuitive understanding is, thus, to associate skewness with where the mean sits in relation to the ideal 'normal' mean (and, most of the time, the median), rather than associating it with the tails of the distribution or thinking about it as the inverse of what our intuition tell us.

References
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