The average, also known as the arithmetic average or the mean of a set of values is a nice and simple number that is easy to calculate and interpret. However, it not valid under all circumstances and must be used with caution.

Given a set of numbers, we cannot blindly calculate the average – take the sum of the numbers and divided by the number of items in the set. I am guilty of doing this too.

## What does the average represent?

Consider a set of students and the marks they have got. What does the average mark represent? The typical performance of the class? Yes, if most people got a mark close to the average? Then we could talk about *above-average* and *below-average*?

Yes, but under certain circumstances.

## The bell curve

Most of us would have encountered the bell curve at some stage in life – eg. performance based increments. It is a probability distribution that has a peak and is symmetric on either side of the peak.

The peak represents the *average* and the fatness of the curve or the spread the *standard deviation*. These two quantities are derived from the probability distribution.

Yes, the arithmetic mean is a derived quantity. No one just came up with its formula.

The point is that the arithmetic mean is valid only for such symmetric distribution. So is the idea of standard distribution.

## Mutual Fund Star Ratings are flawed!

Mutual fund risk and returns (or any security for that matter) *are not normal*. This automatically means quantities like alpha, beta, sharpe, Sortino etc are not valid. And in turn, means star ratings which use these metrics are not valid.

## Mutual Fund Category Average

To elaborate the above point, consider the 5-year return distribution of multi-cap funds.

It should be immediately obvious that the distribution is not symmetric. Therefore a typical or average return is not of much use as it does not divide the distribution equally.

*In fact, one should not even calculate the average in such a case.*

Instead of the average one should use the median. This is a quantity that divides the distribution in half – 50% above the median and 50% below. For a bell curve, the average = median.

The median is also known as the second quartile. Any return above the median return falls in the top 50%.

One can also use the third quartile – top 75% of returns.

The whole point of this post is to suggest using the median and/or3rd quartile instead of the mean. The mean is “typically” invalid.

As mentioned above, I am guilty of using the category average returns too. I have now updated the **mutual fund screener** with median return instead of the mean.

Well described. Using average has serious flaws in MF returns. It’s like saying if one leg is in the refrigerator and another is in the furnace, on an average a person is comfortable!

Yes … it can paint a picture which is quite different from reality and thus can very easily mislead.