The statistics that actually matter

Five numbers describe most of what a set of returns is doing. The trap is that one of them is almost always the wrong one to use.

Centre: two different averages

The arithmetic mean adds and divides:

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$

The geometric mean multiplies and takes the root:

$g = \left[\prod_{i=1}^{n}(1 + x_i)\right]^{1/n} - 1$

For returns, these are not two ways of saying the same thing. Consider +50% then −50%:

• Arithmetic mean: $(50 - 50)/2 = 0\%$
• Geometric mean: $(1.5 \times 0.5)^{1/2} - 1 = -29.3\%$
• ₹1,00,000 becomes ₹1,50,000, then ₹75,000.

You lost a quarter of your money and the arithmetic mean says you broke even. The geometric mean is the only one that reconciles with the balance, because money compounds — the second year's return applies to what the first year left, not to what you started with.

So: the geometric mean is the return you got. The arithmetic mean is the return of a typical single year, which is a different and much less useful question. This is why SEBI's mutual fund disclosure rules require compounded annualised returns for periods beyond a year rather than a simple average.

The two are related. For returns that are not wildly dispersed,

$g \approx \bar{x} - \frac{\sigma^2}{2}$

The gap is proportional to variance. That is worth sitting with: volatility does not merely make the ride uncomfortable, it mechanically subtracts from what you keep. Two funds with the same arithmetic mean and different volatility do not end in the same place.

The median, and when to prefer it

The median is the middle value. It ignores how far away the extremes are, which is exactly why it is useful when they are extreme.

Reported fund returns, income distributions and company profits are all pulled around by a few large values. "The average Indian investor" computed as a mean is a number almost nobody resembles; the median is the person in the middle of the queue.

Rule of thumb: if mean and median differ noticeably, the distribution is not symmetric and quoting only the mean is hiding that.

Spread: variance and standard deviation

$\sigma^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$

$\sigma = \sqrt{\sigma^2}$

Squaring does two things: it makes deviations positive so they do not cancel, and it weights large deviations far more than small ones. Standard deviation then takes the square root to return to the units of the thing measured, so $\sigma$ is in percent when returns are.

Why $n-1$ and not $n$. The sample mean is itself estimated from the same data, which uses up one degree of freedom and makes deviations from it slightly too small on average. Dividing by $n-1$ corrects it. With 20 observations the difference is about 2.6%; with 2,000 it is nothing.

Annualising. Variance scales with time, so standard deviation scales with its square root:

$\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{252}$

using roughly 252 trading days. A daily $\sigma$ of 1% is about 15.9% annualised. The $\sqrt{t}$ rule assumes returns are independent across days — chapter 13 is about what happens when they are not.

Shape: skew and kurtosis

Skew measures asymmetry. Negative skew means the left tail is longer: many small gains and occasional large losses. Equity index returns are typically negatively skewed, which matters because the thing people check — the standard deviation — treats a 5% fall and a 5% rise identically.

Kurtosis measures how heavy the tails are relative to a normal distribution. Financial returns reliably have excess kurtosis: extreme days happen far more often than a normal distribution predicts. Chapter 6 is about exactly how badly that assumption fails and why it is still used.

Reading five numbers together

What you see	What it suggests
Mean ≫ median	A few large values are doing the work
Large $\sigma$, mean ≈ median	Volatile but symmetric
Negative skew	Losses arrive in bigger pieces than gains
High kurtosis	The rare case is less rare than the model thinks
Arithmetic ≫ geometric	Volatility is eating the compounded result

No single one of these is a verdict. Together they are a description.

The point

The arithmetic mean answers "what was a typical year"; the geometric mean answers "what did I end up with", and only the second reconciles with your balance. They differ by roughly half the variance, so volatility mechanically reduces compounded returns rather than merely making them bumpy. Standard deviation scales with the square root of time, and returns are reliably negatively skewed with fatter tails than a normal distribution allows.