# Statistical Foundations: Mean and Standard Deviation

Many financial calculations and estimations require a statistical analysis of the variability of past market returns. Because we have strong evidence that market returns are approximately normally distributed, we can estimate potential market movements with a given probability using two simple parameters: mean (or average return) and standard deviation (or volatility).

**Mean**

Mean describes where the distribution is centered, and standard deviation describes dispersion from the mean, or uncertainty (for example, the width of the distribution). Conveniently, we only need to estimate the standard deviation for short-term risk prediction, because mean returns tend to be close to 0 (zero).

**Standard Deviation**

From a risk perspective, the most important variable that defines the normal distribution is the standard deviation, because it measures uncertainty. The larger the standard deviation, the larger the uncertainty, or risk.

For example, we could say that an asset with a standard deviation of 10% is twice as risky as an asset with a standard deviation of 5%.

Multiples of standard deviation can be used to estimate lower tail probabilities of loss. Lower tail probabilities of loss refer to the chance of losses exceeding a specified amount. For instance, there is a 16% chance of losses exceeding 1 standard deviation from the mean (we know this because 1 standard deviation included 68% of returns to either side of the mean of a normal distribution, leaving 32%/2 = 16% probability in each tail). Because returns tend to cluster around the mean, larger standard deviation moves have a lower probability of occurring.

We use confidence level multipliers to arrive at the tail probability of loss levels and implied VaR confidence levels (Confidence levels are 1- lower tail probability of loss)

Confidence Level Multiplier | Lower Tail Probability of Loss | Implied VaR Confidence Level |

1 | 16% | 84% |

1.28 | 10 | 90 |

1.65 | 5 | 95 |

2 | 2.5 | 97.5 |

2.33 | 1 | 99 |

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