In financial analysis, a return distribution is commonly described by its expected return and standard deviation. For example, the S&P 500 Index might have an expected return of 10 percent and a standard deviation of 15 percent. By assuming that the returns of the S&P 500 Index conform to a particular distribution, such as a normal distribution, we can infer the entire distribution of returns from the expected return and standard deviation. The expected value of a distribution is referred to as the first moment of the distribution and is measured by the arithmetic mean of the returns. The variance, which equals the standard deviation squared, is called the second central moment or the second moment about the mean. It measures the dispersion of the observations around the mean. The first central moment is not the mean itself, but rather zero, because central moments are measured relative to the mean.
The normal distribution is symmetric around the mean; hence, the median (the middle value of the distribution) and the mode (the most common value of the distribution) are both equal to the mean. Moreover, the normal distribution has a standard degree of peakedness. These properties of the normal distribution explain why just the mean and standard deviation are sufficient to estimate the entire distribution.
Although investment returns usually are assumed to be approximately normally distributed, this assumption is less likely to hold for very short horizons, such as one day, and for long horizons, such as several years. Moreover, certain assets and investment strategies have properties that produce nonnormal distributions over any horizon. Thus, in some cases, to estimate a return distribution, one must go beyond the first moment and the second central moment to the third central moment, which is called skewness, or the fourth central moment, which is called kurtosis.