What are moments in statistics?

In statistics, moments are quantitative measures that describe the shape and dispersion of a probability distribution. They are calculated by taking the expected value of powers of the deviation from the mean of a random variable. Moments can be used to characterize a distribution in terms of its central tendency, spread, and shape.

Why use moments?

Moments are used to summarize and understand the characteristics of a probability distribution. They provide a concise way to describe the distribution without having to consider all the individual data points. Moments are also useful for comparing different distributions and for making inferences about the underlying population.

Most commonly used moments

The four most commonly used moments are:

• First moment: Mean: The mean is the most common measure of central tendency. It is the average of all the values in a dataset.
• Second moment: Variance: The variance is a measure of how spread out the values in a dataset are. It is the average squared deviation from the mean.
• Third moment: Skewness: Skewness is a measure of the asymmetry of a distribution. A positive skewness indicates that the distribution is skewed to the right, while a negative skewness indicates that the distribution is skewed to the left.
• Fourth moment: Kurtosis: Kurtosis is a measure of the peakedness of a distribution. A high kurtosis indicates that the distribution is peaked and has heavy tails, while a low kurtosis indicates that the distribution is flat and has light tails.

Types of distribution – Kurtosis

There are four main types of distributions based on kurtosis:

• Mesokurtic: These distributions have a kurtosis of around 3, which is the kurtosis of a normal distribution.
• Leptokurtic: These distributions have a kurtosis greater than 3, which indicates that they are peaked and have heavy tails.
• Platykurtic: These distributions have a kurtosis less than 3, which indicates that they are flat and have light tails.
• Disordered: These distributions have a kurtosis that is either very high or very low, and they do not fit into any of the other categories.

Skewness: Impact on Mean, Median, and Mode

The skewness of a distribution can affect the relationship between the mean, median, and mode. In a symmetrical distribution, the mean, median, and mode are all equal. However, in a skewed distribution, the mean, median, and mode will be different from each other. The amount of difference between the mean, median, and mode depends on the degree of skewness.

In summary, moments in statistics provide a systematic way to summarize and characterize the features of a probability distribution, including its center, spread, skewness, and kurtosis.