﻿ Harmonic, quadratic, trimmed, and windsorized mean

# Harmonic, quadratic, trimmed, and windsorized mean

## Harmonic mean

Prism computes the harmonic mean and its confidence interval by first transforming all the values to their reciprocals, and then computing the mean of those reciprocals and the CI of that mean. The harmonic mean is the reciprocal of that mean. If the values are all positive, larger numbers effectively get less weight than lower numbers. The harmonic means is most often used to find the average of a set of rates or velocities.

It makes sense to use the harmonic mean when the set of reciprocals of the data form a symmetrical approximately Gaussian distribution.

Prism computes the quadratic mean and its confidence interval by first transforming all the values to their square (value multiplied by itself), and then computing the mean of those squared values and the CI of that mean. The quadratic mean is the square root of that mean. The quadratic mean is also called the root mean square.

It makes sense to use the quadratic mean when the set of squares of the data form a symmetrical approximately Gaussian distribution.

## Trimmed and Winsorized means

The idea of trimmed or Winsorized means is to not let the largest and smallest values have much impact. Before calculating a trimmed or Winsorized mean, you first have to choose how many of the largest and smallest values to ignore or down weight. If you set K to 1, the largest and smallest values are treated differently. If you set K to 2, then the two largest and two smallest values are treated differently. K must be set in advance. Sometimes K is set to 1, other times to some small fraction of the number of values, so K is larger when you have lots of data.

To compute a trimmed mean, simply delete the K smallest and K largest observations, and compute the mean of the remaining data.

To compute a Winsorized mean, replace the K smallest values with the value at the K+1 position, and replace the k largest values with the value at the N-K-1 position. Then take the mean of the data. .

The advantage of trimmed and Winsorized means is that they are not influenced by one (or a few) very high or low values. Prism does not compute these values.

## Mode

The mode is the value that occurs most commonly. It is not useful with measured values assessed with at least several digits of accuracy, as most values will be unique. It can be useful with variables that can only have integer values. While the mode is often included in lists like this, the mode doesn't always assess the center of a distribution. Imagine a medical survey where one of the questions is "How many times have you had surgery?" In many populations, the most common answer will be zero, so that is the mode. In this case, some values will be higher than the mode, but none lower, so the mode is not a way to quantify the center of the distribution.