Arithmetic mean, geometric mean, harmonic mean, least square mean, and trimmed mean

In statistics, a central tendency is a central or typical value for data distribution. Mean (or average) is commonly used to measure the central tendency. However, depending on the data distribution or the special situation, different types of Mean may be used: arithmetic mean, geometric mean, least-squares mean, harmonic mean, and trimmed mean.

The most common Mean is the arithmetic mean. If we say ‘Mean’, it is the default for arithmetic mean.

Arithmetic Mean is calculated as the sum of all measurements (all observations) divided by the number of observations in the data set.

Geometric Mean is the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale. Geometric mean is often used in the data that follows the log-normal distribution (for example, the pharmacokinetics drug concentration data, the antibody titer data...). 

In practice, geometric mean is usually calculated with the following three steps:

• log-transform the original data
• calculate the arithmetic mean of the log-transformed data
• back transform the calculated value to the original scale

Harmonic Mean is the reciprocal the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.

The harmonic mean are calculated with the following steps:

• Add the reciprocals of the numbers in the set. To find a reciprocal, flip the fraction so that the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of 6/1 is 1/6.
• Divide the answer by the number of items in the set.
• Take the reciprocal of the result.

The harmonic mean is not often used in day-to-day statistics but is quite often used in some statistical formula. For example, for two-group t-statistics with unequal sample size in two groups, the t value can be calculated using the following formula with harmonic mean to measure the average sample size.

Least Squares Mean is a mean estimated from a linear model. Least squares means are adjusted for other terms in the model (like covariates), and are less sensitive to missing data. Theoretically, they are better estimates of the true population mean.

In a previous post "Least squares means (marginal means) vs. means", the calculation of least squares mean is compared with the arithmetic mean.

In analyses of clinical trial data, the least-squares mean is more frequently used than the arithmetic mean since it is calculated from the analysis model (for example, analysis of variance, analysis of covariance,...). The difference between two least-squares means is called the ratio of geometric least-squares means (or geometric least-squares mean ratio) - along with its 90% confidence intervals - is the common approach for assessing the bioequivalence. 

Trimmed Mean may also be called truncated mean and is the arithmetic mean of data values after a certain number or proportion of the highest and/or lowest data values have been discarded. The data values to be discarded can be one-sided or two-sided. 

The key for trimmed mean calculation is to determine the percentage of data to be discarded and whether or not the data to be discarded is one-sided or two-sided. The percentage of data to be discarded may be tied to the percentage of missing data. 

Trimmed mean can be calculated and then used to fill in the missing data - a single imputation method for handling the missing data. Trimmed mean as a single imputation method for missing data has its limitations, but it is still used in analyses of clinical trials - usually for sensitivity analyses.

In ICH E9-R1 "Addendum on Estimands and Sensitivity Analysis in Clinical Trials" training material, about the composite strategy to handle the intercurrent event, trimmed mean is mentioned to be an approach in handling the intercurrent event.