The mean, also known as the expected value in Statistics, is a measure of central tendency which represents the average of the data. Generally, is the sum of all observations divided by the number of observations of the data (arithmetic mean). In this tutorial we will review how to calculate the arithmetic mean as well as the trimmed, geometric and weighted means in R.

## Arithmetic mean with the mean function

In order to calculate the **arithmetic mean of a vector** we can make use of the `mean`

function. Consider the following sample vector that represents the exam qualifications of a student during the current year:

`x <- c(2, 4, 3, 6, 3, 7, 5, 8)`

Using the `mean`

function we can calculate the mean of the qualifications of the student:

```
mean(x) # 4.75
# Equivalent to:
sum(x)/lenght(x) # 4.75
```

Note that, if for some reason **some elements of the vector are missing** (the vector contains some `NA`

values), you should set the `na.rm`

argument of the function to `TRUE`

. Otherwise, the output will be an `NA`

.

```
# Vector with NA
x <- c(2, 4, 3, 6, 3, 7, 5, 8, NA)
# If the vector contains an NA value, the result will be NA
mean(x) # NA
# Remove the NA values
mean(x, na.rm = TRUE) # 4.75
```

## Arithmetic trimmed mean in R

The arithmetic trimmed mean **removes a fraction of observations from each end of the vector before the mean is computed**. This is specially interesting when the vector contains outliers of some data we don’t want to be used when calculating the mean. For instance, if we trim our data to the 10% only the 80% of the central data will be used to compute the mean.

```
# Sample vector
y <- c(1, rep(5, 8), 50)
# Arithmetic mean
mean(y) # 9.1
# Arithmetic trimmed mean to the 10%
# (removes the first and the last element on this example)
mean(y, trim = 0.1) # 5
```

## Weighted mean in R with the weighted.mean function

The arithmetic mean considers that each observation has the same relevance than the others. If we want to **assign a different relevance for each observation** we can assign a different weight to each observation (the arithmetic mean considers the same weight for all observations).

In order to assign weights we can make use of the `weighted.mean`

function as follows:

```
# Sample vector
z <- c(5, 7, 8)
# Weights (should sum up to 1)
wts <- c(0.2, 0.2, 0.6)
# Weighted mean
weighted.mean(z, w = wts) # 7.2
```

Note that the latter is equivalent to:

`sum(z * wts) # 7.2`

If your data contains any `NA`

value the function also provides the `na.rm`

argument.

## Geometric mean in R

The geometric mean **is the n-th root of the product of the elements of the vector**. In order to calculate it you can use the `exp`

, `mean`

and `log`

functions or use the `geometric.mean`

function from `psych`

, which includes the `na.rm`

argument if needed.

```
# Sample vector
w <- c(10, 20, 15, 40)
# Geometric mean
exp(mean(log(w))) #18.6121
# Alternative (which includes the na.rm argument)
# install.packages("psych")
library(psych)
geometric.mean(w) #18.6121
```