Arithmetic Mean
The arithmetic mean is obtained by adding all the values in a dataset, x1, x2, ..., xn, and dividing the result by the number of observations, n. $$ \mu = \frac{x_1+x_2+\cdots+x_n}{n} = \frac{1}{n}\sum_{i=1}^{n} x_i $$
The arithmetic mean is the most familiar type of average and one of the most commonly used concepts in statistics.
Whenever people talk about an "average" without specifying a particular type, they are usually referring to the arithmetic mean.
What Is the Arithmetic Mean Used For?
The arithmetic mean is one of the main measures of central tendency.
Its purpose is to summarize a collection of values with a single number that represents the center of the dataset. This makes it easier to understand the overall level of the data at a glance.
Because it is simple to calculate and interpret, the arithmetic mean is widely used in mathematics, statistics, economics, science, and many everyday situations.
Limitations of the Arithmetic Mean. Although the arithmetic mean is a useful summary measure, it does not provide information about how the data are distributed. Two datasets can have the same mean while exhibiting very different patterns. In addition, extreme values (outliers) can have a strong influence on the result, causing the mean to differ substantially from most of the observations. For this reason, the arithmetic mean should often be considered alongside other statistical measures.
A Practical Example
Consider the following dataset:
$$ 1 \ , \ 5 \ , \ 7 \ , \ 3 \ , \ 6 \ , \ 8 $$
Since there are six observations, we have n = 6.
To calculate the arithmetic mean, add all the values and divide the sum by 6:
$$ \mu = \frac{1+5+7+3+6+8}{6} $$
$$ \mu = \frac{30}{6} $$
$$ \mu = 5 $$
Therefore, the arithmetic mean of the dataset is 5.
Example 2
Now consider a different dataset:
$$ 1 \ , \ 2 \ , \ 3 \ , \ 27 \ , \ 28 \ , \ 29 $$
The arithmetic mean is:
$$ \mu = \frac{1+2+3+27+28+29}{6} $$
$$ \mu = \frac{90}{6} $$
$$ \mu = 15 $$
Note. In this example, the mean is 15, yet none of the observations is close to this value. The data are grouped into two separate clusters, one around 2 and the other around 28. As a result, the arithmetic mean falls between the two groups and does not accurately reflect where most of the values are located.
Example 3
Consider a dataset consisting of five observations:
$$ 7 \:, \: 10 \:, \: 6 \:, \: 2 \:, \: 5 $$
The arithmetic mean is calculated as follows:
$$ \mu = \frac{1}{5}\sum_{i=1}^{5} x_i $$
$$ \mu = \frac{1}{5}(7+10+6+2+5) $$
$$ \mu = \frac{1}{5}(30) $$
$$ \mu = 6 $$
Therefore, the arithmetic mean of the dataset is 6.
Further Observations
The arithmetic mean has several useful variations and related concepts.
- Weighted Arithmetic Mean
The weighted arithmetic mean extends the standard arithmetic mean by assigning a weight, wi, to each observation. Values with larger weights have a greater influence on the final result. $$ \mu = \frac{\sum_{i=1}^{n} x_i w_i}{\sum_{i=1}^{n} w_i} $$
In statistics, the arithmetic mean is often used together with other measures of central tendency, such as the median and the mode, to provide a more complete description of a dataset.
