Arithmetic Mean
The average of a sum of numbers
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What is the Arithmetic Mean?
The arithmetic mean is the average of a sum of numbers, which reflects the central tendency of the position of the numbers. It is often used as a parameter in statistical distributions or as a result to summarize the observations of an experiment or a survey.
There are several types of means with different calculation methods. The arithmetic mean is the simplest and most widely-used type. It is applied in finance frequently but is not always the most ideal tool for certain purposes.
Summary
- The arithmetic mean is calculated by dividing the sum of a collection of numbers by the count of the numbers, which reflects the central tendency of that collection.
- The arithmetic mean is not always able to properly identify the “location” of a data set, since it can be skewed by outliers.
- In finance, the arithmetic mean is appropriate to support future estimates.
How to Calculate the Arithmetic Mean
To calculate the arithmetic mean, add a collection of numbers and divide the sum by the count of the numbers in that collection. The mathematical expression is given below:
Where:
- ai – The value of the ith observation
- n – The number of observations
For example, the closing prices of a stock for the last five days are collected respectively: $89, $86, $79, $93, and $88. The arithmetic mean of the stock price is, thus, $87 [(89 + 86 + 79 + 93 + 88) / 5]. The value shows the central tendency of the stock price for the last five days. It reflects the position the current stock price is by comparing it with the 5-day average price.
As its formula shows, the arithmetic mean measures every observation value equally, so it is also known as an unweighted average or equally-weighted average. It is a special case in the concept of a weighted average, where a weight can be assigned to each observation as needed.
All the weights in the collection of observations must sum up to 1. The arithmetic mean assigns a weight of 1/n for each observation, assuming there are n observations in the collection.
Where:
- wi – The weight for the ith observation
Arithmetic Mean, Median, and Mode
The arithmetic mean is frequently used to identify the “central position” of the distribution of a group of data. However, it is not always an ideal indicator. The occasional observations that are significantly greater or smaller than the rest of the group are known as outliers.
Outliers are not representative of a group of data, but they can significantly impact the arithmetic mean. In a positively-skewed collection of data, the extremely large outliers drive the arithmetic mean up; in a negatively-skewed collection of data, the extremely small outliers drive the mean down.
In situations with outliers, the mode or median can better indicate the central tendency of a set of data than the mean. The mode is the value that appears with the highest frequency. The median is the “middle point” that exactly separates the higher half and lower half of the data set. Outliers exert a much smaller impact on the two parameters (especially the mode).
Therefore, the mode and median might be more representative of a collection of data with extremely large or small outliers. In a positively-skewed data set, the median and mode are smaller than the arithmetic mean. In a negatively-skewed data set, the median and mode are larger than the arithmetic mean.
Arithmetic Mean, Geometric Mean, and Harmonic Mean
In addition to the arithmetic mean, the other two types of average that are commonly used in the finance world are the geometric mean and harmonic mean. The different types of means are applied for different purposes.
The arithmetic mean should be used when looking for the average of a set of raw values, such as stock prices. The geometric mean should be used when dealing with a set of percentages, which are derived from raw values, such as the percentage change of stock prices.
Also, the calculation of geometric mean takes into account the compounding effect over periods, which cannot be captured by the arithmetic mean. Therefore, the geometric mean is more appropriate to measure the average historical performance of investment portfolios, especially when dividends and other earnings are reinvested. The arithmetic mean is often used to estimate future performances.
The harmonic mean can deal with fractions with different denominators. Therefore, it is the most appropriate approach to average ratios, e.g., the P/E and EV/EBITDA ratios. Unequal denominators will cause different weights for each data when the arithmetic mean is applied.
The arithmetic mean of P/E ratios is biased unless all the P/E ratios in the group show the same value for the denominator (the same earnings per share), which is rarely the case. The advantage of the harmonic mean is that it assigns equal weights to all the data in the group, regardless of whether the denominators are equal or not.
Related Readings
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