What Is a Mean? Definition in Math and Formula for Calculation

What Is a Mean?

Mean is the simple mathematical average of a set of two or more numbers.

The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series, and the geometric mean method, which is the average of a set of products. However, all the primary methods of computing a simple average produce the same approximate result most of the time.

Key Takeaways

The mean is the mathematical average of a set of two or more numbers.
The arithmetic mean and the geometric mean are two types of mean that can be calculated.
The formula for calculating the arithmetic mean is to add up the numbers in a set and divide by the total quantity of numbers in the set.
The formula for calculating the geometric mean is to multiply all the values in a data set, then take the root of the sum equal to the quantity of values within that data set.
A mean helps you to assess a set of numbers by telling you the average, helping to contextualize each data point.

Understanding Mean

The mean is a statistical indicator that can be used to gauge performance over time. Specific to investing, the mean is used to understand the performance of a company’s stock price over a period of days, months, or years.

An analyst who wants to measure the trajectory of a company’s stock value in, say, the last 10 days would sum up the closing price of the stock in each of the 10 days. The sum total would then be divided by the number of days to get the arithmetic mean. The geometric mean will be calculated by multiplying all of the values together. The nth root of the product total is then taken—in this case, the 10^th root—to get the mean.

Formulas for Arithmetic Mean and Geometric Mean

Calculations for both the arithmetic and geometric means are fairly similar. The calculated amount for one will not substantially vary from another. However, there are subtle differences between the two approaches that do lead to different numbers.

Arithmetic Mean

The formula for calculating the arithmetic mean is to add up all figures and divide by the quantity of figures used. For example, the arithmetic mean of the numbers 4 and 9 is found by adding 4 and 9 together, then dividing by 2 (the quantity of numbers we are using). The arithmetic mean in this example is 6.5.

Arithmetic Mean

Pros

It is easier to calculate.
It is simpler for following along and audit results.
Its calculated value is a finite number.
It has more widespread use in algebraic computations.
It is often the fastest type of mean to calculate.

Cons

It is highly affected by material outliers or extreme numbers outside of a data set.
It is not as useful for skewed distributions.
It is not useful when using time series data (or other series of data with varying basis).
It weighs every item equally, diminishing the importance of more impactful data points.

Geometric Mean

The geometric mean is more complicated and uses a more complex formula. To get the formula for calculating the geometric mean is to multiply all values within a data set. Then, take the root of the sum equal to the quantity of values within that data set. For example, to calculate the geometric of the values 4 and 9, multiply the two numbers together to get 36. Then, take the square root (since there are two values). The geometric mean in this example is 6.

Geometric Mean

Pros

It is less likely to be impacted by extreme outliers.
It returns a more accurate measurement for more volatile data sets.
It considers the effects of compounding.
It is more accurate when using a data set over a long period of time (due to compounding).

Cons

It can’t be used if any value within the data set is 0 or negative.
Its formula is more complex and not easily used.
Its calculation is not transparent and more difficult to audit.
It is less prevalent and not used as much as other methods.

Example Mean Calculations

Let’s put this into practice by examining the price of a stock over a 10-day period. Imagine an investor purchased one share of stock for $148.01. The price of the stock over the next 10 days is also included.

Image by Sabrina Jiang © Investopedia 2022

The arithmetic mean is 0.67%, and is simply the sum total of the returns divided by 10. However, the arithmetic mean of returns is only accurate when there is no volatility, which is nearly impossible with the stock market.

In addition to the arithmetic and geometric means, the harmonic mean is calculated by dividing the number of observations by the reciprocal (one over the value) of each number in the series. Harmonic means are often used in finance to average data that occurs in fractions, ratios, or percentages, such as yields, returns, or price multiples.

The geometric mean factors in compounding and volatility, making it a better metric of average returns. Because it is impossible to take the root of a negative value, add one to all the percentage returns so that the product total yields a positive number. Take the 10^th root of this number and remember to subtract from one to get the percentage figure. The geometric mean of returns for the investor in the last five days is 0.61%. As a mathematical rule, the geometric mean will always be equal to or less than the arithmetic mean.

$\begin{aligned}\text{Arithmetic Mean} &= \tiny{\frac{ (0.0045) + 0.0121 + 0.0726 + ... + 0.0043 + (0.0049) + 0.0376 }{ 10 } } \\&= 0.0067 \\&= 0.67\% \\\end{aligned}$

$\begin{aligned}\text{Geometric Mean} &= \tiny{\sqrt[10]{ 0.9955 \times 1.0121 \times 1.0726 \times ... \times 1.0043 \times 0.9951 \times 1.0376 } - 1} \\&= 0.0061 \\&= 0.61\% \\\end{aligned}$

Analyzing the table shows why the geometric mean provides a better value. When the arithmetic mean of 0.67% is applied to each of the stock prices, the end value is $152.63. However, the stock traded for $157.32 on the last day. This means that the arithmetic mean of returns is understated.

On the other hand, when each of the closing prices is raised by the geometric average return of 0.61%, the exact price of $157.32 is calculated. In this example, and often in many calculations, the geometric mean is a more accurate reflection of the true return of a portfolio.

While the mean is a good tool to evaluate the performance of a company or portfolio, it should also be used with other fundamentals and statistical tools to get a better and broader picture of the investment’s historical and future prospects.

Examples of When Means Are Important in Investing

Within business and investing, mean is used extensively to analyze performance. Examples of situations in which you may encounter mean include:

Determining whether an equity is trading above or below its average over a specified time period.
Looking back to see how comparative trading activity may determine future outcomes. For example, seeing the average rate of return for broad markets during prior recessions may guide decision making in future economic downturns.
Seeing whether trading volume or the quantity of market orders is in line with recent market activity.
Analyzing the operational performance of a company. For instance, some financial ratios like days sales outstanding require determining the average accounts receivable balance for the numerator.
Quantifying macroeconomic data like average unemployment over a period of time to determine general health of an economy.

What is a mean in math?

In mathematics and statistics, the mean refers to the average of a set of values. The mean can be computed in a number of ways, including the simple arithmetic mean (add up the numbers and divide the total by the number of observations), the geometric mean, and the harmonic mean.

How do you find the mean?

The mean is a characteristic of a set of data that describes some sort of average. To find the mean, you can compute it mathematically using one of several methods, depending on the structure of the data and the type of average you need. You can also visually identify the mean in many cases by plotting the data distribution. In a normal distribution, the mean, mode, and median are all the same value that occurs at the center of the plot.

What is the difference between mean, median, and mode?

The mean is the average that appears in a set of data.

The median is the midway point above (below) where 50% of the values in the data sits.

The mode refers to the most frequently observed value in the data (the one that occurs the most).

Why is mean important?

Mean is a valuable statistical measurement that tells you what the expected outcome is when comparing all data points together. Although it doesn’t guarantee future results, the mean helps set the expectation of a future outcome based on what already has happened.

Is a mean the same as an average?

Yes. A mean is the mathematical average of a set of two or more numbers.

The Bottom Line

The mean is another word for a mathematical "average." The simple or arithmetic mean is the average calculated by summing up the values of some observations and dividing by the number of observations. The geometric mean is calculated by multiplying all the numbers in a dataset and then taking the n^th root of the product, where n is the total number of values in the dataset. The geometric mean is particularly useful when dealing with quantities that have a multiplicative or exponential relationship, such as growth rates, percentages, or ratios. The harmonic mean is calculated by dividing the total number of values in the dataset by the sum of the reciprocals of the individual values. It is also used when dealing with rates, ratios, or situations where the relationship between the values is inversely proportional.

The mean is an important descriptive statistic, but should not be interpreted in isolation. One should also keep in mind the shape of the data distribution and other metrics like the standard deviation, median, and mode.