Mean Definition

What is an average?

The average is the simple mathematical average of a set of two or more numbers. The average of a given set of numbers can be calculated in several ways, including the arithmetic average 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 major methods of calculating a simple average produce the same approximate result most of the time.

Key points to remember

  • The mean is the mathematical mean of a set of two or more numbers.
  • Arithmetic mean and geometric mean are two types of means that can be calculated.
  • The formula for calculating the arithmetic mean consists of adding the numbers in a set and dividing by the total amount of numbers in the set.
  • The formula for calculating the geometric mean is to multiply all the values ​​in a set of data and then take the root of the sum equal to the amount of values ​​in that set of data.
  • An average helps you evaluate a set of numbers by telling you the average, helping to put each data point in context.

Average comprehension

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

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

Formulas for the arithmetic mean and the geometric mean

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

Arithmetic average

The formula for calculating the arithmetic mean is to add all the digits together and divide by the amount of digits 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 amount of numbers we use). The arithmetic mean in this example is 6.5.

Arithmetic average

Advantages

  • It’s easier to calculate.

  • It’s easier to track and check the results.

  • Its calculated value is a finite number.

  • It has more widespread use in algebraic calculations.

  • This is often the fastest type of average to calculate.

The inconvenients

  • It is strongly affected by material outliers or extreme numbers outside of a data set.

  • It’s not as useful for biased distributions.

  • It is not useful when working with time series data (or other data series with a varying basis).

  • It weighs each element equally, diminishing the importance of more impactful data points.

Geometric mean

The geometric mean is more complicated and uses a more complex formula. The formula for calculating the geometric mean consists of multiplying all the values ​​of a set of data. Then take the root of the sum equal to the amount of values ​​in this data set. For example, to calculate the geometry 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

Advantages

  • It is less likely to be impacted by extreme outliers.

  • It returns a more accurate measurement for more volatile data sets.

  • It takes into account the effects of capitalization.

  • It is more accurate when using a dataset over a long period of time (due to composition).

The inconvenients

  • It cannot be used if any value in the dataset is 0 or negative.

  • Its formula is more complex and not easily usable.

  • Its calculation is not transparent and more difficult to audit.

  • It is less widespread and less used than the other methods.

Example of average calculations

Let’s put this into practice by looking at the price of a stock over a 10-day period. Imagine that an investor buys a stock for $148.01. The stock price 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 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 inverse (one over the value) of each number in the series. Harmonic averages are often used in finance to average data that comes in the form of fractions, ratios, or percentages, such as yields, returns, or price multiples.

The geometric mean takes into account composition and volatility, making it a better measure of average returns. Since it is impossible to take the root of a negative value, add one to all percentage returns so that the product total gives a positive number. Take the 10e root of that number and don’t forget to subtract from one to get the percentage. The geometric mean of investor returns over the past five days is 0.61%. As a mathematical rule, the geometric mean will always be equal to or less than the arithmetic mean.

Image by Sabrina Jiang © Investopedia 2022


Analysis of 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 final value is $152.63. However, the stock traded at $157.32 on the last day. This means that the arithmetic mean of returns is underestimated.

On the other hand, when each of the closing prices is increased by the geometric mean return of 0.61%, the exact price of $157.32 is calculated. In this example and as often in many calculations, the geometric mean more accurately reflects the true performance of a portfolio.

Although the average is a good tool for evaluating the performance of a company or a portfolio, it should also be used with other fundamentals and statistical tools to get a better and broader picture of historical and future investment prospects.

Examples of cases where the means are important to invest

In business and investing, the average is widely used to analyze performance. The following are examples of situations in which you may encounter wickedness:

  • Determine whether a stock is trading above or below its average over a given period.
  • Look back to see how comparative business activity can determine future results. For example, see the average rate of return for extended markets in previous years recessions can guide decision-making during future economic downturns.
  • See if the trading volume or quantity of market orders is consistent with recent market activity.
  • Analyze the operational performance of a company. For example, certain financial ratios such as the Days of incredible deals require determining the average accounts receivable balance for the numerator.
  • Quantification of macroeconomic data like average unemployment over a period of time to determine the general health of an economy.

What is an average in mathematics?

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

How do you find the average?

Mean is a characteristic of a data set that describes a kind of average. To find the mean, you can calculate it mathematically using several methods, depending on the structure of the data and the type of mean you need. You can also visually identify the mean in many cases by plotting the distribution of the data. In a normal distributionthe average, fashionand 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 data set. The median, instead, is the midpoint above (below) where 50% of the values ​​in the data fall. The mode refers to the most frequently observed value in the data (the one that appears most often).

Why is the average important?

The mean is a valuable statistical measure that tells you what the expected result is when you compare all the data points together. Although it does not guarantee future results, the average helps set the expectation of a future result based on what has already happened.

Is an average the same thing as an average?

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

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