What you'll learn:
☞ What is a mean?
A mean is like a regular math average of a group of two or more numbers.
You can find the mean of a set of numbers using different methods, but the main ones are the arithmetic mean and the geometric mean.
The arithmetic mean adds up all the numbers and divides by how many there are, while the geometric mean calculates an average based on products.
However, these methods usually give pretty similar answers.
| What is a mean? | How to calculate arithmetic mean? | How to calculate geometric mean? |
|---|---|---|
| The average of a group of numbers. | Add up numbers and divide by how many there are. | Multiply numbers, then take a special root. |
How is mean used in investing?
The mean is a tool used in statistics to figure out how something changes over time. When it comes to investing, the mean helps us see how a company’s stock price performs over days, months, or years.
Imagine an analyst who wants to track a company’s stock value over the last 10 days. They would add up the closing price of the stock for each of those 10 days.
Then, they’d divide the total by 10 to find the arithmetic mean. Another method is the geometric mean, where they’d multiply all the values together and then take the 10th root (since there are 10 days) to find the mean.
This helps the analyst understand how the stock is doing on average.
☞ Formulas for arithmetic & geometric mean
The calculations for both arithmetic and geometric means are quite similar. The results won’t be very different between the two methods, but they do give slightly different numbers.
How to calculate arithmetic mean
To calculate the arithmetic mean, you add up all the numbers and then divide by how many numbers there are.
For instance, if we have the numbers 4 and 9, the arithmetic mean is found by adding 4 and 9 together, then dividing by 2 (since there are 2 numbers). In this case, the arithmetic mean is 6.5.
Pros & cons of arithmetic mean
| 🥇 Pros | 🥈 Cons |
|---|---|
| Easier to calculate. | Affected by outliers or extreme numbers. |
| Simpler for following and auditing. | Not suitable for skewed distributions. |
| Calculated value is finite. | Not suitable for time series data or varying basis data. |
| Widespread use in algebra. | Weights all items equally, downplays impactful data points. |
| Often faster to calculate. |
How to calculate geometric mean
The geometric mean involves a more intricate process and employs a complex formula. To compute the geometric mean, you need to multiply all the values in a dataset.
Next, you’ll find the root of the sum based on the number of values in that dataset. For instance, if you’re dealing with the values 4 and 9, you’d multiply them to get 36.
Then, take the square root (as there are two values). In this case, the geometric mean is 6.
Pros & cons of geometric mean
| 🥇 Pros | 🥈 Cons |
|---|---|
| Less affected by extreme outliers. | Can’t be used if any value is 0 or negative. |
| More accurate for volatile data sets. | Formula is complex and not easily used. |
| Considers the effects of compounding. | Calculation is not transparent and harder to audit. |
| Accuracy for long-term data sets due to compounding. | Less commonly used compared to other methods. |
☞ Example mean calculations
Let’s apply these concepts practically by examining a stock’s price over a 10-day period. Imagine an investor bought one share of stock for $148.01 and fluctuates up to $157.32.
Here’s a breakdown of calculations using different mean methods:
| Date | Stock Price ($) | Return (%) | Arithmetic Proof ($) | %Returns +1 | Geometric Proof ($) |
|---|---|---|---|---|---|
| 8/4/2023 | 148.01 | ||||
| 8/5/2023 | 147.34 | -0.00453 | 148.99 | 0.99547 | 148.92 |
| 8/6/2023 | 149.12 | 0.01208 | 148.32 | 1.01208 | 149.83 |
| 8/7/2023 | 159.94 | 0.07256 | 150.11 | 1.07256 | 150.74 |
| 8/8/2023 | 149.60 | -0.06465 | 161.00 | 0.93535 | 151.67 |
| 8/9/2023 | 149.97 | 0.00247 | 150.60 | 1.00247 | 152.59 |
| 8/12/2023 | 151.40 | 0.00954 | 150.97 | 1.00954 | 153.53 |
| 8/13/2023 | 151.72 | 0.00211 | 152.41 | 1.00211 | 154.47 |
| 8/14/2023 | 152.37 | 0.00428 | 152.73 | 1.00428 | 155.41 |
| 8/15/2023 | 151.62 | -0.00492 | 153.38 | 0.99508 | 156.36 |
| 8/16/2023 | 157.32 | 0.03759 | 152.63 | 1.03759 | 157.32 |
| Arithmetic: 0.67% | 0.0067 | Geometric: 0.61% | 1.00612 |
Arithmetic Mean: 0.67%
Arithmetic Mean = (0.0045 + 0.0121 + 0.0726 + … + 0.0043 + 0.0049 + 0.0376) / 10 = 0.0067 = 0.67%
The arithmetic mean, which sums up the returns and divides by 10, provides 0.67%. However, this method assumes no volatility, which is rarely true in the stock market.
Geometric Mean: 0.61%
Geometric Mean = ((0.9955 × 1.0121 × 1.0726 × … × 1.0043 × 0.9951 × 1.0376)^10)^(1/10) – 1 = 0.0061 = 0.61%
The geometric mean factors in compounding and volatility. It’s calculated by modifying the percentage returns, taking their product’s root, and then subtracting one. For our case, the geometric mean is 0.61%.
Analyzing the results shows the advantage of the geometric mean. Using the arithmetic mean, the final value would be $152.63, but the actual stock price was $157.32.
On the other hand, the geometric mean’s accurate reflection yields the exact price of $157.32. It’s often a better measure for true returns.
While the mean helps assess performance, it’s wise to use it alongside other fundamentals and statistical tools. This broader approach gives a clearer historical and future perspective on investments.
☞ When to use means in investing
In the world of business and investing, averages are widely used to understand performance. Here are some situations where you might come across averages:
- Comparing Stock Performance: You can figure out if a company’s stock is trading higher or lower than its usual level over a specific time frame.
- Predicting Future Trends: Looking at past trading patterns helps predict what might happen next. For instance, understanding the average return rate of the market during past economic downturns can guide decisions in similar situations.
- Market Activity Analysis: Checking if the volume of trades or the number of market orders matches recent market trends helps make sense of current activity.
- Company Performance: By calculating averages, you can analyze how a company is doing. Financial ratios like days sales outstanding need the average accounts receivable balance to work with.
- Understanding the Economy: Studying macroeconomic data, like the average unemployment rate over a period, gives insights into how healthy an economy is overall.
☞ Means FAQ
| Question | Answer |
|---|---|
| What is a mean in math? | 🧮📊 The mean in math is the average of a set of values. It can be calculated using methods like arithmetic mean, geometric mean, and harmonic mean. |
| How do you find the mean? | ❓ To find the mean, you can use mathematical methods based on your data structure and the type of average needed. Visualizing the data distribution can also help identify the mean. In a normal distribution, mean, mode, and median are the same value at the center. |
| Difference between mean, median, and mode? | ↔️ The mean is the average in a dataset. The median is the middle point where 50% of values lie above (or below). The mode is the most frequent value observed in the data. |
| Why is mean important? | ⭐ The mean is a crucial statistic that provides an expected outcome when comparing all data points. It helps set future outcome expectations based on past data, though it doesn’t guarantee future results. |
| Is a mean the same as an average? | ✅ Yes, a mean and an average refer to the same thing – the mathematical average of a set of numbers. |
☞ Final thoughts
In the realm of mathematics and statistics, the mean stands as a synonym for the mathematical “average.” The commonly used simple or arithmetic mean is derived by summing up a set of observations and then dividing by the count of observations.
On the other hand, the geometric mean involves multiplying all values within a dataset and then extracting the nth root of the resulting product, where n signifies the total number of values.
The geometric mean proves particularly beneficial when tackling data with a multiplicative or exponential relationship, such as growth rates, percentages, or ratios.
There’s also the harmonic mean, calculated by dividing the total number of dataset values by the sum of the reciprocals of individual values. This measure shines in contexts involving rates, ratios, or cases where the relationship between values is inversely proportional.
While the mean is a vital descriptive statistic, it should not be interpreted in isolation. It’s crucial to consider the data distribution’s shape and other metrics like standard deviation, median, and mode to gain a comprehensive understanding of the data’s characteristics.
By combining these insights, you’ll be better equipped to draw meaningful conclusions from your statistical analyses.
