The Math Behind Crypto Rate Comparisons: Average of Ratios Explained
When Should I Swap tells you that a conversion rate is "above average" or "below average," it is comparing today's rate to a historical average. But how that average is calculated matters more than you might expect. There are two reasonable-sounding ways to compute an average conversion rate, and they give different results. One of them is biased. The other is not.
This article explains both methods, shows why they differ, introduces the mathematical concept behind the difference, and walks through a concrete example so you can see it in action. No math degree required -- just a willingness to follow along with some simple arithmetic.
The Setup: What Are We Averaging?
Suppose you want to know the average conversion rate between Bitcoin (BTC) and Ethereum (ETH) over the past 5 days. You have the following data (these are simplified numbers for illustration):
| Day | BTC Price (USD) | ETH Price (USD) | BTC/ETH Rate |
|---|---|---|---|
| Monday | $100,000 | $5,000 | 20.0 |
| Tuesday | $95,000 | $5,000 | 19.0 |
| Wednesday | $100,000 | $4,000 | 25.0 |
| Thursday | $90,000 | $5,000 | 18.0 |
| Friday | $105,000 | $5,500 | 19.09 |
There are two ways to calculate the "average" BTC/ETH rate from this data. Both sound reasonable. Only one is correct for our purposes.
Method 1: Ratio of Averages
The first approach is to average each cryptocurrency's dollar price separately, then divide:
Average BTC price: ($100,000 + $95,000 + $100,000 + $90,000 + $105,000) / 5 = $98,000
Average ETH price: ($5,000 + $5,000 + $4,000 + $5,000 + $5,500) / 5 = $4,900
Ratio of averages: $98,000 / $4,900 = 20.0
This method says the average BTC/ETH rate is 20.0. It calculates the average price of each asset independently, then computes the ratio of those two averages.
Method 2: Average of Ratios
The second approach computes the conversion rate for each individual day, then averages those daily rates:
Daily rates: 20.0, 19.0, 25.0, 18.0, 19.09
Average of daily rates: (20.0 + 19.0 + 25.0 + 18.0 + 19.09) / 5 = 20.218
This method says the average BTC/ETH rate is 20.218. It calculates the actual conversion rate for each day and then averages those rates directly.
Two Methods, Two Different Answers
The ratio-of-averages method gave us 20.0. The average-of-ratios method gave us 20.218.
That is a difference of about 1.1%. In cryptocurrency markets, where the numbers are larger and the volatility is higher, this difference can be substantially larger -- sometimes several percentage points.
So which one is correct?
Why the Difference Exists: Jensen's Inequality
The mathematical reason these two methods give different results is a concept called Jensen's inequality. Do not worry about the formal definition -- here is the intuitive version.
Division (computing a ratio) is a nonlinear operation. When you divide two numbers, the relationship between them is not proportional in a simple way. Because of this nonlinearity, the average of a bunch of ratios is generally not equal to the ratio of the individual averages.
In simpler terms: averaging the ingredients before combining them gives a different result than combining them first and then averaging. When the operation that combines them is division (as it is with conversion rates), the order matters.
Jensen's inequality tells us specifically that for convex functions (and dividing by a variable is convex when the variable is positive), the ratio-of-averages will systematically differ from the average-of-ratios. The direction and magnitude of the difference depend on the volatility of the data. More volatility means a larger difference.
Cryptocurrency markets are highly volatile. The prices of BTC and ETH can move by large percentages from day to day. This means the Jensen's inequality bias is not a tiny rounding error -- it can be a meaningful distortion.
Which Method Is Correct?
The answer depends on what question you are asking.
If you are asking: "What was the conversion rate on a typical day?" then the average-of-ratios is the right method. It directly averages the actual daily conversion rates, giving you the rate you would have experienced on an average day.
If you are asking: "What is the ratio of the average prices?" then the ratio-of-averages is the right method. But that is a different question, and it does not correspond to any actual conversion rate that existed on any specific day.
For a tool like Should I Swap, where the goal is to compare today's conversion rate to the historical norm, the relevant question is: "On a typical day over the past N days, what was the conversion rate?" That is the average-of-ratios.
To put it concretely: if you had swapped BTC for ETH on a random day during the past 5 days, you would have received the daily rate for that day. The average of those daily rates tells you what the expected rate was. The ratio of the average prices does not correspond to any day's actual rate.
A More Dramatic Example
To make the difference more visible, consider a scenario with higher volatility:
| Day | BTC Price (USD) | ETH Price (USD) | BTC/ETH Rate |
|---|---|---|---|
| Day 1 | $100,000 | $5,000 | 20.0 |
| Day 2 | $100,000 | $2,500 | 40.0 |
| Day 3 | $100,000 | $5,000 | 20.0 |
In this example, BTC stays flat while ETH drops by 50% on Day 2 and recovers on Day 3.
Ratio of averages:
- Avg BTC: $100,000
- Avg ETH: ($5,000 + $2,500 + $5,000) / 3 = $4,167
- Ratio: $100,000 / $4,167 = 24.0
Average of ratios:
- Daily rates: 20.0, 40.0, 20.0
- Average: (20.0 + 40.0 + 20.0) / 3 = 26.67
The difference is now over 10%. The ratio-of-averages gives 24.0, while the average-of-ratios gives 26.67. The average-of-ratios correctly reflects that on one of the three days, the rate was dramatically higher (40.0), pulling the average up. The ratio-of-averages underestimates this because averaging the ETH prices first smooths out the Day 2 crash before the division happens.
If today's rate is 25.0, the ratio-of-averages method would say you are above the 24.0 average (a +4.2% signal). The average-of-ratios method would say you are below the 26.67 average (a -6.3% signal). Same data, opposite signals, depending on which method you use.
Using the wrong method does not just produce a slightly different number -- it can flip the signal entirely.
Why This Matters in Practice
Accurate Signals
The primary reason to care about calculation methodology is accuracy. When Should I Swap tells you a rate is "above average," you need confidence that the average was calculated correctly. Using the biased ratio-of-averages method could produce false signals -- telling you a rate is above average when the correct calculation shows it is actually below average, or vice versa.
Chart Consistency
Should I Swap displays a historical chart with a dashed line representing the average conversion rate over the selected period. That dashed line is calculated using the average-of-ratios method. If the numerical signal used a different method, the signal and the chart would disagree. A rate that visually sits below the dashed line on the chart might be labeled "above average" by the signal, or vice versa.
By using the same method everywhere, the tool ensures internal consistency: the signal, the percentage, and the chart line all tell the same story.
Transparency
Methodology transparency matters. When a tool shows you data, you should be able to understand how it was generated. If someone asked "how is the average calculated?" and the answer was "we average each price separately and divide," that would be easy to understand but would produce biased results. Should I Swap uses the average-of-ratios because it is the correct method, and this article exists so you know exactly what "correct" means in this context.
How Should I Swap Implements This
Under the hood, the calculation works like this:
- The tool fetches up to 365 days of daily price data for both cryptocurrencies from CoinGecko (data provided by CoinGecko).
- For each day, it computes the conversion rate by dividing the price of the "from" cryptocurrency by the price of the "to" cryptocurrency.
- It takes the last N days of those daily rates (where N is your selected period: 7, 14, 30, 90, 180, or 365).
- It averages those N daily rates together. This is the average-of-ratios.
- It compares today's live conversion rate to that average and expresses the difference as a percentage.
The multi-period signal grid repeats this process for all six standard periods simultaneously, giving you the average-of-ratios calculation for each window side by side. For a guide on interpreting those signals, see our article on how to read a multi-period signal summary.
No ratio-of-averages is used anywhere in the tool. Every average conversion rate displayed -- whether in the main signal, the chart dashed line, or the multi-period grid -- uses the average-of-ratios method.
An Analogy: Average Speed
If the math still feels abstract, consider this everyday analogy.
You drive 60 miles in 1 hour, then 60 miles in 2 hours. What was your average speed?
Method 1 (ratio of averages): Average distance = (60 + 60) / 2 = 60 miles. Average time = (1 + 2) / 2 = 1.5 hours. Ratio = 60 / 1.5 = 40 mph.
Method 2 (average of ratios): Speed on trip 1 = 60/1 = 60 mph. Speed on trip 2 = 60/2 = 30 mph. Average = (60 + 30) / 2 = 45 mph.
The correct answer depends on what you are measuring. If you want the average speed for the total journey (120 miles in 3 hours), it is 40 mph. But if someone asked "what speed did you typically drive at?" the answer that matches your experience is 45 mph -- you drove at 60 mph half the time and 30 mph the other half.
For conversion rates, we are asking the "typical rate" question. The average-of-ratios gives us the rate you would have typically experienced across the days in the period.
Common Questions
Does the difference matter for all pairs?
The magnitude of the difference depends on volatility. For stable pairs (two assets that tend to move together), the difference is small. For volatile pairs or during turbulent market periods, the difference can be several percentage points. Since crypto markets are generally volatile, using the correct method matters for most pairs most of the time.
Could you use a different kind of average?
Yes, there are other types of averages. A geometric mean, for instance, has different properties than an arithmetic mean. Should I Swap uses the arithmetic mean of daily ratios because it is the most intuitive ("what was the typical day's rate?") and the easiest to verify. The geometric mean would be more appropriate for compounded returns, which is a different use case.
Why not just show both?
Showing both methods would add complexity without adding value for most users. The average-of-ratios is the correct method for comparing today's rate to the historical norm. Showing the biased method alongside it would create confusion and potentially lead people to rely on the less accurate number.
How big can the difference get?
In extreme cases with very high volatility, the difference between the two methods can exceed 10%. In typical market conditions for major pairs like BTC/ETH, the difference is usually 0.5% to 3%. That may sound small, but it is large enough to flip a signal from "above average" to "near average" or even "below average" -- which is the entire purpose of the tool.
Why Methodology Transparency Matters
In the crypto space, many tools and platforms present data without explaining how it is calculated. You see a number, perhaps a "score" or a "signal," and you are expected to trust it. But when the calculation methodology is not transparent, you have no way to evaluate whether the number is meaningful, whether it is calculated correctly, or whether it is being presented in a misleading way.
Should I Swap takes the opposite approach: the methodology is documented, the source data is attributed (CoinGecko), and this article explains the reasoning behind the choice of calculation method. If you disagree with the approach or want to verify the results, you have enough information to do so.
We believe that transparency in methodology builds trust, and trust is the foundation of any useful tool. If you cannot verify how a number was produced, you cannot know whether to rely on it. If you can, you can make your own judgment about its value.
Summary
- There are two ways to calculate an average conversion rate: ratio-of-averages (average each price, then divide) and average-of-ratios (calculate each day's rate, then average).
- They give different results because of Jensen's inequality, a mathematical property of nonlinear operations like division.
- The average-of-ratios is the correct method for comparing today's rate to the historical norm, because it directly averages the actual daily conversion rates.
- The ratio-of-averages introduces a systematic bias that increases with market volatility and can produce incorrect signals.
- Should I Swap uses the average-of-ratios method consistently across all signals, charts, and multi-period comparisons, ensuring accuracy and internal consistency.
The math behind a tool matters. Even if you never calculate a ratio by hand, knowing that the numbers you see are produced by the correct method gives you well-founded confidence in the data.
Ready to compare rates? Try Should I Swap -- it's free, no account required.
Data provided by CoinGecko. Should I Swap is an informational tool and does not provide financial advice. Past performance does not indicate future results.