One of the necessary aspects of knowing how things work is understanding exactly what’s being measured when you’re presented with data. Get into the habit of asking what is being counted.
Our running example
Suppose it’s an especially nice day outside. You lace up your running shoes and head out. The first three miles are pleasant, but then you decide to walk the next mile. Your shoes are too tight, so you have to fix them but only after you untangle a really bad knot. You realize it’s getting late and push the final mile home more than planned.
- 3 miles in 30:00 minutes of effort level 5 at a leisurely jog
- 1 mile in 20:00 minutes of effort level 3 at a walk
- 0 miles in 5:00 minutes of effort level 1 while we fix the knot
- 1 mile in 8:00 minutes of effort level 9 at a brisk run
Average pace
What is the average pace? Mostly everyone will add up the total time (30 + 20 + 5 + 8 = 63 minutes) and divide by the total distance (3 + 1 + 0 + 1 = 5 miles) to get 63 / 5 = 12:36 minutes per mile. Some overachievers might ignore the part where we’re not moving for (30 + 20 + 8) / 5 = 11:36 minutes per mile. Others only want to log their running portion for 38 / 4 = 9:30 minutes per mile. We can tell you the “average pace” but unless we include how it’s calculated, you have no idea what is being counted.
Average effort
It gets worse! What about our average effort level? We have the same options to cook the numbers as with average pace, but another option emerges here. Any function (e.g. effort level) of the data (e.g. distance and time) can be averaged along any dimension of measurement. The average effort level over time is 5 x 30:00 + 3 x 20:00 + 1 x 5:00 + 9 x 8:00 divided by the total time to get 4.55. The average effort level over distance is 5 x 3 +3 x 1 +1 x 0 + 9 x 1 divided by the total distance to get 5.40.
These aren’t the same number.
What is being counted here? One prioritizes all minutes equally while the other prioritizes all miles equally. Neither method is more correct than the other as they both have their valid uses. Both give a simple summary of how the run felt and since nobody cares how our run felt, there generally isn’t a lot of discussion about methodology.
What is being counted?
Each of those calculations of average effort is called a weighted average. The calculation “over time” distributed the weights across time while the “over distance” version distributed weights across the distance traversed. In a weighted average, we weight the observations by (non-negative) weights, sum them all up, and then divide by the sum of the weights. Often, the weights are chosen to sum to one so the final step can be skipped (no need to divide by one).
Notice the “over distance” calculation did not include the effort level of our shoe-tying break since the lack of any distance traveled means nothing happened of concern. Numerically, this was automatically handled because the weight for that portion was zero.
Finally, we note that every average is a weighted average. The simple averages that we all grew up with give uniform weights to each observation. If there are N observations, then each weight is 1/N, and the sum of the weights is one.
Good old Calculus class
If you’ve had the pleasure, pleasure being a relative term, of taking any calculus courses in high school or college, you have some bigger tools to apply this reasoning. Whenever an integral had a dx at the end, the “integration variable” was the x-axis, that is, distance along the x-axis was given uniform priority. Every distance unit along the x-axis counted for exactly as much “weight” as any other equal distance unit elsewhere on the x-axis.
Later, in Calculus 3, you may remember integrating along time (the integral had a dt) or even arc-length (the integral had a ds). Each of these methods was counting something different. Indeed, what is being counted here?
Summing along time (integrals with a dt) is perfectly analogous to our calculating effort level over time. What is being counted? Units of time provide the weights in the weighted sum.
Summing along arc length (integrals with a ds) is perfectly analogous to our calculating effort level over distance. Our route might be wobbly or circular, so it makes sense to consider our route in this sense. Just what is being counted? Units of distance provide the weights in the weighted sum.
We learn the easiest first
Summing along the x-axis (integrals with a dx) is the easiest, so we teach it first. It’s also the least useful in our everyday real-world example above. If our course was completely parallel to the x-axis, then this would be identical to the arc length method, but the real world is usually much messier.
Now what is being counted with this method? Units of horizontal distance (along the x-axis) provide the weights in the weighted sum. There are times when this is most appropriate, but we have to reach a little for an example.
Your family signs up for the same darn race every year to start their New Year’s resolution. The “run along the x-axis 5k” is a simple and straight 5k that, true to its name, runs exactly along the x-axis. Throughout the race, you have to swerve in and out of hazards. Kids and dogs everywhere. While the exact length of the course is 3.10 miles, you actually run 3.19 miles total (your arc length) because of the extra distance from the swerving.
Since only progress along the x-axis contributes to your goal, you’re tempted to calculate all your data “projected” onto the x-axis. Well, summing along the x-axis does that for you automatically (if you don’t do anything complicated to undo the effect of the projection). Did we mention that any hills lifting and lowering you from the x-axis won’t be counted? Hills extend the route that you actually travel compared to the idealized x-axis. This method throws away a lot of information when you realize the actual race wasn’t quite as idealized as the advertised course.
Influencing what is being counted
Politics, money, fame, honest mistakes. Anything you can imagine can influence how things are being counted. The more complicated the measurement becomes, the more freedom there is to manage the result in a desired direction.
Did they die from a virus or with a virus? The latter weights each instance with a positive test result equally while the former weights instances appropriately based on actual circumstances. The two measurements produce wildly different results.
Jobs
Economic measures are just as tricky. Are we counting participants according to their situation? The extent of their situation? The balance of forces that produce the situation? The underlying cause that eventually produced their situation? The U.S. government estimates the number of jobs worked by Americans. This measurement on a per-job basis made perfect sense 50 years ago, but things are always evolving. The internet has enabled lots of folks to work lots of (W-2) “gig” jobs as desired, and some state and local governments are forcing more and more contractors and gig workers to count as (W-2) employees.
A competing survey counts the number of Americans who have a job. One sums over jobs with equal weights (part-time, full-time, or no-time) while the other sums over people. Does it make sense to put equal weight on a 2-hour/week job as a 40-hour/week job? Exactly what is being counted? Is the goal to measure the economic health of the population or is the goal to get reelected?
News
The news media is romanticized to report on all things newsworthy, so let’s start our own media empire for the next example. We want a fair and balanced representation of life on planet Earth, so we’re going to sum over all people on the planet with each person having an equal weight. What we would find is that the average person is living a boring and routine existence. Nothing exciting is happening within the range of their senses almost all the time. Darn, our media empire just went out of business because nobody watches it. Too boring. The same thing happens if we sum over square miles. Most people and most dry land are pretty calm and peaceful almost all the time.
Wake up
Let’s try again with a remixed media empire. This time, put zero weight on anything boring. We plan to sum over events and put the highest weight on those events that will suck in the viewer so much that they won’t change the channel through the next commercial. Furthermore, let’s put a much lower weight on events that don’t flatter our sponsors. Likewise, we can increase weights on events that do make our sponsors appear more relevant. This makes us a lot of money! Also, as owners, we can also influence the weightings to favor or disfavor events that drive our political goals. Why not? The typical viewer gives no thought as to what is being counted for tonight’s lineup.
The two content weighting schemes produce wildly different programming. One puts equal weight on individual life experiences across the planet while the other assigns seemingly arbitrary weights to the most exciting events. Are you asking yourself why? Realize, dear reader, that you are the product while the sponsors are the customers. Existing media sources cater to the customer by delivering you.
In summary
A collection of data isn’t proof, but it can be evidence. Generally, the data are summarized into bite-sized statistics that can be spoken aloud easily and repeated over dinner. Ask yourself what is being counted. Listen to the analysts who are willing to dig in and talk through what is being counted and how it’s measured (give extra attention if that analysis is being censored).
Trust your senses. Turn off the news. How does it feel? The law of attraction guides you toward everything that aligns with your vibration circumventing everything that doesn’t. We suggest listening to your inner being to determine where to set your vibration because your vibration determines what data flow into your experience.
