# What are (a) the source of and (b) the justification (e.g. experimental) for the concept of "normalized power" used in physical endurance training?

In lots of places, I see the concept of "normalized power" recommended for measuring the intensity of a workout and it is claimed in many of these places to be superior to "average power" for this purpose.

I'm interested in knowing---but have had no luck finding---the following: (1) the origin of the concept as used in physical training. Is there some published paper, for example? (2) what empirical quantitative evidence exists for its usefulness and effectiveness? And where can I find that evidence?

I've seen a lot of subjective anecdotal testimony about its value, but no actual data or even a serious justification for using it.

My understanding is that high-intensity efforts are disproportionately taxing, and NP is a way of reflecting that. That is, if you went for one ride where you maintained a steady 75% FTP output, and another one where your output varied, but averaged out to 75% FTP, the second one would be more taxing.

Andy Coggan introduced the concept in his paper "Training and racing using a power meter" (later expanded to a book under the same title). This mathematically approximates empirical observations of power output, blood-lactate concentration, and other lab observations. He discusses it in the paper. You'll find a few articles about it at the Training Peaks blog (Coggan is one of the principals at Training Peaks).

• I don't think Andy has ever been one of the principals at Training Peaks. The origin of the concept was indeed given in Training and Racing using a Power Meter, which he wrote and released in March 2003, before there was a Training Peaks (or even its precursor, Cycling Peaks). As a historical anecdote, he originally called it "corrected" power. The name "normalized" power was suggested because the algorithm is based on something called the L4 vector norm. The original 2003 document can be found here: ipmultisport.com/ref_lib/Coggan_Power_Meter.pdf Commented Nov 1, 2021 at 22:29
• And, I don't think the claim is that NP is unambiguously superior to average power. The claim is actually a bit subtler: that when a ride is highly variable in intensity, then an intensity-weighted mean comes closer to physiological stress than an unweighted mean. Average power is an unweighted mean, while NP is one type of intensity-weighted mean (there are lots of others, including Skiba's XP, which uses a related but slightly different weighting). Commented Nov 1, 2021 at 22:39
• I stand corrected. Commented Nov 1, 2021 at 22:39
• @R.Chung I can't think if a situation where average power is superior to normalized power. If there are high variations in power output, normalized power certainly does a better job than average power. But in the extreme case when there are literally no variations - such as a mundane trainer session - normalized power just reduces to average power. In other words, normalized power is always at least as good as average power as a measure of how hard a ride was physiologically, and most of the time it's significantly superior. Commented Nov 1, 2021 at 22:48
• @AndrewHenle Average power is superior when you need to measure actual output rather than physiological stress, such as when you're estimating drag, projecting speed for power, or planning nutritional replenishment. Those are situations where the physics is more important than the physiology. In addition, NP is known to be aberrant over short periods of time: during short and variable intervals, NP can be less than AP. Commented Nov 2, 2021 at 7:07

Focusing more on the empirical part of the question, consider the link that R. Chung gave in his first comment to Adam Rice's answer. That's the introductory chapter to Andrew Coggan's book, training and racing with a power meter. I believe Coggan may go into more detail in a later chapter, but I don't have the actual book.

Starting on page 9, Coggan described how normalized power is derived. Average power is one measure of how demanding a ride was, but a ride with very variable efforts is more physiologically demanding than a ride at the same power. In the lab, they can measure blood lactate level. Coggan proposed that blood lactate level is a better measure of physiological demand than power. Coggan explained:

This choice was made because many physiological responses (e.g., muscle glycogen and blood glucose utilization, catecholamine levels, ventilation) tend to parallel changes in blood lactate during exercise – in this context, then, blood lactate levels can be viewed as an overall index of physiological stress.

Note that lactate is a product of aerobic and anaerobic energy production. When you start exercising, your blood lactate level rises slightly. If you make a sustained effort under your threshold power, your blood lactate stabilizes. If you make a sustained effort above your threshold power, it rises rapidly. The theory above seems plausible to me (but I have no bio background).

Coggan and unnamed collaborators had a number of trained cyclists exercise in a lab at various effort levels, above and below their lactate threshold. The power you can produce at your lactate threshold is probably similar to your cycling functional threshold power.

After that, they had a number of datapoints for percent of lactate threshold and power. They used a technique like linear regression to relate the two variables. The relationship was not linear, and Coggan says:

Perhaps not surprisingly, an exponential function provided the best fit, but a power function of the following form proved to be nearly as good:

blood lactate (% of lactate at LT) = power (% of power at LT)3.90; R2 = 0.806, n=76

Based on these data, a 4th-order function was used in the algorithm for determining the IF (the exponent was rounded from 3.90 to 4.00 for simplicity’s sake).

So, to trust the concept of normalized power, you need at least two things. Most important, you need to trust that blood lactate level (normalized to % of lactate threshold) is a good enough proxy for physiological or metabolic cost.

This is probably less important than the above precept, but you also need to be able to trust the equation above that relates power to blood lactate level. It is possible that the equation they use is flawed, or that it is good in most conditions but flawed in some. A term of art used in (at least) statistics and econometrics is that the functional form of the model needs to be correct. Without access to Coggan's raw data and/or to other data obtained the same way, it's hard to refute the equation. Take note of R. Chung's comments in this answer and Adam Rice's; there are other formulae that may be better (NB: I have not assessed the evidence, I'm only saying that alternatives exist.) The R2 value of 0.8ish is high, which lends confidence.

Some other considerations are that the sample of cyclists needs to be representative. I believe Coggan's sample was trained cyclists. You can check the average VO2max of the sample if you can access the peer-reviewed research paper. If it's in the 40s, they're fairly fit. If it's in the 60s, they're very fit. Less trained individuals might have different physiological responses, e.g. maybe they are less efficient at clearing lactic acid, and thus NP might under-state the physiological demand on them. It's not possible to assess things like this from where I sit.

One issue I've become aware of is that, because of the 4th-order exponent, you're weighting power that's a lot above threshold very, very heavily. It's possible for a ride with a lot of sprints to have an NP that's much higher than you could sustain at steady state. This discrepancy may be worse for riders with high anaerobic capacity. Be aware of this. Related but downstream of NP, training load is calculated from an exponentially weighted average of your recent training stress (which relies on NP). Depending on the nature of my recent efforts, I sometimes feel like I needed to be better rested than the optimal training load (suggested by algorithm) would have me train at. So, there's a discrepancy there, but the issue could be the formula for NP. Or the formulae that calculate training loads might be inaccurate, or it could just my own response (e.g. my own curve is, say, 4.05 * percent of LT power vs. the population average 3.90.). Or it could be external factors like my own rest, which isn't observed in these models, and which practitioners understand that you have to account for anyway.

Aside from that, it is hard to answer the second part of the question because I'm not sure what to compare it to aside from straight average power or kJ work done, and what outcomes it might be used to predict (e.g. fatigue, but how do you set up the experiment, how do you measure fatigue, is that measure valid, etc). In general, average cyclists depend on the judgment of the people who created the formulas, and people had to exercise expert judgment all the way through the chain of evidence. It's hard for laypeople to parse the scientific evidence unaided. For example, if you have no subject matter background, can you reliably tell the if an objection raised by some random person who claims to have a PhD in something that may be unrelated is valid or not? Can you differentiate that critique from a person who has expertise in the field and who has a reasoned and nuanced critique?

I am not saying that you should not trust the concept of NP. A healthy stance is to generally accept the formulas, but be aware that they're based on population averages and on proxies for a concept. It seems like some people are contrarian by nature. This orientation can be helpful if you can offer a reasoned critique or alternative. It is possible for most of the field to be wrong, perhaps because that's where the general state of knowledge is (e.g. bloodletting used to be the clinical standard of care for a lot of things, although some cultures may have thought that it wasn't effective against the supernatural - I trust the point is clear).

As an example of different functional forms relating two physiological parameters, consider maximum heart rate, which can be used to pace efforts as well (NB: without a power meter, it would be better to find your threshold heart rate and use that rather than to try to find your max HR.) Many of us know 220-age - this is a formula used to approximate your maximum heart rate. This is something you'd get out of linear regression. When some researchers tried to find the academic source for 220-age, they found that the original source phrased their method a bit vaguely:

The formula maximum heart rate=220–age in years defines a line not far from many of the data points

Despite my last sentence, "not far from many of the data points" should sound a lot like least squares regression if you know that technique - it draws a line that minimizes the total of the squared distances between the line and each data point. In fact, the authors tried to replicate the regression result by approximately reproducing the data points in the article. They got 215.4 – 0.9147*age. Hence, 220-age was probably an approximation chosen for ease of use by non-scientists (e.g. personal trainers). Moreover, the article cites a number of other studies that tried to do the same thing in different samples, all of which gave different formulas.

As another aside, that formula actually gives you the average max HR for a certain age. Most of us will have higher or lower max HRs than the formula indicates. Some of us will have much lower or much higher max HRs. For example, the traditional formula gives a max HR of 180, and the slightly improved formula from the original dataset gives 179. I did a recent ramp power test, and my highest HR during the test was 192. That means that my actual max HR is at least 192. I know from other testing that my lactate threshold heart rate should be around 178-180 beats per minute, which is close to the average max HR for my age from the formula.

Going back to the concept of normalized power, all the formulas for normalizing power to lactate threshold will produce a formula based on the average experience of the people being studied. It might be possible that some of us (even those of us who are trained athletes) have physiological responses considerably different from that mean response. The more of an outlier you are, the less well any formula for NP will work for you. Of course, with NP, it's much harder for a person to determine if they're off from the mean response than with max HR.

• About a year prior to that Robergs citation from 2002, Gina Kolata from the NYTimes tracked down Haskell and Fox and asked them about the 220-age formula. It's very amusing, especially in context of the later Robergs paper: nytimes.com/2001/04/24/health/… Commented Nov 2, 2021 at 6:51
• As I mentioned above, there are many ways to produce an intensity-weighted mean of power -- NP is just one of them, just like there are many ways to calculate various means, medians, modes, trimmed means, for any distribution. NP happens to have become dominant, but there are alternatives if it doesn't do what you want. Commented Nov 2, 2021 at 7:02
• You may find this additional context interesting for your answer: NP is part of an ecosystem of measurements that Coggan developed. Although NP and FTP appear as the "base" of a pyramid of measurements (including IF, TSS, ATL, CTL, and the PMC), Coggan has said that he and Hunter Allen first had the idea for the PMC and then tried to figure out what they needed to get there, so they worked backward down the pyramid to NP. Commented Nov 2, 2021 at 7:12

I was very curious about where the 4th power came from. I thought it might be from some fundamental physics and/or physiology, but it doesn't. It's just the result of a curve fitting/regression exercise described below. My answer overlaps a lot with the one by Weiwen Ng, but since I'd already written it before I'd see that, I decided to go ahead and add it; also there are a few differences.

I read the read the report by Andrew Coggan that R.Chung provided a link to (thanks for that). I notice that is says that it was written as a chapter in the USAC coach's manual.

What is "lactate threshold" (LT)? It's included in the definition of "intensity factor" used in this report. Despite the fact that I know what lactate is, I didn't know the definition of "lactate threshold." I found the following definition or its equivalent multiple places online: "Your lactate threshold is the level at which the intensity of exercise causes lactate to accumulate in the blood at a faster rate than it can be removed, making it the border between low- and high-intensity work."

The author uses blood lactate level as a measure of intensity "intensity factor" (IF) and he includes a rationale for that. Actually he expresses it as a percentage of the "lactate threshold" (LT) for the individual rider. He measures that versus power---also as a percentage of "power at LT." As he explains, he did this for "a large number of trained cyclists exercising at intensities both below and above their LT." This apparently results in 76 data points (blood lactate, power). He then does a type of regression or curve fitting which results in the stated relationship.

Based on my experience doing this type of exercise, I'd guess guess that he does a linear fit (a straight line, in this case) of log(blood lactate level) (bl/bl0) versus log(power) (p/p0), where bl0 and p0 are the values of bl and p at LT). The slope of this fitted line turns out to be 3.90, which he rounds to 4.0. That's where the 4th power of power comes from.

IF = bl/bl0 = (p/p0)^{3.90}

Because bl and p are measured as percentages of their values at LT (call these bl0 and p0), this relationship reduces to "1 = 1" at the LT. (NOTE: The author expresses these two ratios as percentages but that just adds a constant and doesn't change the ultimate result.)

One thing that disappoints me is that he doesn't include the actual direct results of the measurement values he used for this fitting exercise, even in graphical form. I don't know if those can be found anywhere else online. I've ordered his book, so I'll soon know if they are included there. The only thing he includes along these lines is "R2 = 0.806." I assume that R2 really stands for R^2 (R squared), which usually represents the "Coefficient of determination" (see https://en.wikipedia.org/wiki/Coefficient_of_determination) one of the many goodness-of-fit measures. See: https://en.wikipedia.org/wiki/Goodness_of_fit).

A POINT OF TRIVIA: The way the adjective "normalized" is normally used, it's p/p0 that would normally be called "normalized power" and the thing that Coggan's equation computes is what Coggan himself calls "intensity factor," which makes sense. I'm sure that the name will never change however, considering how long it's been in use. That's why I call this comment "trivia."

• In an ideal world, there would have been a study to empirically demonstrate how well NP works as a training tool and that it's superior to other measures. Thinking about what it would take to do an acceptable job of doing that experiment however, I doubt that it will ever be done. I would guess that that NP is quite good for its intended purpose given (a) Coggan's rationale and (b) the numerous anecdotal reports I've seen relating positive experiences and advocating its use. Commented Nov 3, 2021 at 23:57
• The real trivia is that Andy originally called it "corrected" power. After it was pointed out that Andy's algorithm was mainly a version of the Lp-norm (en.wikipedia.org/wiki/Norm_(mathematics)#p-norm) where p=4, someone suggested to Andy that he rename it "normalized" power, and he accepted that suggestion. That someone was, um, me. I was surprised a few years later to learn that TrainingPeaks had trademarked the term since they hadn't come up with it and it was in common use before then. Commented Nov 4, 2021 at 8:21
• In addition, I had done some early analysis of how sensitive NP was to the exponent 4 (as opposed to, say, 3.9), and also how sensitive it was to different types of smoothing. As I've said above, there are several different ways to do intensity-weighting: Skiba's xP is a similar but related alternative. I decided at the time that intensity-weighting made sense: whether the Lp-norm is the "right" form, or whether p should be 4 was a detail I thought would eventually be resolved. I don't think it has been, yet. Commented Nov 4, 2021 at 8:25
• @R.Chung Do you have any sense why Coggan et al chose to use the exponent 4 vs 3.9, then? My naive guess is to ease computations by users in the field, like the max HR formula. The thing is, in the field in 2003, I assume they would have exported this to Excel, which won't explode if asked to compute x^3.90. Same with modern head units. I glanced at the Wiki article on Lp-norm, and while my understanding was limited, it didn't seem like p could only be integers. Commented Dec 3, 2021 at 0:59
• @WeiwenNg I don't know for sure but I have a supposition about it. That said, NP isn't terribly sensitive to p. Here's a plot I made of parameter sensitivity when Andy first proposed the idea (as mentioned above). The data are from Bjarne Riis' 1997 Amstel Gold race where his NP over the entire 6+ hour race was 356w. anonymous.coward.free.fr/wattage/if-tss/np-parameters.png As you can see, NP is a little more sensitive to width of the smoothing interval than it is to p=3.9 vs. 4.0. This is why Skiba's xP uses different smoothing but leaves p at 4.0. Commented Dec 3, 2021 at 7:14