What Is “Expected Average Heart Rate”?

In 2011 I began to record my average heart rate for nearly every run.  I wasn’t using my heart rate to do zone training or anything else; I just focused on the average heart rate.  It was just another piece of data that was easy to accumulate so, why not?   As I accumulated more heart rate data it occurred to me that I should be able to develop a formula to calculate my expected average heart rate for a run of a given pace.

The intent was that such a formula would provide insight into whether I had “good run” or a “bad run” on any given day.  For example, if my average heart rate was higher than expected for the pace, that would indicate I had a bad running day. While I didn’t have any scientific evidence for this, it just seemed like common sense to me that if my heart was working harder than what it normally does, then my body was not as efficient as normal.  This information could be useful to help me understand fitness prior to a race, possible overtraining, or problems that might cause me to lay off for a while.

The Current Formula

The formula for Expected Heart Rate is comprised of a constant, a pace factor, and a distance factor.  The formula can be written as follows:

Constant factor + pace in mph * pace factor + (distance adjustment) * (distance factor)

The pace factor accounts for the fact that the faster I run, the higher my heart rate will be.  The distance factor takes into account that as I run longer, my heart rate will be elevated for a longer period, thus increasing the average heart rate over the entire run. There is a constant because whatever speed and distance I run, my heart rate will be elevated above its resting rate.

As an example, here is the calculation for a 5 mile run with a pace of 8:20 per mile:

Factor	bpm
Constant	83.5*
Pace factor	62.6**
Distance factor	-1.125***
Total expected heart rate	145.0

* The constant does not vary by pace
** Pace is converted to mph and multiplied by a factor (7.1942 mph * 10.8)
*** The distance adjustment is 1.5*((distance/4) -2)

For a 5 mile run at 8:20 pace, I would expect a heart rate average of 145.0.   I then take the expected heart rate and divide by the actual heart rate to get the expected to actual ratio (I will refer to the ratios as HRR for brevity).  For example, if my actual average heart rate on this run was 144, I would take 145.0/144 to get a ratio of 1.01.  The higher the ratio, the better the run, according to this formula.

Note: The distance adjustment is made up of two calculations.  The first portion takes the distance divided by four, and then subtract 2 from that result.  For example, for a 4 mile run, this formula results in -2.  For a 12 mile run, the result is 1.  This result is then multiplied by the distance adjustment factor, which is 1.5 in the current formula.

Note: To convert from pace to mph, first, put the pace into decimal form.  For example, a run at 8:15/mile pace in decimal form is 8.25.  Then, take the reciprocal for the pace in decimal form and multiply by 60 (60 *(1/pace)).  This gives you the pace in miles per hour.  So, a run at 8:15/mile pace is equal to 7.27 mph.

Evolution of the Formula

The parameters I am using now are my fifth iteration.  When I originally developed the formula, I noticed my average heart rate for a 4 or 5 mile run at my typical 8:00 min/mile pace was normally right around 150 bpm. I wanted a formula that could reproduce that result.

The original formula had only a constant of 30 and a pace factor of 16.1118.

For a run at 8:00 min/mile pace (7.5 mph), this results in an expected HR of 150.8:

30 + 16.1118*7.5 = 150.8.

Using a factor with four decimal places definitely implies a level of precision that does not exist in this context!

Based on my database of average heart rates at the time, the fit looked decent.  When I calculated the ratios of expected average heart rate to the actual average heart rate, most of the ratios were close to 1.00, which is what I hoped for.

Adjusting the Formula for Pace and Distance

As I added to the database, I sorted my results based on pace and noticed that faster runs had higher ratios than slow runs which to me indicated that my formula was biased towards faster paces.  As a result, I tweaked the constant factor and the pace factor so the trend line was flat when sorted by pace.

The following graph shows the HRR vs. pace for the original set of factors and the current set.

When I sorted the results by distance I noticed that shorter runs had higher ratios than longer runs so the formula was biased towards shorter runs.  To compensate for this, I added a distance factor.  That adjustment was not sufficient so I added a multiplier to the distance factor which resulted in a flat trend line when sorted by distance.  The impact of the distance factor is to add .375 bpm per mile in excess of 8 miles or to subtract .375 bpm per mile for distances less than 8 miles.

 

Initial Results

Initially, I graphed the HRR on a daily basis.  However, as the number of data points increased, the “noise” increased due to day to day variations as can be seen in this graph:

I changed the graph to use a 10-day rolling average. I was not as concerned about day to day variations as I was in longer term trends.  The 10-day rolling average, in my opinion, shows long term trends more clearly as can be seen is this graph.

[In Part Two (coming soon) Paul will discuss what he has discovered using his Expected Average Heart Rate  data]

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Paul Grinvalds is a retired actuary living in Iowa. He started running in 8th grade and has run continuously, more or less, since then. Paul has run marathons in all 50 states in under four hours and is currently closing in on his goal of 100 marathons, having just completed #95. The statistics that are generated by running serve as one motivation (of many) to keep running on a regular basis.