When a Straight-Line Trend Reaches Too Far
In “Why Extrapolation Is Risky,” we saw that a fitted line can calculate a prediction beyond the observed \(x\)-range without showing that the relationship continues there. A striking way to examine this problem is to look at athletic records. Record times have improved over many decades, so a line through past performances can suggest what future records might be. But a record trend is not guaranteed to keep improving at the same rate.
A record chart also needs careful interpretation. Each point is a record-setting performance, not a typical athlete’s time in that year. Records change in steps, when someone surpasses the existing mark; many years may pass without a change, followed by a new record. The people competing, training methods, equipment, track or course conditions, and measurement practices can also change. A straight line smooths over these details.
To examine a trend, it is helpful to express times in seconds. For example, \(3\) minutes \(59.4\) seconds is \(3(60)+59.4=239.4\) seconds. This makes subtraction and rates of change straightforward. If times decrease, the slope is negative: the model predicts that the record time becomes faster as years pass.
A line through two record dates is a simple way to illustrate extrapolation, but it is not the same as fitting a least-squares regression line to a broad set of observations. The examples below use historical records as endpoints, show exactly what a constant-rate projection would say, and then compare that claim with other evidence. They are illustrations of the limitations of long-range extrapolation, not claims that a particular official made each projection.
A Four-Minute Mile and a Short Trend That Did Not Continue
For years, running a mile in under four minutes was treated as an extraordinary barrier. Roger Bannister ran the mile in \(3:59.4\) in 1954. John Landy then set a faster record of \(3:58.0\), and Herb Elliott lowered the record to \(3:54.5\) in 1958. Those records show rapid improvement during that period, but they do not establish that the same rate would continue.
The four-minute mark is useful as a reminder that a striking threshold is not automatically a physical limit. Bannister’s performance showed that the threshold could be crossed. At the same time, a few rapid improvements should not be extended indefinitely as if runners would keep gaining the same number of seconds every year.
Worked Example: Extending the Early Mile-Record Trend
Historical example. The mile record was \(3:59.4\) in 1954 and \(3:54.5\) in 1958. Suppose we use just these two records to make a constant-rate projection to 1966. This calculation is a simplified endpoint-line illustration, not a regression based on every mile record.
Convert to seconds and find the rate. The 1954 time is \(3(60)+59.4=239.4\) seconds. The 1958 time is \(3(60)+54.5=234.5\) seconds. The time decreased by \(234.5-239.4=-4.9\) seconds over four years, so the rate is
The arithmetic can be checked by multiplying the annual decrease by four: \(-1.225(4)=-4.9\) seconds, the observed change between the two records.
Project to 1966. That year is eight years after 1958. Continuing the same rate would predict a time of
The calculation checks because eight years at \(1.225\) seconds faster per year implies \(9.8\) seconds faster overall. In 1966, Jim Ryun set a mile world record of \(3:51.3\), or \(3(60)+51.3=231.3\) seconds. The simple projection was \(231.3-224.7=6.6\) seconds faster than that actual record.
Interpretation. The line extends a steep four-year improvement into the next eight years. The later record was faster than the 1958 record, but not nearly as fast as the endpoint-line projection suggested. This is a concrete example of a rate observed over a short period failing to continue at the same pace.
A Long-Range Mile Projection Meets a Long Pause
Another comparison uses a much longer span. The men’s mile world record was \(3:59.4\) in 1954 and \(3:43.13\) in 1999. As of 2026, the record remains \(3:43.13\). These values let us test what a straight-line extension of the 1954-to-1999 change would have predicted for 2026.
Worked Example: Projecting the Mile Record to 2026
Find the endpoint rate. In seconds, the 1954 record was \(239.4\), and the 1999 record was \(3(60)+43.13=223.13\). The change was \(223.13-239.4=-16.27\) seconds over \(1999-1954=45\) years:
A check is \(-0.3616(45)\approx-16.27\) seconds, consistent with the change in the two endpoint times.
Extrapolate 27 years. The year 2026 is 27 years after 1999. Continuing the same rate gives
The calculation is consistent: the model projects an improvement of about \(9.76\) seconds over 27 years. But the actual record in 2026 is still \(223.13\) seconds, \(3:43.13\). The projected time is about \(223.13-213.37=9.76\) seconds faster than the record.
Interpretation. The projection is not evidence that a runner should have run \(3:33.37\) by 2026. It shows what the constant-rate assumption would calculate. In reality, the record did not improve during that period. This comparison does not prove that no future record improvement is possible; it shows that the historical average rate did not continue through 2026.
Using a Recent Record Trend to Stress-Test a Forecast
Long-range predictions can look plausible at first and still rely on an assumption that becomes less credible as the time horizon grows. A marathon projection illustrates how to check that assumption. The calculation below uses two real record times, but its 2050 result is a hypothetical extrapolation—not an established forecast.
Worked Example: Extending a Marathon Record Trend to 2050
Historical endpoints. The men’s marathon record was \(2:04:55\) in 2003 and \(2:01:39\) in 2018. Converting to seconds gives \(2(3600)+4(60)+55=7495\) seconds and \(2(3600)+1(60)+39=7299\) seconds. These times decreased by \(7299-7495=-196\) seconds over 15 years.
Calculate the annual rate.
Checking the rate by multiplying gives \(-13.0667(15)\approx -196.0\) seconds, matching the change between the records.
Project 32 years from 2018. Continuing the rate to 2050 would give
The time conversion checks: \(6880.87=3600+3240+40.87\), which is \(1\) hour, \(54\) minutes, and about \(40.87\) seconds. This is a mathematical consequence of the assumed rate, not evidence that a marathon record near that time will occur in 2050.
Evaluate the extrapolation. The endpoint rate is based on two records in a 15-year period, and the projection extends it another 32 years. Records do not improve by a fixed number of seconds every year, and many factors affecting performances may change. The result therefore depends heavily on extending a short-term average as if it were a stable long-term rate. The responsible conclusion is that the line calculates about \(1:54:41\), but the historical endpoints do not establish that this is a dependable 2050 prediction.
Why Record-Based Projections Can Break Down
These examples point to several reasons that a linear projection can diverge from later outcomes. They are useful context checks when deciding whether extending a trend is sensible.
- The observed rate may be temporary. A burst of improvement over a few years, as in the early mile example, does not show that the same yearly change will persist.
- Records are selective and irregular. A record chart tracks exceptional performances, not the full range of athletes’ times. A record can remain unchanged for years, then be lowered in one performance.
- The conditions can change. Training, competition, equipment, surfaces, and record-keeping rules can affect performances. A single line does not separate these influences or guarantee they stay constant.
- Real processes may have limits. A linear model predicts the same change per year however far it is extended. Human performance cannot be assumed to improve at a constant rate indefinitely.
- The endpoints may hide what happened in between. Two endpoints give an average change, not a complete description of the path between them. Several intermediate patterns can share those endpoints and lead to different projections.
These are reasons to question a distant projection, not proof that every extrapolation will be wrong. A record might improve after a long pause, for example. The important distinction is between what a line calculates and what the available observations justify saying about the future.
Common Mistakes and AP Exam Tips
- Calling every projected value a fact. Say “the line projects” or “the model calculates,” rather than claiming the predicted record will occur.
- Mixing minutes and seconds in a rate calculation. Convert each time to seconds before subtracting and dividing. A change of \(16.27\) seconds is not \(16.27\) minutes.
- Treating two endpoints as a well-tested trend. A line through two records represents their average change. It does not show that the change was steady between those dates or will continue afterward.
- Claiming that one later outcome proves what will happen forever. A stalled record shows that a particular constant-rate projection did not match that period. It does not prove that the record can never improve.
- Forgetting the response units. Record time is measured in seconds or minutes and seconds, not in “records per year.” Include the units when reporting a rate or prediction.
- Giving only the label “extrapolation.” A stronger answer identifies the observed time period, states that the requested year lies beyond it, and explains why a constant rate may not persist.
For a full-credit explanation, separate the calculation from its evaluation. State the projected time and show how it follows from the rate. Then compare the prediction with later observations when available, or identify a contextual reason the trend might change. As in “What \(r\)-squared Does Not Tell You,” a strong fit to observed data does not by itself guarantee a reliable prediction far beyond those data.
Check Your Understanding
Use the examples and principles in this tutorial to evaluate each claim.
- A record time changes from \(4:00.0\) to \(3:56.0\) over five years. Convert both times to seconds and calculate the average change per year.
- Why is a line through two record-setting performances not the same as evidence that the record improved steadily between those performances?
- A projection based on historical mile records gives \(3:33.37\) for 2026, while the record remains \(3:43.13\). What does this comparison show, and what does it not prove?
- Explain one reason why a marathon trend based on a short time period might not support a prediction several decades into the future.
- Write a cautious sentence describing a long-range record projection. Include what the model calculates and one limitation of extending the trend.