Running Critical Speed & D′ Calculator
Enter two or three recent maximal running trials to estimate critical speed and D-prime from the distance-time model. The calculator also predicts a target-distance time and reverses the model for a target-duration interval.
Critical speed (m/s)4.36
- Critical speed (km/h)
- 15.71
- Critical pace
- 3:49/km, 6:09/mi
- D-prime distance capacity
- 204.02
- Model fit
- 99.99%
- Predicted target time
- 18 min 19 sec
- Target-duration distance
- 989.36
- Target-duration pace
- 3:02/km, 4:53/mi
Trial fit
| Trial | Distance | Time | Modeled distance | Residual |
|---|---|---|---|---|
| Short | 1,200 | 3 min 50 sec | 1,207 | -7.42 |
| Middle | 2,400 | 8 min 20 sec | 2,385 | 14.57 |
| Long | 3,600 | 13 min 0 sec | 3,607 | -7.15 |
Fit uses the linear distance-time model d = D-prime + critical speed x time.
How to use this calculator
- Choose whether you have two or three maximal trial results.
- Enter each trial distance in meters and each elapsed time as decimal minutes.
- Set a distance to predict for an all-out effort from the fitted model.
- Set an interval duration to estimate the modeled distance and pace for that effort.
How critical speed and D-prime are calculated
Critical speed is the slope of a runner's distance-time relationship over hard efforts in the severe-intensity range. D-prime is the intercept of that same line. In practical terms, critical speed is the sustainable part of the model, while D-prime is the finite distance capacity above that speed. Both values depend heavily on the trials used to fit them.
This calculator uses the linear model:
d_i = D-prime + CS x t_i
- d_i is trial distance in meters.
- t_i is trial time in seconds.
- CS is critical speed in meters per second.
- D-prime is the fitted intercept in meters.
For two or three trials, the calculator fits ordinary least squares. It first finds the mean time and mean distance, then uses CS = sum((t_i - mean t)(d_i - mean d)) / sum((t_i - mean t)^2). The intercept is D-prime = mean d - CS x mean t. With three trials, it also reports R^2 = 1 - residual sum of squares / total sum of squares. With two trials, the line always passes through both points, so the apparent fit is mechanically perfect and cannot diagnose whether the trials agree.
What moves the result most
The spread of trial durations matters. Trials that are too close together make the slope sensitive to timing error, GPS distance error, and pacing mistakes. A short, middle, and long maximal effort gives the regression more room to detect inconsistency. D-prime is usually less stable than critical speed because it is an intercept, so small changes in trial selection can move it a lot.
The calculator warns when a longer trial does not take longer, when average speed rises substantially with distance, or when a three-trial fit has an R-squared below 0.95. Those checks do not prove a result is good or bad, but they are useful signals to recheck effort quality, terrain, wind, heat, surface, elevation, or data entry.
Predictions and limits
For a target distance, the fitted line is rearranged to t_target = (d_target - D-prime) / CS. For a target duration, it uses d_duration = D-prime + CS x t_duration, then divides duration by distance to show average pace. Predictions far outside the duration range of the trials are flagged as extrapolations.
This model does not account for fatigue during repeated intervals, recovery kinetics, hills, wind, heat, drafting, altitude, or day-to-day changes in running economy. It is best treated as a training estimate from comparable maximal efforts, not as a guarantee of race performance or clearance for maximal exercise.
Worked example
Suppose a runner enters 1,200 m in 3.833 minutes, 2,400 m in 8.333 minutes, and 3,600 m in 13.000 minutes. The fitted critical speed is about 4.36 m/s, or 15.71 km/h. The corresponding critical pace is about 3:49 per kilometer and 6:09 per mile.
The fitted D-prime is about 204 m. A 5,000 m target gives (5000 - 204) / 4.36, or about 18 min 19 sec. A 3-minute interval gives about 989 m, which is roughly 3:02 per kilometer pace for that modeled effort.
Common questions
What are critical speed and D-prime in running?
Critical speed is the slope of the fitted distance-time line from maximal trials, expressed as speed. D-prime is the distance intercept of that line and represents a finite modeled distance capacity above critical speed. Both are model estimates, not direct physiological measurements.
Which distances should I use for critical speed testing?
Use recent maximal efforts that are long enough to stress severe-intensity performance and far enough apart to give the model a real span. Many runners use a short, middle, and longer trial rather than three distances clustered together. Keep terrain, surface, footwear, and weather as comparable as possible.
Can I calculate critical speed from two race times?
Yes, two trials are enough to draw the distance-time line and calculate critical speed and D-prime. The limitation is that two points always fit perfectly, so there is no independent check on whether one result was affected by pacing, measurement, terrain, or conditions.
What R-squared indicates a reliable fit?
For three trials, an R-squared near 1.00 means the three performances sit close to one line. This calculator warns below 0.95 because inconsistent efforts or measurement error can materially change the slope and intercept. A high R-squared is helpful, but it does not make extrapolated race predictions certain.
Why is my D-prime negative or unusually large?
A negative D-prime usually means the trial data are incompatible with the two-parameter model, often because the longer effort is too fast relative to the shorter one. An unusually large D-prime can come from trials that are too close together, inaccurate distances, nonmaximal pacing, or conditions that changed between efforts.
How can I use critical speed to plan intervals?
Use the target-duration calculation to estimate the modeled distance and average pace for a hard interval of that length. Treat it as a starting point, then adjust for recovery, workout purpose, terrain, fatigue, and how repeatable the session needs to be.