# 400m Estimate Using Multiple Linear Regression

### Context

In 2018 I saw a post:

Predicting 800m Times
by Sam Harding

In it he used Multiple Linear Regression to perform more and more accurate 800m estimations. I had done something similiar in my article on 400m estimations:

Understanding Correlations Between Short Sprint Distance Ratios
by C. D. Chester

I found out very recently how to accomplish this in Microsoft Excel. You can watch how to do so here:

### Finished Product

Using data from 142 IAAF athletes (all of which have times for 100m, 200m, and 400m) I created this formula for estimating the 400m: $\large y = b (2.4202 + (3648.91/(625 a - 222 b)) - (13.2512/a)) - 0.4783 a - 11.233 \pm 5.7 \%$

Note: a is the 100m time, while b is the 200m; both are in seconds.

The original equation for this product is actually vastly different from when it started. Originally it had the 400m to 100m ratio and 400m to 200m ratio in it. Using Simple Linear Regression I estimated some simple, accurate equations for them (yields an error maximum of 5.15%). You can find the equations for them in this PDF:

All my other data can be found in these files:

Since I know someone is going to ask me how I came up with the above formula (from the summary Microsoft Excel gave) here's the alternative Wolframalpha produced:

The ± 5.7% comes from the original error in my 400m-200m and 400m-100m Ratio Formula. For instance, using Wayde van Niekerk's PR's my formula produces 45.62 (which is ~ 5.68% off), while my older ratio formula would yield 45.77 (which is ~ 5.97% off).

## Surprising Alternative

I ran the first one using the actual 400m to 200m ratios. Obviously that might annoy some people so I did it again with the estimation formula for the 400m to 200m and an interesting fact occurred: it eliminated the 400m to 200m ratio altogether and was still accurate (99.95%).

Here is that formula:

y=15.15824-1.3617(a)+2.8655(b)-30.1857(b/a)+((69094.902)(b/a))/(10000-3552(b/a))

In comparison to the other formulas this one yields ever so slightly lower than the others.

Original - 45.77 ( ~ 5.97%)
Real 400m to 200m Ratio - 45.62 (~ 5.68%)
Estimated 400m to 200m Ratio - 45.61 (~ 5.66%)

## Conclusion

I have to thank Sam for it was with his article that I produced this. Below is the calculator using the Estimated 400m to 200m Ratio formula.