## Estimation and Error Propogation

After using some photographs and some rough estimations to approximate the speed of a runner, I got to wondering “How important is the accuracy of the various estimations?”

In the first solution, I estimated that the height of the runner was **5.75 feet. **I then compared that to the runner’s height in **pixel measure**, created a **pixel-to-feet **conversion rate, and used this to calculate the distance the runner travelled during the **1.1 seconds **between the two photographs. Once that distance was determined, the runner’s pace could be estimated at around **9.86 minutes **per mile.

But what if I estimate of the runner’s height was off? And how off would it have to be to make a difference? Let’s say that the runner’ s height is actually **6 feet**, and so my original estimate was off by **.25 feet**. How will this affect the final result?

The runner’s rate was originally calculated to be **590 pixels-per-second**, and this is not affected by the estimate of the runner’s height. But if the runner’s real height was **6 feet**, then the pixel-to-feet conversion rate is really **380-to-6, **which means the runner’s rate translates to about **9.32 feet per second **(intead of **8.93 feet per second**, as originally calculated).

Following through with the calculations under this new height approximation, the runner runs one mile in **9.45 minutes**. Therefore, if my height estimate was off by **.25 feet, **then my minutes-per-mile estimate is off by about **.41 minutes**. In both cases, this represents a difference of about **4%**.

It makes sense that the **percentage** change is the same, as all the mathematical operations being done here are **linear**. The change in the input of **4% **just keeps getting passed through every process without alteration, and eventually comes out as a **4% **change in the **output**.

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