How to report results, significantly.

In the natural sciences, no experimental measurement is perfect. There will always be some amount of *experimental error* associated with a result. This does not mean the measurement is necessarily *wrong*, but only that our tools and methods have limited precision. As scientists, we need to be able to explain these limitations and account for them when reporting scientific information.

For example, if you were to guess the age of the person closest to you right now. You would probably guess an interger number, say, 19 years old. But obviously the person is probably not exactly 19.0000 years old. They might be 19 and few months, or 18 with a birthday coming up, or 27 and just look really young. If you were to honestly report your guess, in a scientific fasion, you would want to include the range of possible ages the person might be: 19 plus or minus ($\pm$) 1 means they are likely 19, but you wouldn't be surprised if they were 18 and a bit, or 20 and few months. This is how we'll have to report our measurement from lab. With a *plus or minus* attached to them.
$$\textrm{Measurement} \pm \delta x$$
This $\delta x$ is our experimental uncertainty.

When you report the measurment, you'll need to take care to report it with the proper number of *significant figures*. By following the guidelines below for significant figures, we can be sure that we don't report infomation that we don't know. Some examples

1. The time it takes to fall $d$ meters in free fall will be given by:
$$t = \sqrt{\frac{2 d}{g}}$$
If $d = 3.25 $ m and $g = 9.81$ m/s^{2}, you would that a calculator says $t$ should equal:
$$\bbox[#FFD8DB,10px,border:3px solid red]{t = 0.8139958198 \; \textrm{s}}$$
However, this result has too many digits. Based on the values we have for $d$ and $g$, we should only include 3 significant figures in the results:
$$\bbox[#CFFFD9,10px,border:3px solid green]{t = 0.814\; \textrm{s}} \; \; \textrm{(correct sig. figs.)}$$

2. If you traveled in a car for 2.3541 hours at a rate of 12 miles per hour, you would find the distance traveled by multiplying: $$x = v t $$ The calculator would give: $$\bbox[#FFD8DB,10px,border:3px solid red]{x = 28.2492 \; \textrm{miles}}$$ But, since the time was only measured with 2 significant figures, we can only report 2 sig figs in the final answer: $$\bbox[#CFFFD9,10px,border:3px solid green]{x = 28 \; \textrm{miles}} \; \; \textrm{(correct sig. figs.)}$$