Why Take Averages in Experiments?

Date: 05/04/2003 at 23:15:58
From: Sara
Subject: Independent and dependent variables

I have to do a project using a toy car and a ramp. I have to measure 
the time it takes from when I release the car until it reaches the 
bottom of the (at least 6-foot) ramp. Time=dependent variable.
I must repeat the step at three other specified heights and determine 
the average of the dependent variable at each height.

How do I set this up?

Date: 05/05/2003 at 06:13:56
From: Doctor Luis
Subject: Re: Independent and dependent variables

Hi Sara,

What they want you to do is release the car several times for each
height, and measure the times it takes to go from the top of the ramp
to the bottom. 

The reason you want to repeat the experiment several times at each 
height is to ensure that you get a good value for the actual time it 
takes to come down. Whenever you take repeated measurements in an 
experiment, you'll find that your measurements will vary slightly. 
Many of these variations (errors) will be random, so you'll have no 
control over them. For example, the ramp could be slightly bumpy, so 
your car could slow down in some places, which would lead you to a 
higher time than you would actually get for a perfectly smooth ramp. 
You can't really prevent this kind of random error (we don't live in a 
perfect world with perfectly smooth frictionless ramps). These errors 
are called experimental errors.

Other errors will come from your technique in doing the experiment, so 
you must try to make these errors as small as possible. For example, 
if you're timing using your wristwatch your reaction time will 
contribute to the error in your measurement of time. Also, when you 
release the car you might inadvertantly give it a push that, however 
slight, will speed up your car by a bit and will lead you to a lower 
measurement. Or maybe you don't always release the car from the same 
spot, which will give the car a small head start in the race down, 
leading to a slightly lower measurement. These errors in technique are 
preventable, so you should try your hardest to make them as small as 
possible. These errors are called systematic errors.

So, given that errors will always happen, even if you try to repeat 
the exact same experiment under the exact same conditions (or as close 
as possible), what can you do?

You take averages.

You repeat your experiment lots of times until you reach a 
satisfactory value of your measurement. You see, if systematic errors 
are zero (or very close to zero), the only way you can get rid of 
experimental errors is by using statistical averages. This works 
because experimental errors are truly random, so even though they'll 
affect individual measurements you take, they won't affect the average 
of those measurements. (The average value of those random errors is 
zero.)

So, with your height as the independent variable, draw a table like 
the following:

 height (cm) |  times (sec)                   | average time (sec)
-------------|--------------------------------|------------
     10      |  100.0, 101.0, 99.8, 101, 101  | 105.6
     20      |  73.2, 73.1, 73.2, 73.2, 73.2  | 73.18
     30      |  61.5, 61.5, 61.4, 61.7, 61.4  | 61.5
    ...      |   ....
    ...      |   ....

To calculate the averages, just sum the measurements and divide by the 
number of measurements. For example, for the height 10cm, the average 
time is (100 + 101 + 99.8 + 101 + 101)sec/5 = 105.6 sec

Notice how for each value of the independent variable (height), I made 
five measurements of the dependent variable (time). You can take more 
measurements than that, or fewer. I think five is a good number, but 
you'll have to pick some number with which you feel comfortable.

As a final note, I made up the numbers in the table above. Your 
numbers should look similar but much smaller (I assumed a very long 
ramp). In general, the lower the height, the longer the car will take,
and the greater the height, the faster it should accelerate down the 
ramp. I don't know the length of the ramp you'll be using or the mass
of your toy car, so you'll have to take your own measurements.

I hope this helps you get started. Thanks for your question, and let 
us know if you have any more problems.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/