Experiment 101

Experiment 101.01 REACTION TIME

PRELAB

PURPOSE

You will measure a quantity that is subject to random errors in order to apply some statistical concepts to the analysis of data.

EQUIPMENT

MouseRT application with MacPlus emulation w/mouse, clock or watch.

DISCUSSION

In science, an experiment is used to test the validity of a theory. Any experiment used to test theory usually involves the measurement of quantities whose values are predicted either directly or indirectly by the theory. However, since all measurements are subject to uncertainty, it is not enough to just make the measurements. A detailed evaluation of measurement uncertainties (or errors) and how much uncertainty they produce in the result is necessary to any test of a theory.

In the direct measurement of a physical quantity, such as the length of a rod with a vernier caliper, repeated measurements may give the same result. In that case the uncertainty of the measurement is set by the least count of the measuring instrument. That situation will be studied in Experiment 101.02 on Measurement.

This experiment will test your reaction time,in which repeated measurements do not produce the same result. Furthermore, the variation in the measurements is much greater than the least count of the measuring instrument. If these measurements are to be of any use, you need a way of reporting both the measurements of the reaction time and its variation. A histogram, or frequency plot, is a method of doing this graphically.

A histogram is a diagram drawn by dividing the original set of measurements into intervals of convenient width (or "bins") and counting the number of measurements (or "frequency") within each bin. For instance, when your instructor shows you a distribution of the grades in a class, a histogram will usually be displayed. If the number of measurements becomes very high (approaches infinity) and the bin size becomes very small (approaches zero), the histogram approaches a continuous curve which is called a "distribution curve".

Figure 01-1: Sample Histogram for 100 Measured Values

(bin size = 2.0)

In Figure 01-1, there are 8 events in the range 4 < x <6; 10 in the range 6 < x < 8; 27 in the range 8 < x < 10; etc.The histogram visually presents the measurement distribution but does not directly provide the best value of the measured quantity or the uncertainty. As you might suspect, the best value is near the middle of the distribution and the uncertainty is related to its spread. Statistical theory suggests the best value is simply the average of the measurements, .

where: N = the total number of measurements

x_i= the value of the measurement number i, where i is a number ranging from 1 to N.

EXAMPLE 1

For instance, suppose we have a collection of 10 measurements of a particular quantitity. They may have been measured by different people, or by using different instruments, or simply by the same person repeating the measurement 10 times. The data that are reported are: 33, 65, 72, 78, 64, 75, 86, 91, 52, 73, quite a wide variation. The average is found by adding them all up and then dividing by number of data points, namely 10. Try it --- you should get = 68.9.

The width of the distribution is related to the deviation of the individual readings from the average. In statistics, the variance is a measure of the deviation, and it can be defined :

EXAMPLE 2

This formula says that we must take the difference between each of the 10 grades and the average value of 68.9, then square each difference, then add up all of the squares, and finally divide that sum by the number of points less one, or 9 in this case. The numbers that get added are: (33 - 68.9)² + (65 - 68.9)² + (72 - 68.9)² + (78 - 68.9)²+ , and so forth. The final result for the sum is: 2540.9. Next, divide this result by 9 to get the variance = 282.3

The variance is proportional to the square of each difference and so has a different physical dimension than the average. The best way to remedy this problem is to simply take the square root of the variance to get the standard deviation. This number truly represents the width of the distribution of the data points.

EXAMPLE 3

This formula says that we must simply take the square root of the variance to get the standard deviation. The result is: standard deviation = s = 16.8. So, in this example the average is 68.9 and the other measurements are distributed around this best value with a peak that is about 17 points wide.

You can use your intuition about averages to understand these statistical quantities. Intuition says that is likely to be a "good" value for x if the individual measurements which are averaged are "clustered close together" in value. Secondly, an average generally improves if more measurements are averaged. Therefore, you can expect that the uncertainty in the average that you calculate should decrease as the number of measurements increases and as the width of the distribution decreases. The standard deviation s, is the statistical function which defines how "closely clustered" the measurements are. The value of s is related only to the width of the frequency distribution and is a measure of the average error for each measurement, no matter how many measurements are taken.

However, since increasing the number of measurements does not decrease the average error per measurement, s cannot itself be the uncertainty in the average value. The standard deviation of the mean, s_m, most closely matches the behavior of our intuitive feeling about what the uncertainty in the average value is. It is defined by taking the standard deviation and dividing it by the square root of the number of points taken. A fully mathematical formula is also presented so that you can see what the operations are: find the average ; subtract each data value from the average and square the difference; add up all the squared differences for each data point; divide this sum by the product of the number of data points times itself less one; finally, take the square root of the whole mess.

EXAMPLE 4

The standard deviation of the mean is found by dividing the standard deviation by the square root of 10, or 3.16. The result is: standard deviation of the mean = s_m= 5.3. This figure is an indicator of how much error we have when we quote 68.9 as the "best value" for the measurement.

Since s_m varies in inverse proportion to the square root of the number of data points that you take, we can decrease the uncertainty in the average either by increasing the number of measurements or by improving the individual measurements (decreasing the average error per measurement). The standard deviation of the mean is a good indicator of the "plus or minus" associated with a given set of measurements, that is, the confidence interval for the set of measurements. Statistical theory tells us that if we take the average value, plus or minus s_m, there is a 68% chance that the actual value will be within these limits. This is another way of saying that, statistically speaking, the true value of the measurement lies somewhere between 68.9 - 5.3 = 63.6, and 68.9 + 5.3 = 74.2. If we want more than just 68% confidence, then we can quote a confidence interval of plus or minus twice s_mto obtain 95% confidence. In our example therefore, it is appropriate to state that we have 95% confidence that the actual value of the quantity that we are measuring lies between 68.9 - 10.6 = 58.3 and 68.9 + 10.6 =79.5. These confidence intervals will get smaller as the number of measurements that we make is increased because s_m must necessarily decrease.

There is no need to be intimidated by the formulas given above. In this lab you will learn how to use the spreadsheet functions to do all of the mathematical "grunt" work for you. Just so that you understand what is happening when you choose a particular function, it is a good idea to go through these calculations manually just once.