HW: 20 Measurements
Purpose of HW: To demonstrate the analysis of repeated measurements for standard deviation, normal distribution, blunders, increased precision in the mean.
Assignment:
1. Obtain a small ruler of about 6 inches in length. Choose or mark two well defined points on a wall, table, floor, or any surface about 6 to 8 feet apart from each other.
2. With the ruler, measure this straight line distance with the ruler. With a line of about 6 ft long, you'll have to lay down the ruler about 12 times to get one measurement total for the line. Keep the ruler "on line" by eye as well as possible. Record the total length in inches. When taking the distance reading, read the ruler to the first interpolated digit. For example if the ruler is in eights of an inch, interpolate to sixteenths.
3. Repeat the measuring process 20 times. Convert your measurements to decimal inches. This is your "sample" of 20 repeated measurements of a distance. Inspect your 20 values. No two values should be the same because of the random error of measurement and the fine reading tolerance. Most values should be clustered about the central value.
4. Calculate the sample mean, and standard deviation. Identify any blunders (outliers). If any readings, are outside a rejection limit of +/- 3* sigma, these are to be rejected.
a. Multiply your standard deviation by 3.
b. Add this 3sigma tolerance to your mean for the upper bound.
c. Subtract this 3sigma tolerance from the mean for the lower bound.
d. If you have any readings outside the upper and lower bound, then that is a blunder.
e. Take blunders out of your sample; recompute the sample mean and standard deviation. Any blunders remaining now outside of the new 3 sigma? The mean of this sample (free from blunders) is your best estimate of the actual distance. However, that mean also has a random tolerance.
5. Construct a "histogram" chart showing the distribution of "scatter".
a. Choose an "interval" width so you have 5 "bins" for your data. For example if your maximum reading is 72" and minimum reading is 71", each bin would be (72-71)/5=1/5=0.20"
b. Compute the interval (bin) limits. The central bin lower bound is the MEAN-INTERVAL/2. The central bin upper bound is the MEAN+INTERVAL/2. In the above example if the MEAN is 71.5, the central interval would have upper=71.5+.2/2=71.6. It's lower bound is 71.5-.2/2=71.4. Therefore the central bin goes from: 71.4 to 71.6. Add the interval to these limits for the limits of other bins:
Bin1 Lower 71.0 Upper 71.2
Bin2 Lower 71.2 Upper 71.4
Bin3 Lower 71.4 Upper 71.6 central
Bin4 Lower 71.6 Upper 71.8
Bin5 Lower 71.8 Upper 72.0
c. Count the number of your 20 measurements that fall into each bin – the frequency.
d. Plot the histogram: vertical axis = frequency, horizontal axis = bin limits (mean at center)
n
8
6
4
2
value
Mean
e. Analyze your histogram. Do you have a "normal" distribution?
1. Do you see a "bell shaped" curve in the distribution? (In general, yes, above)
2. Is the shape somewhat symmetrical about the mean? (In general, yes, above)
3. Is the center of the bell in the middle interval? (Yes, above)
6. Demonstrate the "law of compensation in the mean".
a. Calculate the standard deviation of the mean(SIGMAm) in your sample of repeated measurements.
b. Now measure the distance with a long tape to obtain an accurate "true value" of the distance.
c. Subtract your "true value" and your sample mean.
d. Compare this with the estimated "standard error of the mean". The two values should be in the same "ballpark" of value, but not identical.
e. Calculate the answer to the following question, "How many repeated measurements would be required to measure the distance within one one-hundredth of an inch (0.01") of true value?" To do this, solve the equation for standard error of the mean backward, setting the standard error of the mean to .01", setting sigma to your sample's standard deviation, and solving for N.
6. Prepare an MSWord *.doc report of this experiment. Discuss your procedures, calculations and results.
Submit: Attach your *.doc report to the HW assignment in your WebCT upload using the Assignment tool.