Lab Manual Chapter 1-10 outline - University Of Maryland



Exercise 2

Measurements: Their Errors and Uncertainties

I. Purpose

The main purpose of this exercise is to teach you how to estimate the size of random errors and uncertainties when making measurements. You will learn what we mean by the “error in a measurement” and “uncertainty of a measurement” and techniques for reducing these uncertainties.

II. Reference

Read pages 1-21 from Lyons, "A Practical Guide to Data Analysis for Physical Science Students" or Chapters 4 and 5 from Taylor.

III. Equipment

For each person: stop watch, paper, PC with Excel

For the class as a whole: two setups of a pendulum with L about 1 meter.

IV. Introduction

When you make a measurement of a physical parameter with a measuring device, the measured value will not be perfectly accurate, nor will it be perfectly precise. A measurement is said to be accurate if the value is close to the true value, and a measurement is said to be precise if it is specified to many digits. The measured value will have some error and some uncertainty. The error means that it is not perfectly accurate, and the uncertainty means that it is not perfectly precise.

In this exercise, you will use a stopwatch to time some events. The stopwatch may run too slowly or too fast. Such behavior leads to an error in the time intervals that you measure with this stopwatch. This kind of error is called a systematic error because it originates in the system [the device or procedure] that you are using to make the measurement and it is not random from one measurement to the next. This is not the only systematic error that can happen. You, the operator of the stopwatch, will also not operate the watch perfectly - perhaps you always start the watch a little late - and so you may introduce additional systematic errors.

There also are randomly acting influences that can create uncertainty in the measured results, causing them to be too small sometimes and too large at other times. One type of random uncertainty comes from the limited precision of your measuring device. For example, in this exercise the stopwatch only displays the measured interval to a precision of 0.01 seconds. This limited precision leads to a random uncertainty in the measured time interval since you don't know the value of any of the digits beyond the hundredth place. Usually many random influences affect the measurement.

You can do something about random uncertainty and systematic errors. You can reduce the random uncertainty by doing the measurement many times and calculating the average value of the measurement. This average value has a smaller uncertainty than the individual measurements. You can also estimate the size of random uncertainties by taking many measurements and looking at the spread.

You can reduce the systematic error by calibration. By comparing your stopwatch with a clock that is known to be more accurate, you can find a correction factor that would allow you to correct the measurements that you made with your stopwatch, so reducing the systematic error. However, the correction factor will always have some uncertainty, so you will never be able to completely correct for systematic errors.

In this exercise you will not deal with the systematic errors or their uncertainties, but will concentrate on the random uncertainties. It should always be remembered though that, just because you are not dealing with the systematic errors, that does not mean that they have gone away. They are always there, even if you don’t check for them.

Open an EXCEL Spreadsheet and go to Sheet1.

V. Measurements

You’ll measure how long it takes for a pendulum to swing back and forth for three complete periods. In introductory physics, you may have learned that the period of a pendulum depends only on its length L and on the value of g, the acceleration due to gravity:

[pic]

Therefore, a measurement of the pendulum’s period should be pretty reproducible and shouldn’t depend much on things like the starting angle (for small initial angles).

A- Precision of Readings and the Random Uncertainty

Precision: Your measurements will have an uncertainty that is limited by the precision of the device you are using. For a digital device, the precision is approximately 1/2 of the increment in the last digit that the device exhibits. For example, suppose the last digit on a watch corresponds to seconds, and the watch displays every second (i.e. the increment in the last digit is one second). In this case, the precision is 0.5 seconds. For an analog device, a rough rule of thumb is that the precision is about 1/3 of the smallest division on the scale, as we will see in later labs.

You will be using a stopwatch to take measurements. Take a look at your stopwatch and figure out how to use it. Try starting and stopping it.

QUESTION 1: What is the precision of your stopwatch?

(Please answer “QUESTIONS” in your EXCEL spreadsheet. Put the answer in a “textbox”. To make one, look for a tool bar with an icon with the letter “A” and some horizontal wavy lines. If you do not see this icon, look under “View->Toolbars” and make sure “Drawing” is checked. )

Random Variations in Operation and the Random Uncertainty

Another source of uncertainty is the irreproducible behavior of the operator when using the device. This may have to do with your reflexes, your eyesight (good or bad), etc.

Try timing five 10 second intervals on the Lab clock with the stopwatch and see how variable your results are. Record your results in sheet one of your spreadsheet. Label your work.

QUESTION 2: Suppose you made several trials of the pendulum measurement. Would all your times be identical? Why or why not? If you and another student made measurements with the same pendulum swings, would your two measurements be identical? Why or why not?

QUESTION 3 Based on your answer to question 1, and your observations with the stopwatch, estimate the typical size of the difference between an individual measurement and the average of all the measurements? Explain, in 1-2 sentences, how you arrived at this answer, and why you chose this particular method.

B- Taking the Measurements

Your instructor will have a pendulum somewhere in the room. You will measure the time it takes the pendulum to complete three full periods, starting from the time it reaches its first maximum.

- Altogether, you’ll want 25 measurements. Try to take the best data you can for each measurement. Put your data into sheet 1 of your spreadsheet.

- Get your checksheet initialed now.

C. Analysis of Data

Go back to Sheet 1.

Visualizing the data

Make a histogram of your 25 trials. This will give you a feeling for the spread in your measurements. Follow the “Tip for Experts” from Exercise 1 to choose about 6 to 8 bins that cover the full range of your data. If you have an outlying data point at either the beginning or the end, ignore it when you are choosing your bins. The idea is to get a picture that lets you see the distribution of your data, something like the one below.

[pic]

From the graph, we can make a quick estimate of the uncertainty in the experimental data by looking at the “half width at half maximum” of the spread in the measurements. In the histogram, we look to see in which bins the frequency has dropped to about half of the peak. Here it’s about at 6.95 s and 7.15 s. We then take half of the difference between these two bins, or 0.1 s. This is a good first guess.

QUESTION 4: Without doing any calculations, look at your histogram and by eye estimate the size of the experimental uncertainty in your data.

Get your checksheet initialized now.

Average

(1) Find the average of all 25 measurements.

QUESTION 5: We call the average of all the measurements the “best estimate of the true value for the time interval”. Why is this average a better estimate of the time elapsed for 3 pendulum periods than any individual measurement?

Get your checksheet initialed now

Experimental Uncertainty

- By examining a data set numerically, you can get a better estimate of the uncertainty in a measurement. This is a useful skill, especially if you are looking at someone else's data and trying to figure out how precise it is.

- Make a copy of your list of 25 measurements, then sort the data from the one with the smallest difference from the average to the largest difference from the average. To do this, create a new column which contains, for each measurement, the absolute value of the difference between that measurement and the average value. Now we are going to sort the data according to distance from the mean. First, highlight the names, time intervals, and differences. Then, go to the DATA menu and choose “SORT”. Choose “no header row” and make sure you are sorting on the “differences” column. Once the data is sorted, to find the uncertainty, start with the data point closest to the average value and go to longer differences from the average time until you have bracketed 2/3 the total number of data points. The difference in time between this point and the average is an estimate of the uncertainty in the time interval measurements for trial 1. This procedure is very much like looking at one side of the histogram you made above.

QUESTION 6: Using the 2/3 rule described above, what is the experimental uncertainty for your measurements?

QUESTION 7: Compare your answer here to your answer to Question 3. Are they similar? If not, why not? If not, which do you think is a better estimate of how much the typical measurement deviates from the “true” time it takes the cart to traverse the ramp? If you had to do Question 3 over, how could you have improved your estimate?

- Get your checksheet initialed now

Deviation

A measurement’s deviation is the difference between it and the average [pic] (note: no absolute value here), where [pic] is the average time. Calculate the deviations for your measurements by inserting an additional column into your table. See the example on the next page.

- Calculate the average deviation

QUESTION 8: Is the deviation a good estimate of experimental uncertainty? Why or why not?

Get your checksheet initialed now

Root-Mean-Square Deviation (rms)

- Insert an additional column. In the new column calculate the squares of the deviations. Then calculate the average of the value of the squares of the deviation. The root mean square deviation (RMS) is defined as the square root of the average of the squares of the deviations. Calculate it using the SQRT function in Excel. The result should look as shown in the table on the next page.

|Pendulum Data | | |

|trial # |time (s) |Deviation |Dev2 |

|1 |7.06 |0.0612 |0.0037 |

|2 |7.03 |0.0312 |0.0010 |

|3 |6.78 |-0.2188 |0.0479 |

|4 |6.82 |-0.1788 |0.0320 |

|5 |6.97 |-0.0288 |0.0008 |

|6 |6.90 |-0.0988 |0.0098 |

|7 |7.00 |0.0012 |0.0000 |

|8 |7.00 |0.0012 |0.0000 |

|9 |6.93 |-0.0688 |0.0047 |

|10 |7.12 |0.1212 |0.0147 |

|11 |7.09 |0.0912 |0.0083 |

|12 |6.93 |-0.0688 |0.0047 |

|13 |6.97 |-0.0288 |0.0008 |

|14 |7.03 |0.0312 |0.0010 |

|15 |7.09 |0.0912 |0.0083 |

|16 |6.97 |-0.0288 |0.0008 |

|17 |7.03 |0.0312 |0.0010 |

|18 |7.03 |0.0312 |0.0010 |

|19 |6.97 |-0.0288 |0.0008 |

|20 |7.04 |0.0412 |0.0017 |

|21 |7.06 |0.0612 |0.0037 |

|22 |7.09 |0.0912 |0.0083 |

|23 |7.03 |0.0312 |0.0010 |

|24 |6.84 |-0.1588 |0.0252 |

|25 |7.19 |0.1912 |0.0366 |

|average |6.9988 | |RMS |

| | | |0.093 |

QUESTION 9: Compare your root mean square deviation to your result in question 5 and to the result you got in question 3. Is the root-mean-square a good estimator for the experimental uncertainty?

- It took a fair amount of work to get the standard deviation. Fortunately, Excel has a function called STDEV that calculates the standard deviation automatically, so you don’t need to go through the above steps. Figure out how to use STDEV, and then use it to calculate the standard deviation of your time intervals in Sheet 1. Note that EXCEL defines the function STDEV as [pic]. How does this compare to how you calculated it?

QUESTION 10: How does the value you got for the RMS in question 8 compare to the value you got using STDEV? How do these values compare to the value you got from your estimate from the 2/3 rule in question 5?

- Get your checksheet initialed now

Reducing the uncertainty by averaging:

In QUESTION 5, you probably answered that the average of several measurements is a better estimate of the true time for 3 pendulum swings than any individual measurement. Now we’re going to show that explicitly, and how much better your result gets when you average together 5 measurements.

Make a column numbered 1-5. In each bin adjacent to number 1, compute the average of the first 5 measurements, next to number 2, compute the average of measurements 6-10, etc., until you have 5 results, each of which is an average of 5 measurements.

Now compute the standard deviation of these 5 averages, using the STDEV function. We won’t prove this in Physics 174 (see your Lyons section 1.7.1 (d), equation 1.23, p 25, or Taylor section 4.4, p. 102), but if the uncertainties in the measurements are purely random, then the uncertainty in the average of N measurements gets reduced by the square root of the number of measurements.

[pic]

Thus, the uncertainty in each of your 5 averages should be smaller than the uncertainty in one measurement by [pic] . If you took the average of all 25 measurements, the uncertainty in the average of all 25 will be smaller by a factor of 5 than the uncertainty in any one of your measurements. Please note some texts and EXCEL define it as σaverage = σindividual / [pic].

QUESTION 11: How big to you expect the uncertainty in any one of your 5 averages to be? How does this compare to the STDEV of the 5 averages?

- email your spreadsheet to yourself and submit a copy to WebCT.

- Get your checksheet initialed now

VIII. Homework for Exercise 2

(Submit your completed spreadsheet, including homework to WebCT, If you run into trouble call or e-mail your instructor ASAP)

Complete any parts of the above exercise which you did not finish in class. Your homework should include any spreadsheets and plots that you create. It should also include the answers to any of the questions that start with QUESTION X (where X is a number). Please put the answers to these questions at the bottom of your spreadsheet, labeled clearly with the question number. Please use complete sentences when answering the questions. Also, answer the following questions.

QUESTION 12: A student does 25 measurements of the length of a table, and gets 3.12 m with a standard deviation of 0.02m. You do the measurement once and get 3.15 m. Is your measurement “wrong”? Why or why not?

QUESTION 13: We assume that the average of all the measurements is our best estimate for the “true” value of the quantity that was measured. Suppose one student does 5 measurements of some quantity, and another student does 25 measurements of the same quantity. Which student will have a better estimate of the “true” value? Why?

QUESTION 14: A student takes 25 measurements of the length of a desk, and gets distributions of lengths whose mean is 3.12 m, and whose standard deviation is 0.02 m. What is the uncertainty of the average? How many measurements would she have to make to get an uncertainty of 0.0002 m in the length of the desk? What types of systematic errors might influence this measurement. How will they affect her ability to achieve an uncertainty of 0.0002 m?

Mathematical Endnote for Experts

When you calculate the deviation, you calculated: [pic]

You then squared these deviations and got: [pic]

When you calculated the average, the sum of the squares was calculated: [pic]

Then the sum was divided by the number of students to get the average: [pic]

Finally the square root of this quantity was taken to get the RMS: [pic]

Actually this quantity is not quite correct. What we want is the standard deviation, which is discussed in Section 1.4 of Lyons and defined in equation (1.3’) on page 12. It’s very similar, and [pic]

The difference is not critical for our work in Physics 174 and you can use either. However, if you are being careful, you will want to be aware of the difference.

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