Table of Contents

Chapter 2: Error Analysis

2.8 Exercises

  1. In each of the following cases, guess reasonable limits of uncertainty of the measurement process. In each case, is the limitation due primarily to a systematic or a random error? In each case, what would you do to improve the precision? The accuracy?

    2. (a) You measure the distance from New York, NY to Los Angeles, CA on your car's odometer.

    (b) You measure your pulse rate by counting heartbeats for 15 seconds.

    (c) You estimate an object's weight (its mass is something like a kilogram or two) by comparing its apparent weight, in your hand, to that of a "known" mass.

  2. A student measures a quantity 8 times, with the results

    3. 6.5   6.7   6.9   6.7   6.8   6.4   6.8   6.7

    Calculate the mean, the standard deviation, and the standard error of these data. (If the calculation is wired into your calculator, use it.)

  3. Calculate the mean, standard deviation, and standard error of the following ten measurements of the force constant of a certain spring:

    4. 86 85 84 89 86 88 88 85 83 85 N/m

  4. Ten measurements of the breaking strength of a steel wire yielded the following results (in tons weight):

    5. 5.3 5.6 5.7 5.9 5.6 5.4 5.4 5.6 5.9 5.7 N

    What is your best estimate of the breaking strength of this wire? What precision do you attach to this value? What is the precision of a single such measurement?

  5. Show that, if just two trials of a measurement are made,

  6. Measurements of the actual resistance of commercial, nominally 1200, resistors gave 1230, 1199, 1212, 1238, and 1187. Calculate the mean and the standard deviation of these values. Based on these values, how many more resistors from the same source would you expect to have to measure, in order to be able to quote the mean value with a precision of ± 0.5%?
  7. You make several measurements of the AC line voltage with a digital voltmeter, getting results which you summarize as 119.6 ± 1.3 V. Several readings with a second meter yield the result 123.1 ± 0.9 V. Discuss how you would proceed if you needed a result good to ± 0.5 V.
  8. An observer takes directional bearings (expressed as degrees east of true north) on a distant light source. His results are

    9. 11.05o 10.96o 11.36o 11.02o 11.16o 11.26o

    11.13o 10.85o 10.95o 10.92o 11.18o

    Compare the actual standard deviation of these data to the estimates of the standard deviation derived from the average absolute deviation and from the range of the data.

  9. The next night, the observer in Exercise 8 takes the following bearings:

    10. 10.80o 10.49o 10.91o 11.01o 10.68o 10.40o 11.24o

    Is he justified, do you think, in concluding that the light source has moved? Explain your answer.

  10. Another rule sometimes used to guesstimate the standard deviation of a data set, besides those discussed in this chapter, is as follows: if the data are distributed "normally," 86% of results should lie in an interval 3 r wide (that is within ± 1.5 of the mean). Thus if we strike out the most "extreme" 14% of the values, we can estimate s as 1/3 the range of what's left.  For the following set of time measurements

    11. 8.16 8.14 8.12 8.16 8.18 8.10 8.18 8.18

    8.24 8.16 8.14 8.17 8.21 8.17 8.12 8.15

    8.06 8.10 8.14 8.09 sec

    use this rule to estimate the standard deviation. Compare what you get to the standard deviation calculated directly from Equation (2).

  11. The diameter of a sphere has been measured as 0.89 ± 0.02 cm. Calculate thecorresponding value, and its standard deviation, of (a) the circumference (b) the surface area (c) the volume of the sphere.
  12. A land speed trial is conducted on a track whose length is 1.0000 ± 0.0002 miles. The time required to traverse the track is 13.591 ± 0.006 seconds. What is the speed of the car?  How precisely has it been measured? If you wanted to improve the measurement, would you concentrate on the timing equipment, or on the track length measurement?
  13. Ten measurements, with micrometer calipers, of the diameter of a straight round steel rod gave

    14. 0.452 0.450 0.454 0.446 0.451 0.455 0.448 0.451 0.449 0.450 cm

    and seven measurements of its length gave

    1.80 1.74 1.82 1.72 1.87 1.69 1.83 cm

    Find the volume of the rod, with its standard error.

  14. A certain quantity z = 4x, and you've measured x. One student, using Equation(8a), concludes that

    15. Another, noting that z = x + x + x + x , concludes from (11) that

    Which one of these guys is wrong, and what's wrong with his reasoning?

  15. One alternative index of the error in a result, that's sometimes used, is the "probable error" p. This is the deviation value such that the probability of a result falling within p of the mean value is just 50%. For normally distributed data, find p as a multiple of . Is p the most probable value of a deviation from the mean? If not, what is?
  16. Find 90% and 99% confidence intervals for the value of the diameter of the steel rod in Exercise 13, above.
  17. For the twenty resistance values given in Section 3.6, assuming a normal distribution, how many values are expected to fall in the range from 465 to 473? How many values actually do?
  18. The height of the mercury column in a manometer was measured ten times, giving

    19. 75.06 74.99 74.86 75.02 75.01 75.03 75.00 74.97 74.98 75.02 cm

    Test these data using Chauvenet's criterion to determine which, if any, should be discarded. After discarding any such, recalculate the mean value, and compare the change in the mean to the standard error of the data set.

  19. We have data on two quantities, x and y, between which we think there ought to be a linear relationship:

    20. x = 0.00 0.98 2.05 3.01 4.03 4.97

    y = 1.01 3.16 5.27 6.99 8.19 9.81

    Find the slope of the straight line y vs. x, and estimate the error in the slope.

Answers to the Exercises