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9.4 SYSTEMATIC ERROR (BIAS) Sometimes when we are measuring, a systematic error will creep into our results. For example, suppose we are using a scale that overweighs. That is, the scale systematically shows a weight that is more (maybe several ounces more) than the actual weight of the object we are weighing. In that case our measurements may be very carefully obtained (and hence their standard deviation may be small), but they are biased—the true weight is consistently lower than what is given by the scale. Thus the average of many such measurements, in spite of having a very small standard error, will display (a possibly large) measurement error due to bias. Consider the standard deviation of a set of numbers. Suppose a bias of 15 is added to each number to form a new set of numbers. Is there more spread or variation in this new set of data? Certainly not! Indeed, a little algebra can convince us that the new standard deviation of the biased measurements is the same as the old. First, note that average of (x Ⳮ 15)’s ⳱ (average of x’s) Ⳮ 15 Then the key reason is seen to be that the deviation values used to compute the standard deviation are unchanged: x ⫺ average of x’s ⳱ (x Ⳮ 15) ⫺ (average of x’s) Ⳮ 15 ⳱ x Ⳮ 15 ⫺ average of (x Ⳮ 15)⬘s As we recall from Chapter 2, the standard deviation (the square root of the variance) is computed by adding up the squares of the deviation values, which, as the above equation shows, are unchanged by the addition of 15 to each original value. Thus the sum of the deviation values squared for the x’s is the same as that for the the (x Ⳮ 15)’s. It then follows that the variances and the standard deviations are also the same. Suppose, for example, a newly minted penny is weighed on an unbiased scale (it does not measure consistently high or low) on 10 occasions, with these results (in decigrams): Stem Leaf 30 31 5, 6, 6 0, 0, 0, 2, 3, 4, 5 Key: “30 5” stands for 30.5 decigrams. Mean ⳱ 31.0 decigrams Standard deviation ⳱ 0.4 decigram If we then use another scale with a constant error of Ⳮ2 decigrams to weigh the penny, we would expect to get these results: Mean ⳱ 33.0 decigrams Standard deviation ⳱ 0.4 decigram The mean is increased by the constant error of 2 decigrams, but, as expected, the standard deviation of measurements remains the same. It thus becomes necessary to distinguish precision from accuracy, which is defined below in Section 9.5. The precision of a set of measurements is simply their standard deviation (either the theoretical standard deviation of the population of measurements or the estimated standard deviation of the set of measurements). If a sample mean of measurements is being used to estimate the true value of the quantity being measured, then the standard error of the mean becomes the precision of the mean. Note, however, that if a constant error, or bias, is present in a set of measurements, as with a defective scale, the precision of measurement will not be affected. The mean of the weights given by the scale will be increased by the constant error, but its standard error (that is, the precision of the mean) will remain the same. How, then, can we check a measuring device for bias? We cannot do it statistically. The sample mean and the standard error of the mean are both totally silent on this vital issue. One way to assess bias is to compare the measurement it gives with that given by a device known to be free of bias. For example, the time tone given by the Canadian Broadcasting Corporation from the Dominion Observatory in Ottawa provides an accurate reference point for correct time for the entire country. Another procedure for determining bias is to use more than one measuring device. A laboratory analyzing the amount of impurity in a drug might use two scales. If the two scales agree, that is a good indication (but not a proof) that both are free of bias. Both could, of course, have the same systematic error, but this is an unlikely situation and can usually be ignored. SECTION 9.4 EXERCISES 1. A thermometer with a known constant error of Ⳮ0.5⬚ F (which measures too high on average) is used to give 10 measurements of the temperature of a greenhouse. 80.9 78.7 79.7 79.5 79 81.1 80.6 79.2 78.9 80.4 Find the mean and standard deviation of these temperatures. Then correct the mean for the constant error of the thermometer. 2. A second thermometer is used to measure the temperature of the greenhouse of Exercise 1 and gives these results on 10 occasions (assume that the greenhouse temperature remains constant): 81.7 80.7 81.9 81.3 81 79.7 81.2 81.1 80.2 80.7 Which of the two thermometers appears to be more precise (have less error)? Does it appear that the second thermometer is biased? Hint: Recall that the thermometer in Exercise 1 is biased by 0.5⬚ F. 3. A chemist is not sure whether a certain scale is biased, so she uses two scales when weighing the amount of precipitate found in a flask. Her measurements are in the following table. Which of the scales appears to have greater precision? What do you believe to be the true weight of the precipitate? Why? Scale A 49.2 50.6 50.4 48.0 46.5 50.5 51.5 52.0 49.5 52.5 49.0 55.4 51.9 49.5 48.4 48.0 49.8 51.2 50.2 48.5 48.0 Scale B 54.1 52.3 49.6 50.5 48.6 52.0 49.6 51.2 51.1 4. Twenty measurements are made of the length of an insect (in millimeters): 41 38 37 38 40 39 39 40 42 39 38 37 42 40 38 42 35 40 39 38 What is the insect’s estimated true length? What is the estimated standard error? How could the precision of these measurements be improved? 9.5 ACCURACY After all this discussion about measuring an object, it is reasonable to ask, What is the real, or actual, height of the stack of books? What is the actual weight of the penny? What is the true value of the acceleration due to gravity in Ottawa? The actual length of an object can only be estimated. We can estimate more and more precisely by using a sample mean of measurements, but do we know that bias has been acceptably reduced or eliminated? No! Accuracy is defined to be the degree of closeness of a measurement to the true value. Accuracy is not the same as precision. Precision merely tells us that repeated measurements will all be quite close to each other (and that if we use a sample mean, it will be even more precise in the sense that if we imagined repeatedly constructing such sample means of observations, they would be even closer to each other). But if bias exists, the measurements may be unnacceptably far from the true value. In understanding the various contributions to measurement error from a statistical viewpoint, the following basic formula is very useful: Measurement ⳱ true value Ⳮ bias Ⳮ random error (random noise) Here the “true value” and the “bias” are fixed numbers; the “random error” has a theoretical mean of 0, and its standard deviation is the standard deviation of the population of measurement random errors, which, because the true value and the bias are not random, is the same as the standard deviation of the population of measurements.