Introduction to Estimation—Mean
In the real world, characteristics such as the mean and standard deviation can only be approximated on the basis of experimentation. Such approximation is called estimation.
Estimating the Mean
Imagine there are millions of 550-watt sodium-vapor lamps in the world. Think about the set of all such lamps designed to operate at 120 volts AC, the standard household utility voltage in the USA. Imagine we connect each and every one of these bulbs to a 90-volt AC source, and measure the current each bulb draws at this below-normal voltage. Suppose we get current readings of around 3.6 amperes, as the ammeter in Fig. 5-3 shows. After rejecting obviously defective bulbs, each bulb we test produces a slightly different reading on the ammeter because of imprecision in the production process, just as jelly beans in a bag of candy vary slightly in size.
We do not have the time or resources to test millions of bulbs. So, suppose we select 1000 bulbs at random and test them. We obtain a high-precision digital AC ammeter that can resolve current readings down to the thousandth of an ampere, and we obtain a power supply that is close enough to 90 volts to be considered exact. In this way we get, in effect, a "perfect lab," with measurement equipment that does its job to perfection and eliminates human interpolation error. Variations in current readings therefore represent actual differences in the current drawn by different lamps. We plot the results of the experiment as a graph, smooth it out with curve fitting, and end up with a normal distribution such as the one shown in Fig. 5-4.
Suppose we average all the current measurements and come up with 3.600 amperes, accurate to the nearest thousandth of an ampere. This is an estimate of the mean current drawn by all the 550-watt, 120-volt bulbs in the world when they are subjected to 90 volts. This is only an estimate, not the true mean, because we have tested only a small sample of the bulbs, not the whole population. The estimate could be made more accurate by testing more bulbs (say 10,000 instead of 1000). Time and money could be saved by testing fewer bulbs (say 100 instead of 1000), but this would produce a less accurate estimate. But we could never claim to know the true mean current drawn by this particular type of bulb at 90 volts, unless we tested every single one of them in the whole world.
Estimating the Standard Deviation
In the above imaginary situation, we can say that there is a true distribution representing a graphical plot of the current drawn by each and every one of the 120-volt, 550-watt bulbs in the world when subjected to 90 volts. The fact that we lack the resources to find it (we can't test every bulb in the world) does not mean that it does not exist. As the size of the sample increases, Fig. 5-4 approaches the true distribution, and the average current (estimate of the mean current) approaches the true mean current.
True distributions can exist even for populations so large that they can be considered infinite. For example, suppose we want to estimate the mean power output of the stars in the known universe! All the kings, presidents, emperors, generals, judges, sheriffs, professors, astronomers, mathematicians, and statisticians on earth, with all the money in all the economies of humankind, cannot obtain an actual figure for the mean power output of all the stars in the universe. The cosmos is too vast; the number of stars too large. But no one can say that the set of all stars in the known universe does not exist! Thus, the true mean power output of all the stars in the universe is a real thing too.
The rules concerning estimation accuracy that apply to finite populations also apply to infinite populations. As the size of the sample set increases, the accuracy of the estimation improves. As the sample set becomes gigantic, the estimated value approaches the true value.
The mean current drawn at 90 volts is not the only characteristic of the light-bulb distribution we can estimate. We can also estimate the standard deviation of the curve. Figure 5-5 illustrates this. We derive the curve by plotting the points, based on all 1000 individual tests, and then smoothing out the results with curve fitting. Once we have a graph of the curve, we can use a computer to calculate the standard deviation.
From Fig. 5-5, it appears that the standard deviation, σ, is approximately 0.23 amperes either side of the mean. If we test 10,000 bulbs, we'll get a more accurate estimate of σ. If we test only 100 bulbs, our estimate will not be as accurate.
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