What is average?

Overview

What is average? That's a common question, and the answer is not always so obvious as it might seem. There are several different kinds of average, and all of them are valid in the right situations. Various areas of biomedical equipment technology use different types of averages in different cases.

The word average refers to the most typical value, or most expected value, in a collection of numerical data. When you collect data, the results can vary from one observation to another in a number of ways (even when conditions are supposed to be the same).

First, of course, there is old-fashioned measurement and observational error. Not all rulers are truly the same, and not all applications of the same ruler to the same object turn out the same. Nor is it probable that even the same pair of perfect eyes will correctly read the scale every time a measurement is taken. In short, there will always be some variability in the measurements from one trial to another.

Next, there will be some actual variability in the events being recorded. Natural phenomena do, in fact, vary for one reason or another; blood pressure and heart rate may change slightly from heartbeat to heartbeat quite apart from any external factor. One way to handle these variations is to find the most typical value for the lot. Consider the case in which a student observed a red berry bush over a period of time. At one point, the observer counted 28 bunches of berries and found one to eight berries in the different bunches. What does average mean in this case? There are actually several different kinds of average, but the most commonly encountered are the arithmetic mean (usually called simply the mean), the median, and the mode. These are each different from the others, and all of them are correct averages when used in the right context. Let's look at the data a little more closely (Table 3-5).

The arithmetic mean is the type of average that most people use everyday. The mean is the sum of all values divided by the number (n) of different values. Or, to put it in proper form:The sum of all 28 values in Table 3-5 is 125, so what is the average?The mean average is 4.46, but don't expect to actually find a 0.46 berry. This average is the arithmetic mean.

The median is another type of average: it is the middle value in the data set; that is, the value at the point where exactly half of the values are above it and half are below it. In our berry example there are 28 values, an even number, so the median will be midway between two of them (with 14 above and 14 below). Figure 3-1 shows the data distribution in a crude kind of bar graph. Count the X's in each category from one end to the middle, and then the other end. Note that there are 14 values between 0 and 4 and 14 values from 5 to 9. That means the median value will be half-way between 4 and 5, or 4.5. If there were an odd number of data points, then the middle point—the median—would be the actual data point that has an equal number of points above it and below it.

The mode is also an average of sorts and is defined as the most frequently occurring value in the data set. In the above data (Table 3-5), the mode is easily seen in the X-chart. There were more bunches with five berries than any other number, so 5 is the mode. So now we have a arithmetic mean of 4.46, a median of 4.5, and mode of 5, and all are the average of the same data set depending on the definition of average.


Figure 3-1 Data distribution (X chart) for 28 data values. Source: Carr, J. J., Elements of Electronic Instrumentation and Measurement, (EEIM), Prentice Hall (Englewood Cliffs, N.J., 1996).

Different averages are used for different situations. If the data is perfectly symmetrical, then the mean, median, and mode are the same number. In fact, that's nearly the case in the data above. If th mean, median, and mode are not the same, the the data is not symmetrical around the mean, and the difference is a test of that symmetry. In th berry bush data, the distribution is nearly symmetrical, so the mean could be used. But in other situations the mean is not terribly useful, especially if one or two data points have very large or very small values compared with the rest of the data.

The mean is best used when the measuremer data is symmetrical, the median is often used when the data is highly asymmetrical due to outliers, and the mode is used to answer questions such as, What is the most common cause of death? or What is the most popular TV show on Friday night?

Other averages include the geometric mean and the harmonic mean. The geometric mean is often used when the data is not very symmetrical, especially in biological studies. For example, suppose you have $48 to spend, and you spend half of your available money each day for five days. The data would tabulate as shown in Table 3-6 The arithmetic mean of the five values is:
If we graph these values (Figure 3-2a), we note that the line connecting the tops of the bar graphs is not straight. To find the geometric mean, we need to find the logarithm of each value, add the logarithms up, and then take the logarithmic mean. Then we take the antilog of the log-mean. The log-mean is:Now take the antilog of the answer:The logarithmic chart (Figure 3-2a) is not a straight line. If we want to straighten that line, we use semilog paper (Figure 3-1b) or make the translation with a computer graphing program.

The harmonic mean is a bit more complicated and is used when the data is expressed in ratios, such as miles per hour or dollars per dozen. The expression for harmonic mean (H.M.) reflects the fact that it is the reciprocal of the mean of the reciprocals of the data:Suppose we compare the price of eggs in the local store over one past month (Table 3-7):
The arithmetic mean is:but the harmonic mean is:

3-7-1 Root mean square and root sum squares average
Other averages are sometimes seen in biomedical science, engineering, and technology: integrated average, root-mean-square (rms) and root-sum-squares (rss). The integrated average is the area under the curve of the function (Figure 3-3) divided by the segment of the range over which the average is taken:The integrated average is often found in electronic circuits in which either a resistor-capacitor (RC) low-pass filter or an RC operational amplifier circuit, called a Miller integrator, is used. This circuit has an RC product much greater than the period of the applied input waveform. The output of the circuit is proportional to the time average of the input signal. In computer-based instruments the integrated average is found either by using a numerical integration algorithm or by adding up a large number of sequential samples, spaced equally in time, and then dividing by the total time period.


The rms value is used extensively in electrical circuits and certain other technologies. For example, a sinewave alternating current (ac) wave may be compared with the direct current (dc) voltage level that will produce the same amount of heating in an electrical resistance. The value of the ac wave that is the dc heating equivalent is the rms value. The definition of rms is:where

For the special case of the sinewave, the rms value of voltage is

where V_P is the peak voltage (Figure 3-4). For waveshapes other than sinusoidal, however. Equation 3-7 will evaluate differently.


The rss average is used when pieces of different data are combined to form a single number even though they are in no way correlated with each other. For example, noise signals in electronic circuits are errors and may come from several different independent sources. Suppose we have n independent noise voltage sources (vn₁, vn₁, ••• vn_n). Where these sources are truly independent of each other, they are decorrelated and therefore cannot be simply combined in a linear additive manner but rather must be combined using the rss method:The rss method is used sometimes to define a single valued error term from a number of uncorrelated error terms or, alternatively, to find a single standard error of a number of measurements of the same value.

Example 3-1
An electronic amplifier circuit contains five independent noise sources that produce the following decorrelated noise signal voltage levels: VN1 = 25 nanovolts (nV); VN2 = 56 nV; VN3 = -33 nV; VN4 = -10 nV; and VN5 = 62 nV. What is the rss value of a composite noise signal?Note that the rss value in the example above is not the same as the summation of the components.

Although it is common practice in biomedical measurements, and in science experiments in general, to quote the average value of the data acquired, one must be cautious in using the correct average (or the most reasonable average) and to correctly interpret what average means in context.

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