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VACETS Regular Technical Column

The VACETS Technical Column is contributed by various members , especially those of the VACETS Technical Affairs Committe. Articles are posted regulary on vacets@peak.org forum. Please send questions, comments and suggestions to vacets-ta@vacets.org

October 22, 1996

Statistical Notes #3 - On the Average

In the last two articles, having considered statistics as a tool or a formal system, I alluded to the abuse of statistics in the media and scientific research. Here, I will be more specific about the misuse (or potential abuse) of one of the most popular statistics, the average.

The word "average" has a root in Latin "havaria", which means the amount of compensational money, which each individual was called upon to pay in the event of a ship being wrecked. In modern English, "average" acts both as a verb and as a noun. Nowadays, the average is usually referred to as the arithmetic MEAN or simply the "mean" by scientists and statisticians.

More than a century ago, a French mathematician, Pierre Louis developed and advocated his "numerical method" for the appraisal of therapy. However, he was opposed by most of the leading clinicians of the day. Claude Bernard, arguably the father of modern experimental medicine, was critical of the application of mathematics in medicine; he declared: mathematicians AVERAGE too much and reason about phenomena as they construct them in their minds, but not as they exist in nature". He went on to urged doctors "to reject statistics as a foundation for experimental therapeutic and pathological science". Ironically, almost 100 years later, his disciples have abandoned his words almost completely. Statistics has become an indispensable part in medical research. Almost every medical publication nowadays includes "statistical method" section as to show its credence. So, what was/is wrong with the AVERAGE?

Regardless of what is wrong with this statistic, when confronted by a mass of numbers, many people find confortable refuge in the average. The average, as we realise, is meant to measure the central position of a group of individual values. People are quite happy that the average will represent the set of numbers (or observations); it can be calculated without any fancy arithmetic. The idea of average is so handy that it is not surprising that several kinds of average have been invented so that as wide a field as possible may be covered with the minimum misrepresentation. The average is so popular that every body understands averages - until they come to use them.

The average can be misleading even when it is correctly calculated. When all the values in a data set vary within a narrow range, the average is the best representative value. On the other hand, when values vary greatly among themselves, an average is paradoxically meaningless. The mean of a set of values:

        (a) 6, 5, 7, 6, 6, 5, 6, 7

is 6, and this value is clearly representative of the set. For the different set

        (b) 1, 2, 10, 9, 10, 4

the mean is alo 6, but this value is not truly representative at all. Of a further set

        (c) 1, 1, 2, 3, 2, 1, 11

the mean is 3, a value which entirely conceals the abnormal value 11.

The mean, by itself, fails to show up any information about abnormal values; and in many branches of science, it is often the abnormal values in which the statistician may be particularly interested. Nevertheless, if a mean is sufficiently representative of a set, then any sudden divergence from it will be emphasized by comparing each actual value to the mean value. In other words, the mean does provide a basis whereby abnormal values may be recognised.

The mean is a representative of set (a) because it is the result of averaging similar quantities; that is within a reasonably small range. The mean of (b) is not representative of the whole set because it is the result of averaging dissimilar quantities. The arithmetic is correct in each case, but the resulting mean of set (c) is meaningless as a representative value. Similarly, only the measurements of things with common identity should be averaged. It is not easy to define exactly what constitute common identity in this context and it is really a matter of common sense. For example, tigers and domestic cats are both members of the cat family, but it would be riduculous to compare statistics of one with the other or to aggregate the two sets of data, calculate the mean and then claim that the result constitute a statistical mean for the cat family in general.

In the same way, it is as meaningless to quote the value $500 as the mean income of three individuals whose separate earnings are $1000, $300 and $200. The earners are clearly in different classes and the mean quoted would appear to place them all in another quite different class. It is misleading in such circumstances to consider income as one indivisible subject for statistical analysis since it has too wide a connotation. Much the same principle is involved in averaging the shareholdings of the members of a limited company where there is a wide difference between the sizes of some of the holdings. A company whose share capital is $2000 and which has 100 members would have a mean holding of $20, but the real facts could be as follows:

        
        1 shareholder holds 1010 shares = 1010
        99 shareholders hold 10 shares each = 990

To quote the mean value of $20 is to suggest that the capital of the company is mainly held by small holders. The truth, however, is that one member holds more than 50% of the capital and therefore has control of the company. Furthermore, the calculated mean is twice as great as the individual holdings of 99% of all the members.

The average should never be accepted as significant without supporting credentials. The following table shows the trading results of a company by separate departments over two different years

   Department     Profit (+) and (-)
                  Year 1  Year 2
        --------------------------------
        A         +30     +15
        B         +20     -20
        C         -15     +1
        D         +40     +60
        E         -10     +9

        Average   +13     +13

If one accepts the two yearly means as showing that no changes have occurred between the two years, there is a shock awaiting anyone who looks back at the departmental details from which the means were derived. There has been a shifting of profitability as between departments and the real position is very different; yet comparison of the two means hides the change entirely.

The average can not be blindly employed in all sets of data. It can not, for example, be used to average rates of growth or rates of speed. A mean cycles to a point one km away at 2 km/h and returns over the same distance at 6 km/h. He therefore averages 4 km/h - or does he? Well he does not. It took him 30 minutes to cycle the first km and 10 minutes to return, so that in fact he cycled 2 km in 40 minutes and his average speed is therefore 3 km/h. The calculation of 4 km/h was derived by taking the mean of the two speeds (2 km/h and 6 km/h). The result is wrong because this in effect averages distance related to time, but the time element is different in each case. A correct method, however, is by calculating the HARMONIC mean, which is calculated by dividing the number of values by the sum of the reciprocals of the individual values. The correct average speed may therefore be calculated as being equal to:

        2 / (1/6 + 1/2) = 3 km/h.

The harmonic mean is always appropriate mean for rates and prices.

Still, there is another average called GEOMETRIC MEAN, which as the name implies, is appropriate for data that behave in geometric law. If a population was 2 million in 1980 and had grown to 4 million in 1990, it will clearly have doubled itself in 10 years. What would the population be in 1985? It would not be 3 million (= average of 2 mil and 4 mil), because the increase in population would have been more rapid in the later years of the period than it would have been in the earlier period. The real figure is approximately 2.8 million. This estimate is calculated from the geometric mean as follows:

        sqrt(2 x 4) = sqrt(8) = 2.8 mil (approx.)

The geometric mean is the result of an arithmetic process, but it is not the same process. The geometric mean is calculated by multiplying together all the values in a set and then extracting the nth root, where n equals the number of values. Thus, the square root is required to obtain the geometric mean of the above example.

At this point, it is worth to note that there are at least three kinds of averages, which are appropriate to different kinds of data. In fact, there are two other averages: the median and the mode, which are not mentioned in this article due to their simplicity. However, the inappropriate use of statistics in general, and average in particular, is very prevalenet in either popular media and scientific research. The medical literature is filled with an immense statistical information and endless contradictory findings. There are good evidences suggesting that the inappropriate use and wild manipulation of statistics have contributed significantly to this tragedy of confusion of knowledge. People are writing books and papers based on inappropriate application of statistics. Some of these authors are very popular because they are not afraid to provide solutions to problems that have not yet been solved. However, some investigators do not realise that they have made statistical errors due to either ignorance or lack of statistical knowledge. Whatever the reason, to rely on the analysis of data, the nature of which one does not understand, is the first step in losing intellectual honesty. Therefore, understanding the principle behind a statistical analysis is critical to the interpretation of data. I hope this little article helps you a little bit.

See you next time.

Technical Notes:

  1. The MODE is the sample values that occur most frequently. There may be more than one mode. There may be no mode at all if all the observations occur in equal frequency.
  2. The MEDIAN of a sample of N observations (data points) is that value of the variable with rank (50/100)(1+N). If the rank is not an integer, it is rounded yo the nearest half rank. In other words, it is a middle of a distribution. The median is thus a value where half of the observations fall below and half of the observations fall above it. For example, for the following 15 observations:

    16 22 23 26 26 27 28 30 31 35 36 37 48 50 52,

  3. the mean is 32.15 and the mode is 26 (it occurs twice). The median is the value with rank (50/100)(15+1) = 8; in this case the 8th observation is 30, i.e. the median is thus 30.

  4. There is an approximate relation between mean, median and mode. If the frequency distribution has one peak (unimodal) and moderately asymmetric or skewed, the following relationshsip holds:

    MEAN - MODE = 3(MEAN - MEDIAN)

    or MODE = 3*MEDIAN - 2*MEAN

    or MEDIAN = (MODE + 2*MEAN) / 3

    or MEAN = (3*MEDIAN - MODE) / 2.



  5. T V. Nguyen, Ph.D.
    t.nguyen@garvan.unsw.edu.au

    For discussion on this column, join vacets-tech@teleport.com


    Copyright © 1996 by VACETS and T V Nguyen

    :

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