Although this is not exact because of the use of average prices for intervals, it does closely represent the average price relative to time. There are two other averages for which time is an important element-the geometric and harmonic means.

Geometric Mean

The geometric mean represents a growth function in which a price change from 50 to 100 is as important as a change from 100 to 200.

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To solve this mathematically, rather than usmg a spreaddieet, the preceding equation can be changed to either of two forms:

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The two solutions are equivalent. Using the price levels in Table 2-1, disregarding the time intervals, and substituting into the first equation:

Had one of the periods been a loss, that value would simply be negative. We now perform the arithmetic to solve the equation.

The geometric mean has advantages in application to economics and prices. A classic example is to compare a tenfold rise in price from 100 to 1,000 with a fail to one-tenth from 100 to 10, An arithmetic mean of 10 and 1,000 is 505, while the geometric mean gives

which shows the relative distribution as a function of comparable growth. Due to this property the geometric mean is the best choice when averaging ratios that can be either fractions or percentages

Quadratic Mean

The quadratic mean is as calculated

The square root of the mean of the square of the items (root-mean-square) is most well known as the basis for the standard deviation. This will be discussed later, in the section "Dispersion and Skewness."

The harmonic mean is more of a time-weighted average, not biased toward higher or lower values as in the geometric mean. A simple example is to consider the average speed of a car that fravels 4 miles at 20 mph, then 4 miles

at 30mph. An arithmetic mean would result in 25 mph, without considering that 12 minutes were spent at 20 mph and 8 minutes at 30 mph. The weighted average would give

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This allows the solution pattern to be seen. For the 20 and 30 mph rates of speed, the solution is

which is the same answer as the weighted average Considering the original set of numbers again, the basic form harmonic mean can be applied:

We might apply the harmonic mean to price swings, in which the first swing moved 20 points over 12 daj-s, and the second swing moved 30 points over 8 daj-s

DISTRIBUTION

The measurement of distribution is very important because it tells you generally what to expect. We cannoi know what tomorrows S&P trading range will be, but we have a high level of confidence that it will fall between 300 and 800 points. We have a slightly lower confidence that it will varj from 400 to 600 points. We have virtually no chance of picking the exact range. The following measurements of distribution allow you to put a value on the chance of an event occurring.

Frequency Distributions

The frequency distribution can give a good picture of the characteriatics of the data. To know how often sugai prices were at different price levels, divide prices into 10 increments (e.g., 5.01 to 6.00, 6.01 to 7.00, etc.), and count the number of times that prices fall into each interval. The result will be a distribution of prices as shown in Figure 2-2 It should be expected that the distribution of prices for a phj-sical commodity interest rates (jield). or index markets, will be skewed toward the left-hand side (lower prices or yields) and have a long tail toward higher prices on the right-hand side. This is because prices remain at higher levels for only a short time relative to their long-term characteristics. Commodity prices tend to be bounded on the lower end, limited in their downside movement by production costs and resistance of the suppliers to sell at prices that represent a loss. On the higher end, there is not such a clear point of limitation; therefore, prices move much fiirther up during periods of extreme shortage relative to demand.

The measures of cenfral tendency discussed in the previous section are used to qualify the shape and extremes of price movement shown in the frequency distribution. The general relationship between the results when using the three principal means is

arithmetic mean > geometric mean > harmonic mean Median and Mode