Stein[25] in central Asia makes it clear that the contrast between the water supply about 200 B. C. and in the preceding and following centuries was greater than was supposed on the basis of the scanty evidence available when the dotted line of Fig. 4 was drawn in 1910.
Fig. 5. Changes in California climate for 2000 years, as measured by growth of Sequoia trees.
Fig. 5 is the same as the later portion of Fig. 4, except that the vertical scale has been magnified threefold. It seems probable that the dotted line at the right is more nearly correct than the solid line. During the thirty years since the end of the curve the general tendency appears in general to have been somewhat upward.
Since the curve of the California trees is the only continuous and detailed record yet available for the climate of the last three thousand years, it deserves most careful study. It is especially necessary to determine the degree of accuracy with which the growth of the trees represents (1) the local rainfall and (2) the rainfall of remote regions such as Palestine. Perhaps the best way to determine these matters is the standard mathematical method of correlation coefficients. If two phenomena vary in perfect unison, as in the case of the turning of the wheels and the progress of an automobile when the brakes are not applied, the correlation coefficient is 1.00, being positive when the automobile goes forward and negative when it goes backward. If there is no relation between two phenomena, as in the case of the number of miles run by a given automobile each year and the number of chickens hatched in the same period, the coefficient is zero. A partial relationship where other factors enter into the matter is represented by a coefficient between zero and one, as in the case of the movement of the automobile and the consumption of gasoline. In this case the relation is very obvious, but is modified by other factors, including the roughness and grade of the road, the amount of traffic, the number of stops, the skill of the driver, the condition and load of the automobile, and the state of the weather. Such partial relationships are the kind for which correlation coefficients are most useful, for the size of the coefficients shows the relative importance
of the various factors. A correlation coefficient four times the probable error, which can always be determined by a formula well known to mathematicians, is generally considered to afford evidence of some kind of relation between two phenomena. When the ratio between coefficient and error rises to six, the relationship is regarded as strong.
Few people would question that there is a connection between tree growth and rainfall, especially in a climate with a long summer dry season like that of California. But the growth of the trees also depends on their position, the amount of shading, the temperature, insect pests, blights, the wind with its tendency to break the branches, and a number of other factors. Moreover, while rain commonly favors growth, great extremes are relatively less helpful than more moderate amounts. Again, the roots of a tree may tap such deep sources of water that neither drought nor excessive rain produces much effect for several years. Hence in comparing the growth of the huge sequoias with the rainfall we should expect a correlation coefficient high enough to be convincing, but decidedly below 1.00. Unfortunately there is no record of the rainfall where the sequoias grow, the nearest long record being that of Sacramento, nearly 200 miles to the northwest and close to sea level instead of at an altitude of about 6000 feet.
Applying the method of correlation coefficients to the annual rainfall of Sacramento and the growth of the sequoias from 1863 to 1910, we obtain the results shown in Table 3. The trees of Section A of the table grew in moderately dry locations although the soil was fairly deep, a condition which seems to be essential to sequoias. In this case, as in all the others, the rainfall is reckoned from July to June, which practically means from October to May, since there is almost no summer rain. Thus the tree growth in 1861 is compared with the rainfall of the preceding rainy season, 1860-1861, or of several preceding rainy seasons as the table indicates.
| [TABLE 3] | |||||||
|---|---|---|---|---|---|---|---|
| CORRELATION COEFFICIENTS BETWEEN RAINFALL AND GROWTH OF SEQUOIAS IN CALIFORNIA[26] | |||||||
| (r)=Correlation coefficient (e)=Probable error (r/e)=Ratio of coefficient to probable error | |||||||
| A. Sacramento Rainfall and Growth of 18 Sequoias in DryLocations, 1861-1910 | |||||||
| (r) | (e) | (r/e) | |||||
| 1 year of rainfall | −0.059 | ±0.096 | 0.6 | ||||
| 2 years of rainfall | +0.288 | ±0.090 | 3.2 | ||||
| 3 years of rainfall | +0.570 | ±0.066 | 8.7 | ||||
| 4 years of rainfall | +0.470 | ±0.076 | 6.2 | ||||
| B. Sacramento Rainfall and Growth of 112 Sequoias Mostly inMoist Locations, 1861-1910 | |||||||
| 3 years of rainfall | +0.340 | ±0.087 | 3.9 | ||||
| 4 years of rainfall | +0.371 | ±0.084 | 4.5 | ||||
| 5 years of rainfall | +0.398 | ±0.082 | 4.9 | ||||
| 6 years of rainfall | +0.418 | ±0.079 | 5.3 | ||||
| 7 years of rainfall | +0.471 | ±0.076 | 6.2 | ||||
| 8 years of rainfall | (+0.520) | ±0.071 | 7.3 | ||||
| 9 years of rainfall | +0.575 | ±0.065 | 8.8 | ||||
| 10 years of rainfall | +0.577 | ±0.065 | 8.8 | ||||
| C. Sacramento Rainfall and Growth of 80 Sequoias in MoistLocations, 1861-1910 | |||||||
| 10 years of rainfall | +0.605 | ±0.062 | 9.8 | ||||
| D. Annual Sequoia Growth and Rainfall of Preceding 5 YearsAt Stations on Southern Pacific Railroad | |||||||
| Years | Altitude (feet) | Rainfall (inches) | Approximate distance from sequoias (miles) | (r) | (e) | (r/e) | |
| Sacramento, | 1861-1910 | 70 | 19.40 | 200 | +0.398 | ±0.081 | 4.9 |
| Colfax, | 1871-1909 | 2400 | 48.94 | 200 | +0.122 | ±0.113 | 1.1 |
| Summit, | 1871-1909 | 7000 | 48.07 | 200 | +0.148 | ±0.113 | 1.3 |
| Truckee, | 1871-1909 | 5800 | 27.12 | 200 | +0.300 | ±0.105 | 2.9 |
| Boca, | 1871-1909 | 5500 | 20.34 | 200 | +0.604 | ±0.076 | 8.0 |
| Winnemucca, | 1871-1909 | 4300 | 8.65 | 300 | +0.492 | ±0.089 | 5.5 |