One of the organizations which has compiled the historic data on temperatures around the world and generated global averages is the Hadley Climate Research Unit (CRU) of the University of East Anglia. The data is presented in the form of anomalies. In meteorology and climatology anomaly just means the deviation from some base level, usually a long term average. Hadley CRU gives its global temperature anomalies in degrees Celsius.

In the graph there appears to be several episodes of increasing or decreasing temperature. These episodes appear to be linear. Furthermore the slopes of the upswings appear to be more or less equal. Likewise the slopes of the downswings appear to be equal. The periods of the upswings appear to be longer than the periods of the downswings. Underlying the upswings and downswings there seems to a long term trend.

A bent line can be fitted to the data using regression analysis. Such a regression function so fitted explains 85.1 percent of the variation in the average global temperature over the period 1850 to 2008. In this regression the slopes of the lines for the various episodes can be of any values.

The difference in the slopes for the upswings is not significantly different from zero at the 95 percent level of confidence. and likewise for the slopes for the downswings. Since the differences are not significantly different from zero it is appropriate to do another regression in which the slopes are constrained to be exactly equal for the upswings and also for the downswings. The graph of the data along with the regression estimates is shown below.

The coefficient of determination (R²) for this regression is 83.5 percent, nearly as high as the value for the unconstrained regression. This means the correlation between the regression estimate and the actual temperature anomaly is 0.914.

The measure of the statistical significance of the cycle is the t-ratio for the coefficient for the cyclic variable. The t-ratio is the regression coefficient estimate divided by the standard deviation. The value of that t-ratio for the cyclic variable is 14.2. (The t-ratio for the long term trend is 19.6.) The t-ratio has approximately a normal distribution for the 156 degrees of freedom for the data set. The probability of getting a t-ratio as large or larger from a situation in which its true value is zero is so infinitesimally small it is hard to describe it. In other words there is virtually no chance whatsoever that the 158 year cycle pattern arose just from chance.

Note the downturn for the last few years. If the cyclic pattern of the last 158 years continues, and there is no evidence that it will not, there will be a period of about 32 years during which the average global temperature will decline. The decline will be about 0.2°C over that period. The data and the projection forward to 2100 of the long term trend along with the cycle are shown below.

There is a long term trend. Its value is found by computing the slope of the line between two low points in the cycle or two high points. That value is 0.0054°C per year. This is equivalent to about 0.054°C per decade and 0.54°C per century.

The slopes for the two types of episodes are the sums of the cycle trend and the long term trend. Thus the slope during an upswing is 0.0147°C which is the sum of 0.0054°C for the long term trend and 0.0093°C for the cycle trend. Thus the upswing cycle trend slope is nearly twice as large the long term trend. The slope for a downswing is -0.0043°C which is the net result of a slope of -0.0093°C for the cycle trend and +0.0054°C for the long term trend.

The projection on the basis of the past cycle in global average temperatures is then a downswing from 2005 until about 2037 when the global temperature will be 0.0767°C below the 2008 level. From 2037 the temperature will rise until about 2069 when the temperature anomaly will be about 0.721°C, about 0.394°C above the 2008 level. From 2069 the temperature will decline and by 2100 the temperature will be only about 0.25°C above the 2008 level.

The computation of a margin of error for the projections is more complicated and will be dealt with in another webpage, but it can be said that the standard error σ of the estimate for the regression equation is 0.104°C, so ±2σ is ±0.209°C.

Conclusion

The Hadley CRU data show a cycle and long term trend for average global temperature that goes back 158 years. The long term trend is about 0.54°C per century. The cycle consists of upswings and downswings each of about thirty two years in length. The projection of the trend and cycle to the year 2100 indicates a moderate, noncatastrophic increase in global temperature.