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Statistics: Decisions Through Data

UNIT 10: Exponential Growth

Summary of the Video

Some things—plants, small children—grow linearly; they add a fixed amount in each time period. But other things—like animal populations—grow faster; they are multiplied by a fixed amount in each time period. After the introduction, the video gives an account of the recurring gypsy moth outbreaks in the northeast. A graphic of the reproductive potential of a single pair of moths illustrates multiplicative growth. Actual data on acres of trees defoliated during an outbreak shows the same pattern: an exponential curve . After several years, natural limits (a viral disease) decimated the gypsy moth population and the exponential growth stopped.

A cartoon recounts an old fable to illustrate the eventually explosive nature of exponential growth. A clever courtier in ancient Persia invented the game of chess. The king was so pleased that he allowed the courtier to name his reward. He asked one grain of rice on the first square of the chessboard, doubling for each square. The king was surprised by the 20th square or so. An animated graphic then compares linear with exponential growth, using the chessboard as basis. One dollar on the first square and doubling each square thereafter catches a billion dollars added at each square by the 37th of the 64 squares.

Exponential growth appears in a more surprising setting: The growth of world oil production in this century was nearly exponential until the oil boycott and price shock of 1973. The curve looks exponential, but how can we tell? Taking the logarithm of the observations turns exponential growth into linear growth, and our eyes recognize a straight line more easily than they recognize an exponential curve.

Real data, like oil production, aren't exactly exponential. The exponential curve gives the overall pattern, but deviations from this pattern can be informative. Look at the graph of the logarithms. Fit a straight line to this graph. Now find the vertical distances of the actual data points from the fitted line. These are the residuals . A plot of the residuals against time clearly shows departures from the overall line (that is, from the overall exponential pattern in the original data). The Great Depression and World War II slowed the increase of oil production.

 

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