# The time series of the number of goals scored by England against Scotland in soccer matches played…

Goals scored by England against Scotland

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The time series of the number of goals scored by England against Scotland in soccer matches played at Hampden Park in Glasgow is graphed in Figure 8.8. The matches have been played nearly every second year, with interruptions during the war years. We will treat the data y1, . . . , y52 as coming from an equally spaced time series model {Yt}. Since the number of goals scored is small (see the frequency histogram in Figure 8.9), a model based on the Poisson distribution might be deemed appropriate. The observed relative frequencies and those based on a Poisson distribution with mean equal to  = 1.269 are contained in Table 8.2. The standard chi-squared goodness of fit test, comparing the observed frequencies with expected frequencies based on a Poisson model, has a p-value of .02. The lack of fit with a Poisson distribution is hardly unexpected, since the sample variance (1.652) is much larger than the sample mean, while the mean and variance of the Poisson distribution are equal. In this case the data are said to be over dispersed in the sense that there is more variability in the data than one would expect from a sample of independent Poisson-distributed variables. Over dispersion can sometimes be explained by serial dependence in the data.

Table 8.2 Relative frequency and fitted Poisson distribution of goals scored by England against Scotland

 Number of goals 0 1 2 3 4 5 Relative frequency .288 .423 .154 .019 .096 .019 Poisson distribution .281 .356 .226 .096 .030 .008