Practice question

For a spatially random Poisson pattern, expected variance equals the mean. The observed values, mean 5.3 and variance 5.05, give a variance-to-mean ratio of about 0.95, which is close to 1 and therefore consistent with random dispersion. A uniform distribution would show clear underdispersion, with variance substantially below the mean, while a clumped distribution would show overdispersion, with variance exceeding the mean. Small departures from unity occur through sampling error, so a ratio need not equal exactly 1 in finite data. A formal index-of-dispersion test can determine whether the deviation is statistically meaningful given the number of quadrats. Random placement implies that one individual’s location is largely independent of another’s at the scale studied, as might occur when resources are homogeneous and attraction or repulsion is weak. Spatial scale remains critical: a pattern classified as random with one quadrat size may reveal structure with another. Here the near equality of variance and mean supplies the intended evidence for randomness rather than regularity or aggregation.