Statistics for Environmental Engineers

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4.3 EWMA. Show that the exponentially weighted moving average really is an average in the sense that if a constant, say a = 2.5, is added to each value, the EWMA increases by 2.5.

5

Seeing the Shape of a Distribution


KEY WORDS dot diagram, histogram, probability distribution, cumulative probability distribution, frequency diagram.


The data in a sample have some frequency distribution, perhaps symmetrical or perhaps skewed. The statistics (mean, variance, etc.) computed from these data also have some distribution. For example, if the problem is to establish a 95% confidence interval on the mean, it is not important that the sample is normally distributed because the distribution of the mean tends to be normal regardless of the sample’s distribution. In contrast, if the problem is to estimate how frequently a certain value will be exceeded, it is essential to base the estimate on the correct distribution of the sample. This chapter is about the shape of the distribution of the data in the sample and not the distribution of statistics computed from the sample.


Many times the first analysis done on a set of data is to compute the mean and standard deviation. These two statistics fully characterize a normal distribution. They do not fully describe other distributions. We should not assume that environmental data will be normally distributed. Experience shows that stream quality data, wastewater treatment plant influent and effluent data, soil properties, and air quality data typically do not have normal distributions. They are more likely to have a long tail skewed toward high values (positive skewness). Fortunately, one need not assume the distribution. It can be discovered from the data.

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