Statistical Data Analysis using "Sum" Function

Question

Most commonly when I hear descriptive data analysis using statistics these following functions are often inclded:

• Mean
• Standard Deviation
• Variance
• Range
• Mode
• Median etc.

Is the function "Sum" regarded as data analysis function or not? Any example of data analysis performed using this function alone?

Answer 1

Surely, the sum can be an important statistic or data point. One person's summary can be another person's data point.

Much depends on the nature of the variable, what is interesting or important, and what summation does or doesn't correspond to physically (biologically, economically, whatever).

Examples follow.

Population The population of a country is the sum of the populations of its constituent areas.

Income The sum of 3 incomes in one group and that of 7 incomes in another group could be interesting if group means say household and total household income is the focus. Otherwise, we might be as or more likely to focus on mean income in a group because the total depends, as a matter of definition, on the mean and the sample size. Knowing that the incomes of 1000 people exceed in sum those of another 500 people isn't enough information: mean personal income might be more important to an analysis.

Temperature The sum of 30 or so daily mean temperatures at a meteorological station isn't helpful except as a step to the mean temperature for a month, as temperature isn't additive.

Turn and turn about, there are critical temperatures, roughly speaking, for plants to start growing or for people to switch on the heating or air conditioning, so for temperatures $T$ and threshold temperatures $T'$ sums of the form $\sum |T - T'| \times [T > T']$ or $\sum |T - T'| \times [T < T']$ are used variously in ecology and engineering: the units of measurement would be degree-days, say $^\circ$C days. Here for example $[T > T']$ evaluates as 1 if true and 0 if false.

Rainfall More simply, the sum of 30 or so daily rainfalls is interesting and important as a step to the total rainfall for a month.

Answer 2

An example of a summary statistic that is only based on the sum without normalization is the Akaike Information Criterion: $$AIC = 2k - 2\ln(L) = 2k - 2\sum_{i=1}^n \ln(f(x_i))$$ where $f$ is the probability density of a model with $k$ parameters that has been fitted to the data. It is meaningless in itself, though, and only useful for comparing different models.

Another example is the Deviance for logistic regression: $$D=-2\sum_{i=1}^n \ln(P(Y=1|x_i))$$ where $P(Y=1|x_i)$ is computed from a model fitted to the data. In order to interpret the value (e.g. as a "pseudo $R^2$"), however, some normalization is subsequently necessary.

Answer 3

The summary statistics, like the ones that you mentioned, can be used for two purposes:

• to summarize the data sample that you gathered,
• you can use them as an estimate to infer something about the population the observations were sampled from.

The sum may be a useful quantity to calculate if you are interested in summarizing the data that you have. Sample sum would not be a good estimate of the population estimate, for example, if we are talking about non-negative numbers, their sum will increase with growing sample size, so the sum from a smaller sample would always be an underestimate. Notice that the other summary statistics that you mentioned (mean, mode, median, range, variance, etc) do not have this problem. If you would like to know what would be the sum of the population, you either need to collect the data for all of the population or have a statistical model that would extrapolate it from the sample, but the sample sum by itself won't help you to answer such question.