To your first question: In the sense that a trendline is a geometrical representation of certain summary information about data, the line itself constitutes "statistics" about the data, and those statistics are generated under some set of assumptions (e.g. ordinary least squares). Even when you are only interested in describing your data, you should be mindful of the assumptions your visualizations are built upon, and the (sometimes quite different) assumptions about your results that they are likely to engender in your readers.

Furthermore, there is an important point to make here about generality. If your focus is a single countable population of animals, and *only* those animals, then the use of estimation procedures is not necessary. However, most of the time, statistical undertakings assume that populations are infinite even when all present population members are observable. This is because treating all observed individuals as a "sample" allows you to make inferences about the *possible* populations (e.g. if the current generation had grown up under slightly different conditions). Ideally, we would like our theories to generalize to, say "all members of species X in region Y," and not merely to "the group of members observed between the following dates."

In keeping with this theme, and to answer your second question, you might consider looking into some of the non-parametric options for time series. Smoothing algorithms such as LOESS have their faults, but will do the job of "providing an aid to the eye" at least as well as fitting a trendline with no further inferential analyses. Kernel rate estimation (the time-series version of kernel density estimation) is another option. If you would like to put a confidence interval on your time series models, Bayesian Adaptive Regression Splines (BARS) provides a reasonably efficient alternative. Insofar as these methods are non-parametric or pseudo-parametric, they don't commit you as strongly to a particular set of model assumptions, making them better suited to the task than ordinary regression tools.

'Statistics' makes little sense for total counts only if the counts are REALLY TOTAL, i.e. you are 100% sure you have counted, say, ALL the antelopes in the national park. But if you in fact only SAMPLED the real quantity of interest (e.g., counted the antelopes in a given 10km*10km square) and try to extrapolate the measurement (e.g., to estimate the number of antelopes in the whole park by multiplying the antelope density by the park's area) then 'statistics' really kicks in and you have to estimate the possible sampling errors.

So the need for 'statistics' really depends upon what you measure and, more importantly, what you're going to use the measurements for. Even counting all the antelopes in the park may need 'statistics' if you are about to use the measurement not as the number of antelopes in the park, but as the size of the antelope herd (because the park is not walled-in on perimeter and antelopes may move in and out).

To your first question: In the sense that a trendline is a geometrical representation of certain summary information about data, the line itself constitutes "statistics" about the data, and those statistics are generated under some set of assumptions (e.g. ordinary least squares). Even when you are only interested in describing your data, you should be mindful of the assumptions your visualizations are built upon, and the (sometimes quite different) assumptions about your results that they are likely to engender in your readers.

Furthermore, there is an important point to make here about generality. If your focus is a single countable population of animals, and *only* those animals, then the use of estimation procedures is not necessary. However, most of the time, statistical undertakings assume that populations are infinite even when all present population members are observable. This is because treating all observed individuals as a "sample" allows you to make inferences about the *possible* populations (e.g. if the current generation had grown up under slightly different conditions). Ideally, we would like our theories to generalize to, say "all members of species X in region Y," and not merely to "the group of members observed between the following dates."

In keeping with this theme, and to answer your second question, you might consider looking into some of the non-parametric options for time series. Smoothing algorithms such as LOESS have their faults, but will do the job of "providing an aid to the eye" at least as well as fitting a trendline with no further inferential analyses. Kernel rate estimation (the time-series version of kernel density estimation) is another option. If you would like to put a confidence interval on your time series models, Bayesian Adaptive Regression Splines (BARS) provides a reasonably efficient alternative. Insofar as these methods are non-parametric or pseudo-parametric, they don't commit you as strongly to a particular set of model assumptions, making them better suited to the task than ordinary regression tools.

Interesting project.  The question does not mention whether you considered autoregressive moving average (ARMA) approach.

Dear Paul, your question(s) is(are) difficult.

When you study the total counts, it is like studying the cumulative growth, so maybe you can find an inflection point in your growth curve. I have studied such problems, see my article here:

https://www.researchgate.net/publication/228059400_Developing_methods_for_identifying_the_inflection_point_of_aconvexconcave_curve

and the revised version 2 here:

http://arxiv.org/abs/1206.5478v2

You can also use the R package inflection:

http://cran.r-project.org/web/packages/inflection/index.html

or you can download a relevant R code here:

https://www.researchgate.net/publication/235791114_Find_inflection_point_using_R

The two methods, ESE & EDE, are statistical estimators, so you use statistics, although it is not so evident (They are consistent estimators of the inflection point)

I think that in population analysis, consistency is the minimum requirement from our estimators.

As for the use of extended methods, I am a practical guy and, if I was in your position I'd prefer the consistent estimators than a complicated regression analysis.

For estimating maximum or minimum of a curve, see here:

https://www.researchgate.net/publication/261562841_Roots_extrema_and_inflection_points_by_using_a_proper_Taylor_regression_procedure?ev=prf_pub

I have not yet developed the relevant R package for that method. It is very simple and based in regression, if you prefer so.

For example, for the Table 1. Breeding parameters of a Black-headed Heron colony in Waza-Logone, N Cameroon it is found an inlfection point at ~1997.5, see attachment.

I found your work interesting,

My best regards,

Demetris Christopoulos

Article Developing methods for identifying the inflection point of a...

Data Find inflection point using R

Article Roots, extrema and inflection points by using a proper Taylo...

Answer withdrawn. To the downvoter, Just teach and apply the "wrong concept.

Since the referred post was withdrawn and the answer on this question was withdrawn, too, I withdraw the content of this post, too :)

Answer withdrawn. To the downvoter, forever live in your wrong view.

Paul, I believe what the critics of your work refer to as "statistics" describes ranges and model quality parameters.  Excel may not be the best tool for this, as by default it only gives you the R^2, and you have to compute the residuals and then get their distribution, and project that in order to have a good representation of the range of your forecasts.  I (we) have a paper on this topic in CMG'14 International Conference proceedings titled "Time Series Analysis: Forecasting + Regression: 'And' or 'Or'?"   Also on of my patents is dealing with model quality.  Links to both publications are in my profile.