# Benford’s Law for Telemetry Data of Wildlife

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{10}is decadic logarithm):

## 2. Materials and Methods

#### 2.1. Data

^{2}at altitudes between 880 and 3000 m above sea level in the valley of Kaprun (province of Salzburg, Austria). It was adjacent to the core area of Upper Tauern National Park.

_{i}, y

_{i}, z

_{i}), the time (t

_{i}), and (starting with row 2) the distance (2) to the previous position. The leading nonzero digits of the successive distances were also computed in the spreadsheet. If the time between two consecutive measurements exceeded one day, this distance was discarded. Distance 0 (one observation) was ignored.

#### 2.2. Statistics

^{2}, defined by the equation:

^{2}and for each simulated BL-distributed logbook, we computed the p-value from the chi-square distribution with eight degrees of freedom and used it to check if BL was refuted (i.e., p-value below 0.05). To this end, we counted how many of the 1132 simulated logbooks refuted BL. We repeated this simulation 500 times. We thereby obtained a sample of 500 counts. We compared this sample with the observed count of refutations of BL for our logbook data. To this end, we estimated a Poisson distribution for the sample of 500 counts, verified its goodness of fit using probability plots [45] and statistical tests [46] implemented in a Mathematica [37] function (DistributionFitTest), and used this distribution to compute the p-value for the hypothesis of a higher number of refutations than observed for the animal logbooks. For p < 0.05, we concluded that the observed data were unlikely to fulfil BL.

## 3. Results

^{−30}for a higher χ

^{2}than observed), whereas in terms of MAD = 0.0048, the deviation of the data from BL was barely discernible. However, in a simulation of 500 BL-distributed samples of size 73,393 each, much smaller values of MAD were observed; 95% of the simulated MAD-values were below 0.00131. Fitting a lognormal distribution to the sample of simulated MAD-values, MAD > 0.0048 was unlikely (p-value 4.6 × 10

^{−10}).

^{−7}). Thus, although the chi-square test supported BL for most weekly samples, the number of observed refutations was too large to be explained by spurious tests alone.

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Probability plots comparing the simulated counts of refutations for BL-distributed logbooks with a Poisson distribution for (

**a**) refutations using χ

^{2}; (

**b**) refutations using YCC; plots using Mathematica [30].

Digit: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Observed: | 21,473 | 13,461 | 9516 | 7669 | 5941 | 4691 | 3976 | 3576 | 3090 |

Theoretical: | 22,093.5 | 12,923.8 | 9169.6 | 7112.5 | 5811.4 | 4913.4 | 4256.2 | 3754.2 | 3358.3 |

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**MDPI and ACS Style**

Pröger, L.; Griesberger, P.; Hackländer, K.; Brunner, N.; Kühleitner, M.
Benford’s Law for Telemetry Data of Wildlife. *Stats* **2021**, *4*, 943-949.
https://doi.org/10.3390/stats4040055

**AMA Style**

Pröger L, Griesberger P, Hackländer K, Brunner N, Kühleitner M.
Benford’s Law for Telemetry Data of Wildlife. *Stats*. 2021; 4(4):943-949.
https://doi.org/10.3390/stats4040055

**Chicago/Turabian Style**

Pröger, Lasse, Paul Griesberger, Klaus Hackländer, Norbert Brunner, and Manfred Kühleitner.
2021. "Benford’s Law for Telemetry Data of Wildlife" *Stats* 4, no. 4: 943-949.
https://doi.org/10.3390/stats4040055