Empirical Evaluation of Normality Tests and Heavy-Tailed Error Models for NMEA-Derived GNSS Positioning Data from a Low-Cost Receiver

Assessing whether datasets follow a normal distribution is essential for valid statistical inference, yet GNSS positioning errors often deviate from Gaussian assumptions under real conditions. Despite the widespread use of normality tests, their performance on GNSS data remains insufficiently characterized, especially under heavy-tailed regimes. This study investigates long-duration GNSS latitude residuals as a one-dimensional case using stationary observations collected over a continuous 48-h period. Normality is evaluated using graphical diagnostics, descriptive statistics, and hypothesis testing, including histograms, box plots, Q–Q plots, skewness, kurtosis, and seven parametric tests. To assess robustness and sensitivity, a Monte Carlo bootstrapping procedure is applied to more than 160,000 latitude samples. For multiple sample sizes and significance levels, 10,000 replicates are used to estimate empirical power and median p-values. The results indicate clear departures from normality, with Shapiro–Wilk and D’Agostino showing the highest sensitivity, particularly for small and moderate samples. GNSS latitude errors are additionally modeled using Student’s t distribution and a Gaussian–Student’s t mixture. These models better represent the empirical distribution, especially in the upper tail, than the Gaussian model. The findings confirm that Gaussian assumptions may underestimate uncertainty in GNSS analysis. They also show that combining normality diagnostics with flexible statistical models improves error characterization under non-Gaussian conditions.