Let’s talk about two of the common approaches to statistics today.

1. Probability:
   • Frequentist Statistics (FS), probability is interpreted as nothing but the long-run frequency of events in a repeated, hypothetical infinite sequence of trials. It’s based on the idea of objective randomness.
   • Bayesian statistics (BS) views probability as a measure of belief or uncertainty. It incorporates prior beliefs and updates them based on new evidence using Bayes’ theorem.
2. Parameter estimation:
   • FS: The focus is on estimating fixed, unknown parameters from observed data. This estimation is done using methods like maximum likelihood estimation (MLE).
   • BS: Bayesian inference provides a probability distribution for the parameters, incorporating prior knowledge and updating it with observed data to get a posterior distribution.
3. Hypothesis testing:
   • FS: Frequentist hypothesis testing involves making decisions about population parameters based on sample data. It often uses p-values to determine the level of significance.
   • BS: Bayesian hypothesis testing involves comparing the probabilities of different hypotheses given the data. It uses posterior probabilities and Bayes factors to make decisions.

I used a Bayesian t-test strategy to take this prior knowledge into account because I firmly believe that the difference in average ages is approximately 7 and that it is statistically significant. The results, however, revealed an intriguing discrepancy: the observed difference was located towards the posterior distribution’s tail. I did not appreciate this disparity and how it demonstrated how sensitive Bayesian analysis is to previous specifications

To investigate the difference in average ages between black and white guys further, I used frequentist statistical methods such as Welch’s t-test in addition to the Bayesian approaches. The Welch’s t-test findings showed a very important finding:

 

In addition to the Bayesian studies, this frequentist method consistently and clearly indicates a large difference in mean ages. My confidence in the observed difference in average ages between the two groups is increased by the convergence of results obtained using frequentist and Bayesian approaches.

The investigation of average age differences reveals a complex terrain when comparing the Bayesian and frequentist approaches. Frequentist methods give clear interpretability, while Bayesian methods allow flexibility and measurement of uncertainty. The tension seen in the Bayesian t-test highlights how crucial it is to carefully specify the model and take previous data into account. Future analyses can gain from a comprehensive strategy that combines frequentist and Bayesian methodology’s advantages.

The problem in visualising a big collection of geocoordinated police shooting incidents is to effectively transmit information without overloading the viewer. First, I tried using the Folium library to create separate markers on a map to represent each incidence. But as the dataset expanded, the computing cost of making markers for each data point increased.

I decided to use a HeatMap as a more effective solution to this problem. With the help of the HeatMap, incident concentration may be shown more succinctly and the distribution of events on the map can be seen more clearly. I improved the heatmap’s interpretability by controlling its size and intensity using settings like blur and radius.

Alongside the HeatMap, I also used a MarkerCluster layer to further refine the visualisation but later decided not to. Although the general legibility of the map is increased by the MarkerCluster, which clusters nearby incidences together. The decision to exclude MarkerCluster was made to maintain simplicity and reduce the processing time, especially when dealing with a substantial dataset. The HeatMap alone provides a more concise representation of incident concentration while addressing the computational challenges associated with handling a large number of individual markers. Within these clusters, individual markers stay accessible, enabling users to zoom in and examine individual instances.

The outcome of this thorough process was an engaging and educational map. While the MarkerCluster layer allowed for the examination of individual episodes within clusters, the HeatMap offered a visual summary of incidence concentration. This visualisation method takes into account both specific occurrences and large patterns in the dataset, providing a nuanced perspective on the geographical distribution of police shooting incidents while maintaining efficiency and detail.

 

An alternative to the conventional ANOVA that does not require equal variances is called Welch’s ANOVA. When there is variable variance, it may be more resilient.

When the premise of equal variances is broken, Welch’s ANOVA is made to be resilient. By adjusting the conventional F-statistic to take unequal variances into account, it achieves this robustness. The ratio of the mean squares is used to compute the Welch F-statistic, which accounts for various group variances.

In my analysis earlier during ANOVA, the Levene’s test revealed significant differences in variances between ethnic groups, casting doubt on the fundamental premise of equal variances in conventional ANOVA. I chose Welch’s ANOVA as a substitute because I understood how its unique design addressed different variances by modifying the F-statistic and degrees of freedom for greater precision. However, the Shapiro-Wilk tests indicated that the age distribution was not normal in the majority of racial groups. I recognise that large deviations from normalcy may affect results, even though Welch’s ANOVA can withstand mild deviations. The degree of these violations and the particulars of my dataset will determine whether I use Welch’s ANOVA or look into other options. I might think about using non-parametric tests in situations when there is extreme non-normality or small sample numbers.