Explain the fundamental differences between inferential and descriptive statistics, and how each is applied in data analysis.

Descriptive statistics and inferential statistics are two branches of statistical analysis, each serving distinct purposes in the realm of data understanding. Descriptive statistics focuses on summarizing and describing the main features of a dataset, without making any generalizations about a larger population. It's concerned with presenting information about the data that has been gathered in a clear, concise manner. On the other hand, inferential statistics aims to make inferences or predictions about a larger population based on the data obtained from a sample of that population. It goes beyond just describing the sample and seeks to draw conclusions about the wider group it represents.

Descriptive statistics uses methods such as measures of central tendency (mean, median, mode) to identify a typical value for the data. For example, if we have the exam scores of 50 students, descriptive statistics can tell us the average score (mean), the middle score (median) or the most frequent score (mode). In addition, measures of dispersion or spread such as range, variance, and standard deviation are used to indicate how much the data is scattered around the central value. For the same 50 students, we could use standard deviation to understand how widely the scores vary from the average. Histograms, bar charts, and other visual tools are also part of descriptive statistics to summarize the data in a comprehensible way. For example, a histogram might show the distribution of scores, indicating how many students achieved scores within specific ranges. The purpose here is solely to understand, represent and communicate the attributes of this particular group of 50 students, nothing more.

In contrast, inferential statistics takes the information from the sample and tries to generalize it to the population from which that sample was taken. For example, suppose that you conduct a survey of 1000 people from a city to determine their political preferences, this is our sample. Inferential statistics would then be employed to use that sample data to make inferences about the entire city's voting population which is our population. This usually involves hypothesis testing, which means using the sample data to make a decision about claims made about the population and confidence intervals, which provide a range of values within which the actual population parameter is likely to fall. For instance, we might use a hypothesis test to determine if the support for a political party is significantly different between two age groups. We could also calculate a confidence interval to estimate the true proportion of voters in the whole city who support that party, based on the sample data. Regression analysis, another element of inferential statistics, helps determine the relationship between different variables in the population and predict outcomes.

In data analysis, both descriptive and inferential statistics play crucial roles but at different stages. Initially, descriptive statistics are used to understand the basic features of the data; identifying any trends, patterns, and outliers present in the data sample. Before making any assumptions or predictions, one must always understand what is on hand. Once this fundamental understanding is established, inferential statistics become important for making data-driven decisions. For example, in a medical study, descriptive statistics might show the average reduction in blood pressure for a group of patients who have received a new drug. Inferential statistics, on the other hand, could be used to determine whether this reduction in blood pressure is statistically significant and if this effect can be generalized to all individuals with that particular health condition. Both are useful in different ways and it is important to understand their roles to use them correctly. Descriptive statistics gives us the ‘what’ while inferential statistics gives us the ‘so what?’.