first: Mathematics is the broader discipline that provides the theoretical foundation, while statistics is its applied subset, focusing on making sense of data and uncertainty in real-world problems. Together, they complement each other, with statistics relying on mathematical principles for its development and application.
very good , but Modern theories say that statistics is an independent
Mathematical statistics is independent of Mathematics with many concepts, the first and most important of which is that all concepts in statistics are based on the phenomenon of randomness, unlike other branches of mathematics whose data are based on deterministic phenomena. This is what makes statistics unique in many of its concepts and special methods, but despite that it cannot be made a science independent of mathematics, at least the backbone of mathematical statistics is mathematical analysis.
There are multiple issues to unwrap here. First mathematics is not a science. Mathematics does not directly study the natural world through observation and experimentation. Instead mathematics relies on logic and deduction to establish truths based on axioms and definitions.
Secondly, Probability theory is an area of mathematical study with plenty of areas on its own such as ergodic theory, stochastic processes, stochastic differential equations, Markov processes. But again Probability theory is a subset of the mathematical discipline of real analysis.
Statistics uses the foundations of probability theory in collecting, analyzing, interpreting, and presenting data, relying heavily on mathematical principles to draw conclusions from information gathered through observation or experimentation. So in this case Statistics can be considered a science since in practice it is governed by the scientific method dealing with the analysis of the real world.
On the other hand the area often call mathematical statistics is similar to the area known as mathematical physics. They are mathematical disciplines dealing with advanced the mathematics question in their respective areas, statistics and physics, using the logical and axiomatic approach of mathematics.
So the answer is while Mathematical Statistics is a mathematical discipline which develops the mathematical theory arising from the mathematical questions that arise in statistics just as Mathematical Physics is the mathematical discipline which develops the mathematical theory arising from the mathematical questions that arise in physics, Statistics itself is the application of mathematics in addressing experimentation in the real world so a science.
Mathematics is a tool of statistics.
Statistics is used to estimate probabilities of outcomes.
Assumptions must be made on the type of distributions being studied. Probabilistic results are estimated based on the assumed distributions using parametric statistics. Using nonparametric statistics researchers are somewhat freed of assumptions about distributions. In either case results are in terms of probability. Statistics uses models to determine relationships among variables. Mathematical models use mathematics but I would not consider them to be mathematics. The study of climate using modeling is not mathematics and it may not even be science.
Mathematics is the abstract study of numbers, structures, and patterns, while statistics focuses on data collection, analysis, interpretation, and inference to make decisions under uncertainty.
Mathematics is the study of abstract structures, patterns, and logic, while statistics applies mathematical tools—especially probability—to analyze data and make decisions under uncertainty. In short, mathematics builds the language, and statistics uses that language to understand the real world.
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