Final answer:
The 'mean policeman theorem' seems like a typo; it likely refers to a confusion between the 'Mean Value Theorem' of calculus and the 'Central Limit Theorem' of statistics. The Mean Value Theorem deals with the instantaneous rate of change on an interval, while the Central Limit Theorem states the distribution of the sum or mean of many variables approaches a normal distribution.
Step-by-step explanation:
The term “mean policeman theorem” appears to be a misunderstanding or typographical error. It is likely that you meant to ask about the difference between the “Mean Value Theorem” and the “Central Limit Theorem”, both important concepts in mathematics. Let’s clarify these terms.
The Mean Value Theorem is a fundamental theorem in calculus that states that for any function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in the open interval (a, b) such that the instantaneous rate of change (the derivative) at c is equal to the average rate of change over the interval [a, b]. In other words, the slope of the tangent line at the point c equals the slope of the secant line that connects the endpoints of the curve over the interval.
The Central Limit Theorem, on the other hand, is a key concept in statistics which states that the distribution of the sum (or mean) of a large number of independent, identically distributed variables will tend to approximate a normal distribution, regardless of the underlying distribution. This theorem is foundational because it allows us to use normal probability models for means and sums of random variables even when the original variables are not normally distributed. Both theorems are essential for different fields and have different applications.
We are concerned with means because they provide a central measure of a dataset which can be compared across different samples or populations, and they are relatively easy to compute. The central limit theorem is particularly important because it allows us to apply inferential statistics, making it possible to draw conclusions about a population based on a sample.