Question

1. How is statistics different from mathematics? In a lesson on the mean, what mathematical questions and what sta- tistical questions might you ask?

Answer 1

The concept of measurement is important difference between statistics and mathematics.
In mathematics, measurement typically refers to understanding units and precision in problems that deal with most concrete measures such as length, area, and volume.

But, in statistics, measurement can be a bit more abstract.

As an example how we can measure a preference, IQ (Intelligence Quotient), EQ (Emotional Quotient), Life Expectancy in a City. There is no particular method to evaluate in mathematical way.
In that case researchers and statisticians depend on data to reach any answer. Variability and the uncertainty of conclusions is another major difference between statistics and mathematics.

In mathematics, results are usually reached by means of deduction, use of formulae, logical proof, or mathematical induction and typically there is one correct answer. Statistics, however, utilizes inductive reasoning and conclusions are always uncertain.

This is largely due to the interpretation of the context and methods surrounding the data collection and analysis.
Mathematics serves as a tool to help investigate statistical questions, but not the only end of the statistics itself.

According to Mathematics "Mean" refers to the average value calculated from some numbers. Where weightages are same.
Whereas in Statistics "Mean" can refer to expected value from some data and the weightages of the data (corresponding probabilities) are different.

As an example for three numbers 5, 8, 9 the average is (5 + 8 + 9)/3 = 5*(1/3) + 8*(1/3) + 9*(1/3) = 7.33. Here weightages are same that are 1/3 for every data point.
For expected value corresponding probabilities can be given as P(x = 5) = 1/2, P(x = 8) = 1/8, P(x = 9) = 1/3 and mean will become 5*(1/2) + 8*(1/8) + 9*(1/3) = 6.5.

Some possible Qns:

1. What is the mean of the probability distribution given by P(X = x) = 1/(x² + 1), where x = 1, 2, 3, 4, 5?

2. What is the mean of the data {3, 4, 7, 3, 5, 6}?