Relative Frequency and Probability, What's The Difference?

 Relative frequency and probability are related concepts, but they are not exactly the same. Relative frequency refers to the proportion or fraction of times that an event occurs in a given set of data. It is calculated by dividing the frequency of the event by the total number of observations in the data set. For example, if we observe 50 heads in 100 coin tosses, the relative frequency of getting heads is 50/100 = 0.5 or 50%. Probability, on the other hand, refers to the likelihood or chance of an event occurring. It is a measure of how likely or unlikely an event is, and it is usually expressed as a number between 0 and 1 (or between 0% and 100%). Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of getting heads on a fair coin is 0.5 or 50%.

The difference between relative frequency and probability is that relative frequency is based on observed data, while probability is based on a theoretical or assumed model of the underlying process. Probability is a mathematical concept that allows us to reason about the likelihood of events, even when we don't have access to the data or when the data is incomplete. Relative frequency, on the other hand, is a tool for analyzing the distribution of data and making inferences based on the observed patterns.

It is also incorrect for finding the expectation of a random variable from a relative frequency table. To find the expectation of a random variable, we need to multiply each value of the random variable by its corresponding probability and sum up the products. In a relative frequency table, the probabilities are given by the relative frequencies, which are obtained by dividing the frequency of each value by the total number of observations. Therefore, to find the expectation of a random variable from a relative frequency table, we need to first convert the table into a probability table by dividing the frequencies by the total number of observations.

The difference between x̄ (x-bar) and μ (mu)

 In statistics, the symbol x̄ (x-bar) represents the sample mean or average of a set of data. It is calculated by adding up all of the values in the sample and dividing by the total number of values in the sample. The Greek letter mu (μ) represents the population mean or average of a larger group of data that the sample is drawn from. It is calculated in the same way as x-bar, but it represents the true mean of the entire population, rather than just the sample.

The difference between x̄ (x-bar) and μ (mu) is that x-bar represents the average of a sample of data, while mu represents the true average of the entire population from which the sample is drawn. Because samples are inherently imperfect and may not perfectly reflect the larger population, x-bar and mu may be different from each other.

Reference
Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.

The Reasons Why Measurements May Not Be Perfectly Reproducible

About the reproducible of a sample, I reckon that there are several reasons why measurements may not be perfectly reproducible, even when the same phenomenon is being measured under apparently identical conditions:

i. Measurement errors

All measuring instruments have some degree of imprecision or error associated with them. For example, a ruler may not be exactly straight, or a thermometer may not be calibrated perfectly. These errors can accumulate over repeated measurements and contribute to variability in the outcomes.

ii. Environmental factors

Even seemingly small differences in the environment can affect measurements. For example, changes in temperature, humidity, or air pressure can influence the behavior of some measuring instruments.

iii. Human factors

The people conducting the measurements may introduce variability due to their own limitations. For example, they may have slightly different visual acuity or reaction times, or they may interpret the results differently.

iv. Inherent variability

Some phenomena are inherently variable, and measurements of them will naturally vary. For example, in biology, there may be natural variation in the characteristics of organisms, even within a single population.

v. Random chance: Finally, there is always an element of chance involved in any measurement. Even if all sources of variability were eliminated, there would still be some residual randomness that would make it impossible to achieve perfectly reproducible results.

Overall, it is important to recognize that variability in measurements is a natural and unavoidable aspect of scientific research. However, scientists use statistical methods to quantify and manage this variability, in order to draw reliable conclusions from their data.