Population vs. Sample: Key Differences

1. Population vs. Sample: The Whole and The Part

In statistics, these terms have very precise meanings:

• A Population is the entire group of individuals, items, or events that we are interested in studying. It's the "whole."
• A Sample is a subset of the population, selected for analysis. It's a "part" of the whole.

Why Do We Use Samples?

You might wonder, "Why not just study the entire population?" In most real-world scenarios, especially in machine learning, studying the entire population is impractical or impossible due to several reasons:

• Cost: Collecting data from everyone or everything is prohibitively expensive.
• Time: It would take too long. Imagine trying to survey every voter before an election.
• Accessibility: The entire population may not be accessible. For example, the population of "all emails" includes future emails that haven't been written yet.
• Destructive Testing: Sometimes, the act of collecting data destroys the item. A factory can't test the lifespan of every light bulb it produces.

The following video provides some excellent real-world examples to clarify these ideas.

As you can see, the central challenge of data science is to use a manageable sample to learn something meaningful about the vast, often unknowable, population.

2. Parameter vs. Statistic: Describing the Data

Now that we have distinguished between the whole group (population) and the smaller part (sample), let's look at the numbers we use to describe them.

• A Parameter is a numerical value that describes a characteristic of a Population.
• A Statistic is a numerical value that describes a characteristic of a Sample.

A simple mnemonic to remember this is:

• Parameter -> Population
• Statistic -> Sample

This brings us back to what we learned in the last lesson. The population mean ($\mu$) and population standard deviation ($\sigma$) are parameters. The sample mean ($\bar{x}$) and sample standard deviation ($s$) are statistics.

The video below does an excellent job of defining these terms and, importantly, introducing the standard notation used for each.

Here is a summary of the symbols discussed in the video, which you will see constantly in machine learning literature.

To solidify your understanding, consider these examples:

• Scenario 1: A study reports the average GPA of all 30,000 students at a university is 3.1.
  • This is a parameter ($\mu$), because it describes the entire population of students at that university.
• Scenario 2: A news poll surveys 1,000 voters and finds that 55% plan to vote for a certain candidate.
  • This is a statistic ($\hat{p}$), because it describes a sample of voters, not all of them. The goal is to use this statistic to estimate the true population parameter ($P$), which is the proportion of all voters who support the candidate.

3. The Goal: Using Statistics to Estimate Parameters

So, why is this distinction so important? Because the ultimate goal of most data analysis and machine learning is inference.

Inferential Statistics is the process of using data from a sample to make inferences or draw conclusions about the population from which the sample was drawn.

We calculate a statistic from our sample because we hope it's a good estimate of the corresponding parameter in the population, which we can't measure directly. For example, we use the sample mean ($\bar{x}$) to estimate the population mean ($\mu$).

The following article explains this relationship clearly.

For this process of inference to be reliable, the sample must be representative of the population. If we want to know the average height of all adults in a country, we can't just sample professional basketball players. Statisticians have developed various sampling methods to help ensure samples are as representative as possible, such as simple random sampling, where every member of the population has an equal chance of being selected.

Conclusion

Today, you've mastered a fundamental concept in statistics that underpins the entire field of machine learning. Understanding the relationship between a sample and a population is the first step toward building models that generalize to new, unseen data.

Key Takeaways:

• A Population is the complete set of interest, while a Sample is a manageable subset of that population.
• We use samples because studying entire populations is often impractical or impossible.
• A Parameter is a numerical summary of a population (e.g., population mean $\mu$).
• A Statistic is a numerical summary of a sample (e.g., sample mean $\bar{x}$).
• The primary goal of inferential statistics is to use sample statistics to estimate population parameters.
• This distinction is reflected in the mathematical notation used (e.g., $\sigma$ for population standard deviation, $s$ for sample standard deviation).

Preview of the Next Lesson:
We've established that we use a statistic (like the sample mean) to estimate a parameter (the population mean). But how much confidence can we have in that estimate? If our sample mean is 45, can we confidently say the population mean is also 45? Or could it be 42, or 50? This leads us into the world of hypothesis testing. In our next lesson, "Perform a t-test and interpret the p-value," we will learn our first formal method for testing a hypothesis about a population parameter using data from a sample.