Statistics 101 Populations, Variables, and Probability

Eric Marquette

Let’s start by unraveling one of the core ideas in statistics—understanding populations and samples. Now, when we talk about a population in statistics, we’re referring to every single individual in a given group of interest. For example, if we wanted to know the average age of students in a particular classroom, observing every single student in the room would give us population data.

Eric Marquette

But—and here’s the thing—it’s often not practical to study an entire population. Say we’re analyzing all university students in the country to determine their average age—that’s a tall order! Instead, we work with a smaller subset, or a sample, which represents the population. Think of it as getting a snapshot that helps us see the entire picture. Of course, how well the sample represents the population depends on how we gather that data.

Eric Marquette

Which leads us right into variables. Variables are the pieces of information we’re looking to measure in our data. They come in two flavors: qualitative and quantitative. Qualitative variables, also called categorical variables, might include things like eye color—blue, brown, green—or customer opinions in a survey, like "satisfied" or "unsatisfied." Pretty straightforward, right?

Eric Marquette

Quantitative variables, on the other hand, are numerical. These can be anything from the number of items sold in a day to someone’s weight or income. Even here, there are two subtypes: discrete variables, which are whole numbers like how many products a store sold yesterday, and continuous variables, which can have fractional values like a person’s height measured to the nearest centimeter.

Eric Marquette

Now, let’s tackle how we collect these data points or, more specifically, how we capture a sample that represents the broader population. Random sampling is the gold standard—every individual has an equal chance of being picked. Picture pulling names out of a hat. Stratified sampling, though, takes it one step further by dividing the population into subgroups—say, age or gender—and sampling from each subgroup proportionally. This can give us better representation. Then there’s cluster sampling, where we divide the population into clusters, like neighborhoods, and select entire clusters at random. It’s quick but can be risky if clusters don’t truly represent the whole population.

Eric Marquette

These strategies—random, stratified, cluster—they’re all tools in the kit to make our sample reflect the population as closely as possible. And getting that representation right? It’s critical if we want our data to tell a reliable story.

Eric Marquette

When it comes to making sense of data, the first step is to summarize it, both numerically and visually. Summarization gives us a clearer picture of what's going on in the data without, you know, overwhelming us with every single detail. Let’s start with the numerical summaries—what we call measures of central tendency and measures of dispersion.

Eric Marquette

Measures of central tendency include the mean, the median, and the mode. Now, the mean is what most of us call the average. It’s the sum of all values divided by the number of entries. Super simple. The median? That’s the middle value when we arrange the data in order—perfect for situations where outliers, those extreme values, might skew the mean. And then there’s the mode, which is just the most frequently occurring value in the data. For example, if we looked at daily sales of a product, say sneakers, the mode might tell us which number of pairs sold was most common.

Eric Marquette

Now, what about dispersion? Dispersion measures tell us how spread out the data is—how much it varies. The range, the variance, and the standard deviation are your go-to tools here. And honestly, they’re pretty useful! The range is the simplest—it’s just the difference between the largest and smallest values. Variance and standard deviation are a little trickier to calculate, but they give a much deeper insight into how data points differ from the average—or mean.

Eric Marquette

Okay, let’s shift gears to visualizing data, which is where things get, well, a little more fun. Imagine you have a dataset about daily sales—pie charts are great for showing proportions, like the share of sales across different products. Bar charts, on the other hand, are amazing for comparing categories, such as sales by region. And histograms? Well, histograms are all about distribution. They’re perfect when you want to see how the frequency of your data spreads, like knowing how often sales were in the 10-to-20-pairs range versus 30-to-40 pairs.

Eric Marquette

But—and this is a big but—visualizing data comes with its own set of pitfalls. Take wealth distribution graphs, for example. They can sometimes look way more balanced than they actually are. A bar chart might mislead you if the width of wealth categories isn’t consistent. Interpreting these graphs without understanding their context could lead to some very flawed conclusions.

Eric Marquette

So, whether you’re summarizing data through numbers or representing it visually, the key is to always, always consider what story the data is telling you—and whether that story is grounded in solid methods.

Eric Marquette

Probability. It’s everywhere—it's how we deal with uncertainty, like tossing a coin or predicting tomorrow’s weather. So, let’s break this down into three core types of probability: theoretical, empirical, and subjective.

Eric Marquette

Theoretical probability—you might call it the simplest form—relies purely on logic. Take a coin toss: we know there are two possible outcomes—heads or tails—and assuming the coin is fair, each has an equal chance. That’s a theoretical probability of fifty percent for heads. Nice and clean, right?

Eric Marquette

Empirical probability, on the other hand, is based on actual observations or experiments. So, if I flip that coin ten times and it lands on heads seven times, my empirical probability for heads is seventy percent. Real-world data, however, can be messy—run enough coin flips though, and you’ll notice it starting to align more with its theoretical counterpart, thanks to something called the Law of Large Numbers.

Eric Marquette

Finally, subjective probability. This one is more personal; it’s about belief rather than data. Say there’s a twenty percent chance of rain tomorrow... according to a weather app. That number often reflects someone’s judgment—maybe a meteorologist weighing in factors we might not directly see. It’s not exact, but it’s a probability all the same.

Eric Marquette

Now, once you grasp probabilities, you’ll often need to calculate how they work together. Enter the Addition Rule. This rule helps us find the probability of either event A or event B happening. For example, what’s the probability of rolling a five or a six on a die? Well, each outcome independently has a one-in-six chance. Add those together, and you’ve got two-in-six, or about thirty-three percent. But, beware—this works only when the events are mutually exclusive, meaning they can’t overlap. Like, rolling a five and a six at the same time? Not happening on one die.

Eric Marquette

For overlapping or non-mutually exclusive events, a quick fix to avoid double-counting is subtracting the overlap. Like drawing a King or a Heart from a deck of cards—there’s overlap because the King of Hearts fits both events. It’s a bit of a common-sense adjustment, really, and super handy when probabilities aren’t separated into neat little boxes.

Eric Marquette

Complementing that is the Multiplication Rule, which helps calculate the probability of two events both happening, like rolling back-to-back sixes. Here, probabilities get multiplied. With a fair die, the chance of rolling one six is one-sixth. Do it again and you’re multiplying one-sixth by one-sixth, giving you, well, one in thirty-six. And this assumes the rolls are independent—one event doesn’t affect the other. Things get trickier with dependent events, like drawing cards without replacing them. Pull an Ace first, and now the deck’s changed. Fewer cards, fewer Aces, and the probabilities shift. For these situations, we use conditional probabilities instead of assuming independence.

Eric Marquette

Which brings me to the pièce de résistance of probability: Bayes’ Theorem. This is really about refining our predictions as we gather new evidence. Let’s consider another everyday scenario—rain forecasts. Imagine waking up to a cloudy sky. Does that instantly mean a seventy percent chance of rain? Not enough info. But add in that you checked yesterday, and it showed a cold front moving in, and now your odds shift based on this combined context. That’s Bayes, in essence—taking prior knowledge and updating it with what’s happening now.

Eric Marquette

And that’s all for today’s journey—stepping from probabilities to rules and ending with Bayes, the rule that refreshes belief with every added thing we know. Thanks for joining me to dive into the magic and logic of statistics—and hey, learning alongside all of you is always a pleasure. Until next time, keep an eye on the data and an open mind about what it can teach you. Take care!