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Probability Distributions — Discrete and Continuous

The dozen distributions DS papers actually use. When each one shows up. FIND_VIDEO: search 'probability distributions normal binomial poisson' — recommended channel: StatQuest. Aim for 11 min or under.

5 minutesVideo LessonPDF notes

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Key moments

Defining Distribution — A statistical distribution describes how measurements are spread out.
Building a Histogram — Measurements are placed into defined bins to visualize frequency.
Interpreting Likelihood — The histogram shows where measurements are most and least likely to fall.
Precision and Bin Size — Using smaller bins and more data yields a more precise distribution estimate.
Curve Approximation — A smooth curve can be used to approximate the shape shown by the histogram.
Curve Advantages — The curve allows probability calculation for unobserved ranges and is not limited by bin width.
Distribution Summary — Both histograms and curves show how probabilities of measurements are distributed.

PDF notes

Bottom line: You will be able to define a statistical distribution and explain how it represents the likelihood of different measurements occurring. You will learn how histograms and approximating curves visualize distributions and why the curve is often preferred for precise probability calculation.

This lesson on Probability Distributions — Discrete and Continuous is hands-on and example-driven. You will be able to define a statistical distribution and explain how it represents the likelihood of different measurements occurring. You will learn how histograms and approximating curves visualize distributions and why the curve is often preferred for precise probability calculation.

What You'll Be Able To Do

Define a statistical distribution based on measured data.
Explain how binning measurements creates a histogram.
Contrast the advantages of using a curve versus a histogram.
Calculate the region of highest probability from a distribution visualization.
Describe how bin size affects the precision of a distribution estimate.

Topics Covered in Probability Distributions — Discrete and Continuous

Defining Distribution (0:00 - 0:30) — A statistical distribution describes how measurements are spread out.
Building a Histogram (0:30 - 1:15) — Measurements are placed into defined bins to visualize frequency.
Interpreting Likelihood (1:15 - 1:45) — The histogram shows where measurements are most and least likely to fall.
Precision and Bin Size (1:45 - 2:30) — Using smaller bins and more data yields a more precise distribution estimate.
Curve Approximation (2:30 - 3:00) — A smooth curve can be used to approximate the shape shown by the histogram.
Curve Advantages (3:00 - 4:00) — The curve allows probability calculation for unobserved ranges and is not limited by bin width.
Distribution Summary (4:00 - 4:30) — Both histograms and curves show how probabilities of measurements are distributed.

SQL Cheat Sheet

Distribution — Shows how probabilities of measurements are spread out

SELECT height, COUNT(*) FROM measurements GROUP BY 1 ORDER BY 2 DESC;

Histogram — Stacks binned measurements to visualize frequency

SELECT FLOOR(height / 0.5) * 0.5 AS bin, COUNT(*) FROM measurements GROUP BY 1;

Bin Size — The width of the range used to group measurements
```
SELECT 0.5 AS bin_width;
```
Curve Approximation — Smooth line used to estimate probabilities across the range

Probability — The likelihood of measuring a specific value or range

SELECT COUNT(*) FROM measurements WHERE height BETWEEN 5.0 AND 6.0;

Comparison Table

Feature	Histogram	Curve Approximation
Basis	Stacked binned measurements.	Smooth mathematical function.
Precision	Limited by bin width.	Not limited by bin width.
Missing Data	Cannot calculate probability for empty bins.	Can calculate probability for empty bins.
Resource Use	Requires many measurements.	Good with fewer measurements.

Common Pitfalls

Mistake: Assuming an empty histogram bin means that measurement is impossible. Avoid: Use the approximating curve to calculate probability for that range.
Mistake: Using overly large bins, losing precision in the distribution shape. Avoid: Use smaller bins and more measurements for a more accurate estimate.
Mistake: Confusing the height of the curve with the measurement value itself. Avoid: The height shows the likelihood of that measurement occurring.

FAQs

What does the tallest part of the distribution represent? It represents the region where measurements are most likely to occur, often near the average.
Why is the curve approximation better than the histogram? The curve allows calculation of probability for any range, even those not observed, and is not limited by bin size.
How does measuring more people affect the distribution visualization? Measuring more people, combined with smaller bins, leads to a more accurate and precise estimate of the true distribution shape.

Common pitfalls

Mistake: Assuming an empty histogram bin means that measurement is impossible. Avoid: Use the approximating curve to calculate probability for that range.
Mistake: Using overly large bins, losing precision in the distribution shape. Avoid: Use smaller bins and more measurements for a more accurate estimate.
Mistake: Confusing the height of the curve with the measurement value itself. Avoid: The height shows the likelihood of that measurement occurring.

Hello, and welcome to StatQuest. StatQuest is brought to you by the friendly folks in the Genetics department at the University of North Carolina at Chapel Hill. Today we're going to be talking about what a distribution is. Imagine you're at a wild and crazy dance party and you overhear someone talking about statistics. Chances are they're going to be talking about distributions. D'egg. What is a statistical distribution? Imagine we measured the height of a lot of people. The first person we measured was 5.2 feet tall. So we put that measurement in a bin that spans from 5 feet to 5.5 feet. The second person we measured was 5.8 feet tall. The third person we measured was 5.6 feet tall. The fourth person we measured was 5.9 feet tall. The fifth person we measured was only 5.1 feet tall. And the sixth person we measured was 6.3 feet tall. Okay, you get the idea. We measure a bunch of people and put the measurements in bins. When you stack a bunch of measurements into bins like this, you get a histogram. Most of the measurements come from people between 5 and 6 feet tall. People smaller than 5 feet tall were relatively rare. People taller than 6 feet tall were also relatively rare. In other words, if you picked one measurement at random, there's a good chance it would be between 5 and 6 feet tall. The histogram gives us a sense of how likely we'll measure someone really tall or really short or closer to the average. What if we used smaller bin sizes for our measurements? Now the bins are half as wide as before, and here's how our data stacks up in the smaller bins. Again, most measurements are between 5 and 6 feet tall, but we can be more precise and say half the people are between 5.25 feet tall and 5.75 feet tall. By measuring more people and using smaller bins, we get a more accurate and more precise estimate of how heights are distributed. We can use a curve to approximate the histogram. The curve tells us the same thing that the histogram tells us. There's a low probability that we will measure someone shorter than 5 feet tall. There's a low probability that we will measure someone taller than 6 feet. Lastly, there's a relatively high probability we'll measure someone between 5 and 6 feet tall. However, the curve has a few advantages over the histogram. First, even though we measured a bunch of people, we didn't get a value for this bin. Since we can't calculate that probability with the histogram, does that mean we will never get a measurement that fits into that bin? No. Instead, we can use the curve to calculate the probability. Another advantage is that the curve is not limited by the width of the bins. If we wanted to know the probability of measuring someone between 5.021 and 5.317, we could use calculus or a computer to calculate this, without having to round to the nearest bin size. Lastly, if we don't have enough time or money to get a ton of measurements, the approximate curve, based on the mean and standard deviation of the data that we were able to collect, is usually good enough. Thus, using the curve can save us a lot of time and money. Both the histogram and the curve are distributions. They show us how the probabilities of measurements are distributed. The tallest part of the histogram or curve shows the region where measurements are most likely. The low parts of the histogram or curve show where measurements are less likely. We've been talking about how height measurements are distributed, but there are all kinds of distributions with all kinds of interesting shapes. We'll talk about these in future StatQuests. The end. Tune in next time for another exciting StatQuest.

Frequently asked questions

What does the tallest part of the distribution represent?

It represents the region where measurements are most likely to occur, often near the average.

Why is the curve approximation better than the histogram?

The curve allows calculation of probability for any range, even those not observed, and is not limited by bin size.

How does measuring more people affect the distribution visualization?

Measuring more people, combined with smaller bins, leads to a more accurate and precise estimate of the true distribution shape.