## On randomness

Pick a number between one and ten. You picked the number seven. Was I right? Doesn’t matter, the point is there’s a slim chance you picked one or ten. The reason is simply because humans are bad at random number generating. Fear not! Computers are bad at it too, but that’s a problem for a world that revolves around cyber security. Randomness is so important to our daily lives, that we don’t even notice it… mostly.

(more…)## A five dimensional pain

We’re back and this time working in five dimensions! But wait, we’re not done, we’re going higher and I’m not thrilled about this at all. Working in higher dimensions is a pain especially when we’re stuck in a three dimensional world. So what am I doing in a five dimensional space? Well besides pulling my hair out, I’ve been trying to do some not so basic math to find some significance, in life, but mostly in my data.

(more…)## Working in higher dimensions

Imagine if you will, only being able to move along a straight line. You’re now in one dimensional space. But wait! What if we are allowed, quite literally, to take a left. You are now allowed to move along a square space, this is two dimensional space. We can do better, though. You suddenly can move up and down, traveling in an area that’s the shape of a cube! You’re now in 3D space. Then suddenly you disappear from view, but where did you go? Welcome to the fourth dimension, you can’t see it, you can’t imagine it, but we can do math here and above!

(more…)## Defining parametric tests in statistics

We’ve been throwing around the term a lot in this series. I’ve been saying in parametric statistics this, in parametric statistics that, but I kept putting off giving a definition. It’s not because it’s hard to understand, it’s just that typically when you’re doing statistics you already know if you’re using a parametric test, but because we try to make no assumptions in this series, we’re going to put this to bed once and for all. Today we’re talking about parametric statistics!

(more…)## Independence in statistics

A while back we introduced the central limit theorem, it was a way to take data and make it normal (gaussian) as if by magic, which is one of the assumptions needed for parametric statistics (the most commonly used kind). Today we’re introducing another assumption, that the data are independent. The idea of independent events is probably straightforward, but it’s yet another bedrock of statistics that we should talk about in depth to help us understand why things are the way they are.

(more…)## Variance in statistics

Variance, it’s one of those concepts that get’s explained briefly then you find yourself using it over and over. Now that I have a free moment, I figure it’s about time to revisit the “simple” concept and just take a minute to apricate why we have to deal with variance so often and why we try so hard to minimize it when we’re doing experiments. Just like the discussion about the mean, there’s some subtilty that goes into the idea of variance and it’s square root cousin standard deviation and we skip over it in favor of getting into more complex topics.

(more…)## The Bonferroni correction in statistics

Well we’re doing it, today we’re talking about the Bonferroni correction, which is just one of many different ways to correct your analysis when you’re doing multiple comparisons. There are a lot of reasons you may want to do multiple comparisons and your privacy is our main concern so we won’t ask why. Instead we’re going to talk about how to adjust your alpha (chances of making a type 1 error) so you don’t end up making a mistake.

(more…)## One-tailed vs. two-tailed tests in statistics

Sit right back because we’re telling a troubling tale of tails full of trials, twists, and turns. The real question is, will we run out of words that start with t during this post? It will be tricky, but only time will tell. When do we use a two-tailed test vs. a one-tailed test and what do tails have to do with tests anyway? With a little thought, I think we can tackle the thorny topic. In short, let’s talk tails!

(more…)## The p-value in statistics

We’ve been dancing around the p-value for some time and gave it a good definition early on. The p-value is simply the probability that you’ve made a type one error, the lower the p-value the less chance you have of making a type one error, but you increase your probability that you’ll make a type two error (errors in statistics for more). However, just like with the mean, there’s more than meets the eye when it comes to the p-value so let’s go!

(more…)## The z-score in statistics

Okay, time to get back to statistics, if only for today! P-value, z-score, f-statistic, there are a lot of ways to get information about the sample of data you have. Of course, they all tell you something slightly different about the data and that information is useful when you know what the heck it is even trying to tell you. For that reason we’re diving into the z-score, it’s actually one of the more intuitive (to me anyway) measurements so let’s talk about it!

(more…)## The mean in statistics

Yeah it seems simple, I mean (no pun intended) the mean is just the average! Yet as with so many different things in statistics there’s more to the mean than meets the eye! We’re going to go into why the mean is important, why it’s our best guess, why it may not always be your best option, and why we work so hard to find the mean sometimes! It seems simple, but I promise today we’re answering a lot of the big “why’s” in statistics, so let’s go!

(more…)## The Tukey test in statistics

No, not turkey, Tukey, although they are pronounced very similar (depending on who you ask I guess? I’ve seen people pronounce it “two-key”). Any way, today we’re saving our job and the wrath that comes with failure. The mad scientist boss of ours tasked us with testing mind control devices and determining statistically which one (if any) worked. After the last failure, we now had four new devices to test, so we couldn’t use the same method as before. However, an assistant’s work is never done, we didn’t finish the job! That’s what we’re going to do today.

(more…)## The ANOVA in statistics

Our mad scientist is back and this time they are not taking any chances! After statistical failure in the last example, they created not just one, but four mind control prototypes! We’ve been tasked with determining if they are working or face certain DOOOOOOOM! Sure, working for a mad scientist can be stressful, but we can do this… right? I’ve been dreading this post and you’ll see why, there’s a lot to cover before we solve the mystery, so let’s dive into it!

(more…)## Significance in statistics

That feeling when your p-value is lower than your alpha, aww yeah! But what does it really mean? It’s one thing to say there is significance and on the surface it means the two things are different “enough” to be considered two things, but I think there’s a simpler way to explain it. So today we’re going to talk about what significance actually means in the practical sense. Maybe it’s super obvious, but it never hurts to state it anyway.

(more…)## The f-test in statistics

Yep, we’re getting right back into it. I’m still working out things from yesterday, so we can just talk more statistics. This will be an interesting one and hopefully it will be pretty straightforward. The f-test, which in this case is really the f-test to compare two variances. You may have guessed, but the t-test uses the t-distribution (sort of like the normal), well the f-test uses the f-distribution, which is nothing like the normal! Let’s dive in, shall we?

(more…)## Confidence intervals in statistics

Well since our mad scientist from yesterday’s post is on a short break, today we’re going to fill in some of the gaps that post brought into view. First up is the confidence interval. There are some subtle points here, so this should help clarify a few things that may not have been clear yesterday. We’re going to do a somewhat deep dive into what the heck we’re doing when we talk confidence interval and why the standard deviation of our data is important in determining the values.

(more…)## The t-test in statistics

Welcome fellow mad scientist enthusiasts, the last time we talked statistics, we found ourselves in an interesting situation and we need to figure out if the mind control device that was developed is actually working. We introduced the idea of a two population problem and today we’re going to use something called a t-test to determine if our mad scientist succeeded.

(more…)## Two populations in statistics

As a mad scientist, or maybe just a grumpy scientist, you want to test a new mind control technique! To do this you decide that you want to test this works by having people select one of two objects set in front of them. *Insert evil laugh* Using your mind control technique you want your unwitting participants pick the object on the left. You don’t get 100% success, but suspect it’s working, how do we know for sure?

(more…)## A fair coin in statistics

It has been a busy day, but the show must go on so to speak and I’m here today to tell you I have a nice shiny new coin for us to flip! The catch you may ask? Well it could be a fair coin or I may have just swapped it out for a coin that was not fair! Our goal is to see if I’m being sneaky and to do that we’re going to need some statistics!

(more…)## Errors in statistics

Everyone makes mistakes, that’s okay! In day to day life there are a lot of different ways you and I could make mistakes. In statistics however, there are just two ways for you to make a mistake. That may sound like a good deal, but trust me when I say two ways to make a mistake is two too many. To think, you spent all that time picking the right statistical test, did the experiment, analyzed the data, just to make an error in the end. Don’t worry, it happens to the best of us, but knowing what they are will help you prevent them!

(more…)## Day 62: Two Random Variables A2

Like we did with question 1, this will be the solution to the question we posed in the last post, if you haven’t tried to solve it yet, go give it a shot. If you have and are dying to check your answer, then let’s look at the solution.*

## Day 61: Two Random Variables Q2

For those of you who have been following along, today we are going to post another question and in the next post we will give the solution. This will be another two random variable question and we’ve covered everything you need to solve it in our previous posts. So with that, let’s get to today’s question.*

## Day 60: Two Random Variables A1

Hopefully if you’re reading this you saw our last post, where we gave the question we will solve today. If you haven’t had a chance to try and solve it, please feel free to stop and give it a shot. If you’re ready to see how we solve it, then let’s get started.*

## Day 59: Two Random Variables Q1

Well now that we’ve had a minute to take a breath, let’s try out something new. In this post I will give the question and in the next post we can work out the answer. For those of you playing at home, this will be a good way to check your knowledge and for me, it will give the the chance to do the same.*

## Day 56: One Function of Two Random Variables Example 3

Okay quick example, still not super difficult, but one we can work out to a complete solution. We’ve gone over a few examples now, but we’re going to go over a few more for both my benefit and yours. So let’s dive in.*

## Day 55: One Function of Two Random Variables Example 2

Well our last postÂ we took a break and talked zombies! While I would love to do a whole month of halloween topics, this year is not the time, maybe next year. In any case today we are going to go over another example of a single function of two random variables. This is going to be slightly more complex than our first example, however it won’t be extremely complex (we’re working towards it). So let’s take a look shall we?*

## Day 53: One Function of Two Random Variables Example 1

Hopefully at this point we’ve demystified more than just a few concepts at this point. Today we are going to look at one function of two random variables. Originally I was going to break into a joint CDF example that involved dependent variables, but it turns out my book doesn’t cover that! Oops, guess I should’ve read ahead. In any case let’s talk functions!*

## Day 52: Joint Cumulative Distribution Function Example 1

Well here we are again, today we are talking functions of two random variables. If you’re looking for the beginning, this isn’t it, but you can read the introduction here. If you’ve kept up, then you’re ready to go over the example we have today, so let’s get started.*

## Day 51: Joint Cumulative Distribution Function

As promised, today we are going to talk about two random variables that are not independent. This means that the individual probabilities don’t sum to be equal to the joint probability (like they did yesterday). Like our normal CDF, we can find a CDF for two random variables, but let’s take a look at how this works.*

## Day 50: Intro to Two Random Variables

I was debating about not posting anything today. It’s been a bit rough for me these past few days. However, I’m going to write a little something today and tomorrow to introduce two random variables (so we don’t skip a day). This is going to be a lot like our single random variable examples, but (of course) more complex, let’s take a look at what I mean.*

## Day 44: Functions of One Random Variable Example Part 1

Well maybe yesterday was confusing, maybe it wasn’t. In any case, today *should* clarify some things for you if you are confused and should make things more clear if you are not. Today we are going to go over a quick example of what a function involving one random variable looks like. Now you may notice I keep saying one, that’s because you can technically have as many variables as you want, but since this is fairy complex stuff, let’s just stick with the one for now.*

## Day 43: Introduction to Functions of One Random Variable

Now that we’ve looked at conditional probabilitiesÂ we can talk about other things we can do with random variables. If you’ve been keeping up with us so far, then this shouldn’t be too crazy of an idea, really all we are going to do today is take a random variable and transform it somehow. Interested? Let’s go!*

## Day 42: Conditional Probability

Up to now we’ve been dealing with single variable pdf and the corresponding CDF. We said that these probabilities relied on the fact that our variable of interest was independent. However, what if we knew some property that impacted our probability? Today we are talking conditional probability and that is the question we will be answering. It’s going to be a long, long post so plan accordingly.*

## Day 41: Connecting the Concepts

Maybe we shouldn’t phrase it this way, since there is still quite a few days left of 365DoA, but you made it to the end! No, not THE end, but if you’ve been following along the past few posts we’ve introduced several seemingly disparate concepts and said, “don’t worry they are related,” without telling you how. Well today, like a magician showing you how to pull a rabbit from a hat, let’s connect the dots and explain why we introduced all those concepts!*

## Day 40: The Normal Approximation (Poisson)

## Day 39: The Normal Approximation (De Moivre-Laplace)

## Day 38: The Poisson Distribution

## Day 37: Bayes’ Theorem

## Day 36: The uniform pdf

## Day 35: Example of the Gaussian pdf

## Day 34: Example of the Laplace pdf

## Day 33: Example of the Exponential pdf

Over the past couple of days, I’ve been talking about several different types of pdf and the associated C.D.F. Hopefully, we have a clear understanding of each of those concepts, for those of you scratching your head, I would recommend you start here at this other post. Otherwise, let’s (finally) look at a real life example using the exponential pdf!*

## Day 32: The Laplace pdf

Well here we are again… maybe unless you’re new, in which case welcome. If you are just joining us we are talking p.d.f. no not the file format, the probability density function version. If you’re new, you may want to start back here(ish)Â If not, then let’s talk the strangely similar laplace distribution.*

## Day 31: The Exponential pdf

Well, it has been a week, don’t even get me started. But if you’re here you don’t want to hear me complain about my week, that isn’t why we come together! Well today let’s do a bit of a dive into the exponential p.d.f. I hope you’ve brushed up, because this is going to get interesting.*

## Day 30: Confidence Interval

Day 30 already! Where does the time go? It feels like we just started this whole project and it probably wouldn’t be a good idea to look at the remaining time to completion, so let’s not and just enjoy the nice round 30. We will get back to our p.d.f another day, but today is going to be short. That’s what I usually say before typing out 10 pages worth of information so to avoid that, let’s touch on something important, but something I can do briefly. Today we’re talking about confidence intervals*