Data science jokes

(An OpenAI generated image of some data scientists laughing. There are two reasons why you know it's fake: they're all beautiful and they're all laughing at these jokes.)

Where do data scientists go unplanned camping?
In a random forest.

Who do they bring on their trip?
Their nearest neighbors.

What do zoo keepers and data scientists have in common?
They both import pandas.

Where do data scientists go camping to get away from it all?
In an isolation forest.

What's the different between ML and AI?
If it's written in Python, then it's probably ML.
If it's written in PowerPoint, then it's probably AI.

A Machine Learning algorithm walks into a bar.
The bartender asks, "What'll you have?"
The algorithm says, "What's everyone else having?"

Data science is 80% preparing data, and 20% complaining about preparing data.

A SQL query walks into a bar, walks up to two tables, and asks, “Can I join you?”

How did the data scientist describe their favorite movie? It had a great training set.

Why do data scientists love parks?
Because of all the natural logs!

What’s the difference between an entomologist and a data scientist?
Entomologists classify bugs. Data scientists remove bugs from their classifiers.

Why did the data set go to therapy?
It had too many issues with its relationships!

Why does Python live on land?
Because it's above C-level.

One of these jokes was generated by OpenAI. Can you tell which one?

Everything you wanted to know about the normal distribution but were afraid to ask

Normal is all around you, and so is not-normal

The normal distribution is the most important statistical distribution. In this blog post, I'm going to talk about its properties, where it occurs, and why it's so very important. I'm also going to talk about how using the normal distribution when you shouldn't can lead to disaster and what you can do about it.

(Ainali, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons)

A rose by any other name

The normal distribution has a number of different names in different disciplines:

• Normal distribution. This is the name used by statisticians and data scientists.
• Gaussian distribution. This is what physicists call it.
• The bell curve. The names used by social scientists and by people who don't understand statistics.

I'm going to call it the normal distribution in this blog post, and I'd advise you to call it this too. Even if you're not a data scientist, using the most appropriate name helps with communication.

What it is and what it looks like

When we're measuring things in the real world, we see different values. For example, if we measure the heights of 10 year old boys in a town, we'd see some tall boys, some short boys, and most boys around the "average" height. We can work out what fraction of boys are a certain height and plot a chart of frequency on the y axis and height on the x axis. This gives us a probability or frequency distribution. There are many, many different types of probability distribution, but the normal distribution is the most important.

(As an aside, you may remember making histograms at school. These are "sort-of" probability distributions. For example, you might have recorded the height of all the children in a class, grouped them into height ranges, counted the number of children in each height range, and plotted the chart. The y axis would have been a count of how many children in that height range. To turn this into a probability distribution, the y axis would become the fraction of all children in that height range. )

Here's what a normal probability distribution looks like. Yes, it's the classic bell curve shape which is exactly symmetrical.

The formula describing the curve is quite complex, but all you need to know for now is that it's described by two numbers: the mean (often written \(\mu\)) and a standard deviation (often written \(\sigma\)). The mean tells you where the peak is and the standard deviation gives you a measure of the width of the curve. 

To greatly summarize: values near the mean are the most likely to occur and the further you go from the mean, the less likely they are. This lines up with our boys' heights example: there aren't many very short or very tall boys and most boys are around the mean height.

Obviously, if you change the mean or the standard deviation, you change the curve, for example, you can change the location of the mean or you can make the curve wider or narrower. It turns out changing the mean and standard deviation just scales the curve because of its underlying mathematical properties. Most distributions don't behave like this; changing parameters can greatly change the entire shape of the distribution (for example, the beta distribution wildly changes shape if you change its parameters). The normal scaling property has some profound consequences, but for now, I'll just focus on one. We can easily map all normal distributions to one standard normal distribution. Because the properties of the standard normal are known, we can easily do math on the standard normal. To put it another way, it greatly speeds up what we need to do.

Why the normal distribution is so important

Here are some normal distribution examples from the real world.

Let's say you're producing precision bolts. You need to supply 1,000 bolts of a precise specification to a customer. Your production process has some variability. How many bolts do you need to manufacture to get 1,000 good ones? If you can describe the variability using a normal distribution (which is the case for many manufacturing processes), you can work out how many you need to produce.

Imagine you're outfitting an army and you're buying boots. You want to buy the minimum number of boots while still fitting everyone. You know that many body dimensions follow the normal distribution (most famously, chest circumference), so you can make a good estimate of how many boots of different sizes to buy.

Finally, let's say you've bought some random stocks. What might the daily change in value be? Under usual conditions, the change in value follows a normal distribution, so you can estimate what your portfolio might be worth tomorrow.

It's not just these three examples, many phenomena in different disciplines are well described by the normal distribution.

The normal distribution is also common because of something called the central limit theorem (CLT). Let's say I'm taking measurement samples from a population, e.g. measuring the speed of cars on a freeway. The CLT says that the distribution of the sample means will follow a normal distribution regardless of the underlying distribution.  In the car speed example, I don't know how the speeds are distributed, but I can calculate a mean and know how certain I am that the mean value is the true (population) mean. This sounds a bit abstract but it has profound consequences in statistics and means that normal distribution comes up time and time again.

Finally, it's important because it's so well-known. The math to describe and use the normal distribution has been known for centuries. It's been written about in hundreds of textbooks in different languages. More importantly, it's very widely taught; almost all numerate degrees will cover it and how to use it. 

Let's summarize why it's important:

• It comes up in nature, in finance, in manufacturing etc.
• It comes up because of the CLT.
• The math to use it is standardized and well-known.

What useful things can I do with the normal distribution?

Let's take an example from the insurance world. Imagine an insurance company insures house contents and cars. Now imagine the claim distribution for cars follows a normal distribution and the claims distribution for house contents also follows a normal distribution. Let's say in a typical year the claims distributions look something like this (cars on the left, houses on the right).

(The two charts look identical except for the numbers on the x and y axis. That's expected. I said before that all normal distributions are just scaled versions of the standard normal. Another way of saying this is, all normal distribution plots look the same.)

What does the distribution look like for cars plus houses?

The long winded answer is to use convolution (or even Monte Carlo). But because the house and car distribution are normal, we can just do:

\(\mu_{combined} = \mu_{houses} + \mu_{cars} \)

\(\sigma_{combined}^2 = \sigma_{houses}^2 + \sigma_{cars}^2\)

So we can calculate the combined distribution in a heartbeat. The combined distribution looks like this (another normal distribution, just with a different mean and standard deviation).

To be clear: this only works because the two distributions were normal.

It's not just adding distributions together. The normal distribution allows for shortcuts if we're multiplying or dividing etc. The normal distribution makes things that would otherwise be hard very fast and very easy.

Some properties of the normal distribution

I'm not going to dig into the math here, but I am going to point out a few things about the distribution you should be aware of.

The "standard normal" distribution goes from \(-\infty\) to \(+\infty\). The further away you get from the mean, the lower the probability, and once you go several standard deviations away, the probability is quite small, but never-the-less, it's still present. Of course, you can't show \(\infty\) on a chart, so most people cut off the x-axis at some convenient point. This might give the misleading impression that there's an upper or lower x-value; there isn't. If your data has upper or lower cut-off values, be very careful modeling it using a normal distribution. In this case, you should investigate other distributions like the truncated normal.

The normal distribution models continuous variables, e.g. variables like speed or height that can have any number of decimal places (but see the my previous paragraph on \(\infty\)). However, it's often used to model discrete variables (e.g. number of sheep, number of runs scored, etc.). In practice, this is mostly OK, but again, I suggest caution.

Abuses of the normal distribution and what you can do

Because it's so widely known and so simple to use, people have used it where they really shouldn't. There's a temptation to assume the normal when you really don't know what the underlying distribution is. That can lead to disaster.

In the financial markets, people have used the normal distribution to predict day-to-day variability. The normal distribution predicts that large changes will occur with very low probability; these are often called "black swan events". However, if the distribution isn't normal, "black swan events" can occur far more frequently than the normal distribution would predict. The reality is, financial market distributions are often not normal. This creates opportunities and risks. The assumption of normality has lead to bankruptcies.

Assuming normality can lead to models making weird or impossible predictions. Let's say I assume the numbers of units sold for a product is normally distributed. Using previous years' sales, I forecast unit sales next year to be 1,000 units with a standard deviation of 500 units. I then create a Monte Carlo model to forecast next years' profits. Can you see what can go wrong here? Monte Carlo modeling uses random numbers. In the sales forecast example, there's a 2.28% chance the model will select a negative sales number which is clearly impossible. Given that Monte Carlo models often use tens of thousands of simulations, it's extremely likely the final calculation will have been affected by impossible numbers.  This kind of mistake is insidious and hard to spot and even experienced analysts make it.

If you're a manager, you need to understand how your team has modeled data. 

• Ask what distributions they've used to model their data. 
• Ask them why they've used that distribution and what evidence they have that the data really is distributed that way. 
• Ask them how they're going to check their assumptions. 
• Most importantly, ask them if they have any detection mechanism in place to check for deviation from their expected distribution.

History - where the normal came from

Rather unsatisfactorily, there's no clear "Eureka!" moment for the discovery of the distribution, it seems to have been the accumulation of the work of a number of mathematicians. Abraham de Moivre  kicked off the process in 1733 but didn't formalize the distribution, leaving Gauss to explicitly describe it in 1801 [https://medium.com/@will.a.sundstrom/the-origins-of-the-normal-distribution-f64e1575de29].

Gauss used the normal distribution to model measurement errors and so predict the path of the asteroid Ceres [https://en.wikipedia.org/wiki/Normal_distribution#History]. This sounds a bit esoteric, but there's a point here that's still relevant. Any measurement taking process involves some form of error. Assuming no systemic bias, these errors are well-modeled by the normal distribution. So any unbiased measurement taking today (e.g opinion polling, measurements of particle mass, measurement of precision bolts, etc.) uses the normal distribution to calculate uncertainty.

In 1810, Laplace placed the normal distribution at the center of statistics by formulating the Central Limit Theorem. 

The math

The probability distribution function is given by:

\[f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e ^ {-\frac{1}{2} ( \frac{x - \mu}{\sigma}) ^ 2  }\]

\(\sigma\) is the standard deviation and \(\mu\) is the mean. In the normal distribution, the mean is the same as the mode is the same as the median.

This formula is almost impossible to work with directly, but you don't need to. There are extensive libraries that will do all the calculations for you.

Adding normally distributed parameters is easy:

\(\mu_{combined} = \mu_{houses} + \mu_{cars} \)

\(\sigma_{combined}^2 = \sigma_{houses}^2 + \sigma_{cars}^2\)

Wikipedia has an article on how to combine normally distributed quantities, e.g. addition, multiplication etc. see  https://en.wikipedia.org/wiki/Propagation_of_uncertainty.

Outliers have more fun

What's an outlier and why should you care?

Years ago I worked for a company that gave me t-shirt that said "Outliers have more fun". I've no idea what it meant, but outliers are interesting, and not in a good way. They'll do horrible things to your data and computing costs if you don't get a handle on them.

Simply put, an outlier is one or more data items that are extremely different from your other data items. Here's a joke that explains the idea:

There's a group of office workers drinking in a bar in Seattle. Bill Gates walks in and suddenly, the entire bar starts celebrating. Why? Because on average, they'd all become multi-millionaires.

Obviously, Bill Gates is the outlier in the data set. In this post, I'm going to explain what outliers do to data and what you can do to protect yourself.

(Jessica  Tam, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons)

Outliers and the mean

Let's start with the explaining the joke to death because everyone enjoys that.

Before Bill Gates walks in, there are 10 people in the bar drinking. Their salaries are: $80,000, $81,000, $82,000, $83,000, $84,000, $85,000, $86,000, $87,000, $88,000, and $89,000 giving a mean of $84,500. Let's assume Bill Gates earns $1,000,000,000 a year. Once Bill Gates walks into the bar, the new mean salary is $90,985,909; which is plainly not representative of the bar as a whole. Bill Gates is a massive outlier who's pulled the average way beyond what's representative.

How susceptible your data is to this kind of outlier effect depends on the type and distribution of your data. If your data is scores out of 10, and a "typical" score is 5, the average isn't going to be pulled too far away by an outlier (because the maximum is 10 and the minimum is zero, which are not hugely different from the typical value of 5). If there's no upper or lower limit (e.g salaries, house prices, amount of debt etc.), then you're vulnerable, and you may be even more vulnerable if your distribution is right skewed (e.g. something like a log normal distribution).

What can you do if this is the case? Use the median instead. The median is the middle value. In our Seattle bar example, the median is $84,500 before Bill Gates walks in and $85,000 afterwards. That's not much of a change and is much more representative of the salaries of everyone. This is the reason why you hear "median salaries" reported in government statistics rather than "mean salaries".

If you do use the median, please be aware that it has different mathematical properties from the mean. It's fine as a measure of the average, but if you're doing calculations based on medians, be careful.

Outliers and the standard deviation

The standard deviation is a representation of the spread of the data. The bigger the number, the wider the spread. In our bar example, before Bill Gates walks in, the standard deviation is $2,872. This seems reasonable as the salaries are pretty close together. After Bill Gates walks in, the standard deviation is $287,455,495 which is even bigger than the new mean. This number suggests all the salaries are quite different, which is not the case, only one is.

The standard deviation is susceptible to outliers in the same way the mean is, but for some reason, people often overlook it. I've seen people be very aware of outliers when they're calculating an average, but forget all about it when they're calculating a standard deviation.

What can you do? The answer here's isn't as clear. A good choice is the interquartile range (IQR), but it's not the same measurement. The IQR represents 75% of the data and not the 67% that the standard deviation does. For the bar, the IQR is $4,500 before Bill Gates walks in and $5,000 afterwards.  If you want a measure of 'dispersion', the IQR is a good choice, if you want a drop in replacement for the standard deviation, you'll have to give it more thought. 

Why the median and IQR are not drop in replacements for the mean and standard deviation

The mean and median are subtly different measures and have different mathematical properties. The same applies to standard deviation and IQR. It's important to understand the trade-offs when you use them.

Combining means is easy, we can do it through formula understood for hundreds of years. But we can't combine medians in the same way; the math doesn't work like that. Here's an example, let's imagine we have two bars, one with 10 drinkers earning a mean of $80,000, the other with 10 drinkers earning a mean of $90,000. The mean across the two bars is $85,000. We can do addition, subtraction, multiplication, division, and other operations with means. But if we know the median of the first bar is $81,000 and the median of the second bar is $89,000, we can't combine them. The same is true of the standard deviation and IQR, there are formula to combine standard deviations, but not IQRs.

In the Seattle bar example, we wanted one number to represent the salaries of the people in the bar. The best average is the median and the best measure of spread is the IQR, the reason being outliers. However, if we wanted an average we could apply across multiple bars, or if we wanted to do some calculations using the average and spread, we'd be better off with the mean and standard deviation.

Of course, it all comes down to knowing what you want and why. Like any job, you've got to know your tools.

The effect of more samples

Sometimes, more data will save you. This is especially true if your data is normally distributed and outliers are very rare. If your data distribution is skewed, it might not help that much. I've worked with some data sets with massive skews and the mean can vary widely depending on how many samples you take. Of course, if you have millions of samples, then you'll mostly be OK.

Outliers and calculation costs

This warning won't apply to everyone. I've built systems where the computing cost depends on the range of the data (maximum - minimum). The bigger the range, the more the cost. Outliers in this case can drive computation costs up, especially if there's a possibility of a "Bill Gates" type effect that can massively distort the data. If this applies to your system, you need to detect outliers and take action.

Final advice 

If you have a small sample size (10 or less): use the median and the IQR.

If your data is highly right skewed: use the median and the IQR.

Remember the median and the IQR are not the same as the mean and the standard deviation and be extremely careful using them in calculations.

If your computation time depends on the range of the data, check for outliers.

Using AI (LLM) to generate data science code

What AI offers data science code generation and what it doesn't

Using generative AI for coding support has become increasingly popular for good reason; the productivity gain can be very high. But what are its limits? Can you use code gen for real data science problems?

(I, for one, welcome our new AI overlords. D J Shin, CC BY-SA 3.0 , via Wikimedia Commons)

To investigate, I decided to look at two cases: a 'simple' piece of code generation to build a Streamlit UI, and a technically complicated case that's more typical of data science work. I generated Python code and evaluated it for correctness, structure, and completeness. The results were illuminating, as we'll see, and I think I understand why they came out the way they did.

My setup is pretty standard, I'm using Github copilot in Microsoft Visual Studio and Github Copilot directly from the website. In both cases, I chose the Claude model (more on why later).

Case 1: "commodity" UI code generation

The goal of this experiment was to see if I could automatically generate a good enough complete multi-page Streamlit app. The app was to have multiple dialog boxes on each page and was to be runnable without further modification.

Streamlit provides a simple UI for Python programs. It's several years old and extremely popular (meaning, there are plenty of code examples in Github). I've built apps using Streamlit, so I'm familiar with it and its syntax. 

The specification

The first step was a written English specification. I wrote a one-page Word document detailing what I wanted for every page of the app. I won't reproduce it here for brevity's sake, but here's a brief except:

The second page is called “Load model”. This will allow the user to load an existing model from a file. The page will have some descriptive text on what the page does. There will be a button that allows a user to load a file. The user will only be able to load a single with a file extension “.mdl”. If the user successfully loads a model, the code will load it into a session variable that the other pages can access. The “.mdl” file will be a JSON file and the software will check that the file is valid and follows some rules. The page will tell the user if the file has been successfully loaded or if there’s an error. If there’s an error, the page will tell the user what the error is.

In practice, I had to iterate on the specification a few times to get things right, but it only a took a couple of iterations.

What I got

Code generation was very fast and the results were excellent. I was able to run the application immediately without modification and it did what I wanted it to do.

(A screen shot of part of the generated Streamlit app.)

It produced the necessary Python files, but it also produced:

• a requirements.txt file - which was correct
• a dummy JSON file for my data, inferred from my description
• data validation code
• test code

I didn't ask for any of these things, it just produced them anyway.

There were several downsides though. 

I found the VS Code interface a little awkward to use, for me the Github Copilot web page was a much better experience (except that you have to copy the code).

Slight changes to my specification sometimes caused large changes to the generated code. For example, I added a sentence asking for a new dialog box and the code generation incorrectly dropped a page from my app. 

It seemed to struggle with long "if-then" type paragraphs, for example "If the user has loaded a model ...LONG TEXT... If the user hasn't loaded a model ...LONG TEXT...".

The code was quite old-fashioned in several ways. Code generation created the app pages in a pages folder and prefixed the pages with "1_", "2_" etc. This is how the demos on the Streamlit website are structured, but it's not how I would do it, it's kind of old school and a bit limited. Notably, the code generation didn't use some of the newer features of Streamlit; on the whole it was a year or so behind the curve.

Dependency on engine

I tried this with both Claude 3.5 and GPT 4o. Unequivocally, Claude gave the best answers.

Overall

I'm convinced by code generation here. Yes, it was a little behind the times and a little awkwardly structured, but it worked and it gave me something very close to what I wanted within a few minutes.

I could have written this myself (and I have done before), but I find this kind of coding tedious and time consuming (it would have taken me a day to do what I did using code gen in an hour). 

I will be using code gen for this type of problem in the future.

Case 2: data science code generation

What about a real data science problem, how well does it perform?

I chose to use random variables and quasi-Monte Carlo as something more meaty. The problem was to create two random variables and populate them with samples drawn from a quasi-Monte Carlo "random" number generator with a normal distribution. For each variable, work out the distribution (which we know should be normal). Combine the variables with convolution to create a third variable, and plot the resulting distribution. Finally, calculate the mean and standard deviation of all three variables.

The specification

I won't show it here for brevity, but it was a slightly longer than the description I gave above. Notably, I had to iterate on it several times.

What I got

This was a real mixed bag.

My first pass code generation didn't use quasi Monte Carlo at all. It normalized the distributions before the convolution for no good reason which meant the combined result was wrong. It used a histogram for the distribution which was kind-of OK. It did generate the charts just fine though. Overall, it was the kind of work a junior data scientist might produce.

On my second pass, I told it to use Sobel' sequences and I told it to use kernel density estimation to calculate the distribution. This time it did very well. The code was nicely commented too. Really surprisingly, it used the correct way of generating sequences (using dimensions).

(After some prompting, this was my final chart, which is correct.)

Dependency on engine

I tried this with both Claude 3.5 and GPT 4o. Unequivocally, Claude gave the best answers.

Overall

I had to be much more prescriptive here to get what I wanted, but the results were good, but only because I knew to tell it to use Sobel' and I knew to tell it to use kernel density estimation. 

Again, I'm convinced that code gen works.

Observations

The model

I tried the experiment with both Claude 3.5 and GPT 4o. Claude gave much better results. Other people have reported similar experiences.

Why this works and some fundamental limitations

Github has access to a huge code base, so the LLM is based on the collective wisdom of a vast number of programmers. However, despite appearances, it has no insight; it can't go beyond what others have done. This is why the code it produced for the Streamlit demo was old-fashioned. It's also why I had to be prescriptive for my data science case, for example, it just didn't understand what quasi Monte Carlo meant without additional prompting.

AI is known to hallucinate, and we see see something of that here. You really have to know what you're doing to use AI generated code. If you blindly implement AI generated code, things are going to go badly for you very quickly.

Productivity

Code generation and support is a game changer. It ramps up productivity enormously. I've heard people say, it's like having a (free) senior engineer by your side. I agree. Despite the issues I've come across, code generation works "good enough".

Employment

This has obvious implications for employment. With AI code generation and with AI coding support, you need fewer software engineers/analysts/data scientists. The people you do need are those with more insight and the ability spot where the AI generated code has gone wrong, which is bad news for for more junior people or those entering the workforce. It may well be a serious problem for students seeking internships.

Let me say this plainly: people will lose their jobs because of this technology.

My take on the employment issue and what you can do

There's an old joke that sums things up. "A householder calls in a mechanic because their washing machine had broken down. The mechanic looks at the washing machine and rocks it around a bit. Then the mechanic kicks the machine. It starts working! The mechanic writes a bill for $200. The householder explodes, '$200 to kick a washing machine, this is outrageous!'. The mechanic thinks for a second and says, 'You're quite right. Let me re-write the bill'. The new bill says 'Kicking the washing machine $1, knowing where to kick the washing machine $199'." To put it bluntly, you need to be the kind of mechanic that knows where to kick the machine.

(You've got to know where to kick it. LG전자, CC BY 2.0 , via Wikimedia Commons)

Code generation has no insight. It makes errors. You have to have experience and insight to know when it's gone wrong. Not all human software engineers have that insight.

You should be very concerned if:

• You're junior in your career or you're just entering the workforce.
• You're developing BI-type apps as the main or only thing you do.
• There are many people doing exactly the same software development work as you.

If that applies to you, here's my advice:

• Use code generation and code support. You need to know first hand what it can do and the threat it poses. Remember, it's a productivity boost and the least productive people are the first to go.
• Develop domain knowledge. If your company is in the finance industry, make sure you understand finance, which means knowing the legal framework etc.. If it's a drug discovery, learn the principles of drug discovery. Get some kind of certification (online courses work fine). Apply your knowledge to your work. Make sure your employer knows it.
• Develop specialist skills, e.g. statistics. Use those skills in your work.
• Develop human skills. This means talking to customers, talking to people in other departments.

Some takeaways

• AI generated code is good enough for use, even in more complicated cases.
• It's a substantial productivity boost. You should be using it.
• It's a tool, not a magic wand. It does get things wrong and you need to be skilled enough to spot errors.

Essential business knowledge: the Central Limit Theorem

Knowing the Central Limit Theorem means avoiding costly mistakes

I've spoken to well-meaning analysts who've made significant mistakes because they don't understand the implications of one of the core principles of statistics; the Central Limit Theorem (CLT). These errors weren't trivial either, they affected salesperson compensation and the analysis of A/B tests. More personally, I've interviewed experienced candidates who made fundamental blunders because they didn't understand what this theorem implies.

The CLT is why the mean and standard deviation work pretty much all the time but it's also why they only work when the sample size is "big enough". It's why when you're estimating the population mean it's important to have as large a sample size as you can. It's why we use the Student's t-test for small sample sizes and why other tests are appropriate for large sample sizes. 

In this blog post, I'm going to explain what the CLT is, some of the theory behind it (at a simple level), and how it drives key business statistics. Because I'm trying to communicate some fundamental ideas, I'm going to be imprecise in my language at first and add more precision as I develop the core ideas. As a bonus, I'll throw in a different version of the CLT that has some lesser-known consequences.

How we use a few numbers to represent a lot of numbers

In all areas of life, we use one or two numbers to represent lots of numbers. For example, we talk about the average value of sales, the average number of goals scored per match, average salaries, average life expectancy, and so on. Usually, but not always, we get these numbers through some form of sampling, for example, we might run a salary survey asking thousands of people their salary, and from that data work out a mean (a sampling mean). Technically, the average is something mathematicians call a "measure of central tendency" which we'll come back to later.

We know not everyone will earn the mean salary and that in reality, salaries are spread out. We express the spread of data using the standard deviation. More technically, we use something called a confidence interval which is based on the standard deviation. The standard deviation (or confidence interval) is a measure of how close we think our sampling mean is to the true (population) mean (how confident we are).

In practice, we use standard formula for the mean and standard deviation. These are available as standard functions in spreadsheets and programming languages. Mathematically, this is how they're expressed.

\[sample\; mean\; \bar{x}= \frac{1}{N}\sum_{i=0}^{N}x_i\]

\[sample\; standard\; deviation\; s_N = \sqrt{\frac{1}{N} \sum_{i=0}^{N} {\left ( x_i - \bar{x} \right )} ^ 2 } \]

All of this seems like standard stuff, but there's a reason why it's standard, and that's the central limit theorem (CLT).

The CLT

Let's look at three different data sets with different distributions: uniform, Poisson, and power law as shown in the charts below.

These data sets are very, very different. Surely we have to have different averaging and standard deviation processes for different distributions? Because of the CLT, the answer is no. 

In the real world, we sample from populations and take an average (for example, using a salary survey), so let's do that here. To get going, let's take 100 samples from each distribution and work out a sample mean. We'll do this 10,000 times so we get some kind of estimate for how spread out our sample means are.

The top charts show the original population distribution and the bottom charts show the result of this sampling means process. What do you notice?

The distribution of the sampling means is a normal distribution regardless of the underlying distribution.

This is a very, very simplified version of the CLT and it has some profound consequences, the most important of which is that we can use the same averaging and standard deviation functions all the time.

Some gentle theory

Proving the CLT is very advanced and I'm not going to do that here. I am going to show you through some charts what happens as we increase the sample size.

Imagine I start with a uniform random distribution like the one below. 

I want to know the mean value, so I take some samples and work out a mean for my samples. I do this lots of times and work out a distribution for my mean. Here's what the results look like for a sample size of 2, 3,...10,...20,...30,...40. 

As the sample size gets bigger, the distribution of the means gets closer to a normal distribution. It's important to note that the width of the curve gets narrower with increasing sample size. Once the distribution is "close enough" to the normal distribution (typically, around a sample size of 30), you can use normal distribution methods like the mean and standard deviation.

The standard deviation is a measure of the width of the normal distribution. For small sample sizes, the standard deviation underestimates the width of the distribution, which has important consequences.

Of course, you can do this with almost any underlying distribution, I'm just using a uniform distribution because it's easier to show the results 

Implications for averages

The charts above show how the distribution of the means changes with sample size. At low sample sizes, there are a lot more "extreme" values as the difference between the sample sizes of 2 and 40 shows.  Bear in mind, the width of the distribution is an estimate of the uncertainty in our measurement of the mean.

For small sample sizes, the mean is a poor estimator of the "average" value; it's extremely prone to outliers as the shape of the charts above indicates. There are two choices to fix the problem: either increase the sample size to about 30 or more (which often isn't possible) or use the median instead (the median is much less prone to outliers, but it's harder to calculate).

The standard deviation (and the related confidence interval) is a measure of the uncertainty in the mean value. Once again, it's sensitive to outliers. For small sample sizes, the standard deviation is a poor estimator for the width of the distribution. There are two choices to fix the problem, either increase the sample size to 30 or more (which often isn't possible) or use quartiles instead (for example, the interquartile range, IQR).

If this sounds theoretical, let me bring things down to earth with an example. Imagine you're evaluating salesperson performance based on deals closed in a quarter. In B2B sales, it's rare for a rep to make 30 sales in a quarter, in fact, even half that number might be an outstanding achievement. With a small number of samples, the distribution is very much not normal, and as we've seen in the charts above, it's prone to outliers. So an analysis based on mean sales with a standard deviation isn't a good idea; sales data is notorious for outliers. A much better analysis is the median and IQR. This very much matters if you're using this analysis to compare rep performance.

Implications for statistical tests

A hundred years ago, there were very few large-scale tests, for example, medical tests typically involved small numbers of people. As I showed above, for small sample sizes the CLT doesn't apply. That's why Gosset developed the Student's t-distribution: the sample sizes were too small for the CLT to kick in, so he needed a rigorous analysis procedure to account for the wider-than-normal distributions. The point is, the Student's t-distribution applies when sample sizes are below about 30.

Roll forward 100 years and we're now doing retail A/B testing with tens of thousands of samples or more. In large-scale A/B tests, the z-test is a more appropriate test. Let me put this bluntly: why would you use a test specifically designed for small sample sizes when you have tens of thousands of samples?

It's not exactly wrong to use the Student's t-test for large sample sizes, it's just dumb. The special features of the Student's t-test that enable it to work with small sample sizes become irrelevant. It's a bit like using a spanner as a hammer; if you were paying someone to do construction work on your house and they were using the wrong tool for something simple, would you trust them with something complex?

I've asked about statistical tests at interview and I've been surprised at the response. Many candidates have immediately said Student's t as a knee-jerk response (which is forgivable). Many candidates didn't even know why Student's t was developed and its limitations (not forgivable for senior analytical roles). One or two even insisted that Student's t would still be a good choice even for sample sizes into the hundreds of thousands. It's very hard to progress candidates who insist on using the wrong approach even after it's been pointed out to them.

As a practical matter, you need to know what statistical tools you have available and their limitations.

Implications for sample sizes

I've blithely said that the CLT applies above a sample size of 30. For "most" distributions, a sample size of about 30 is a reasonable rule-of-thumb, but there's no theory behind it. There are cases where a sample size of 30 is insufficient. 

At the time of writing, there's a discussion on the internet about precisely this point. There's a popular article on LessWrong that illustrates how quickly convergence to the normal can happen: https://www.lesswrong.com/posts/YM6Qgiz9RT7EmeFpp/how-long-does-it-take-to-become-gaussian but there's also a counter article that talks about cases where convergence can take much longer: https://two-wrongs.com/it-takes-long-to-become-gaussian. 

The takeaway from this discussion is straightforward. Most of the time, using a sample size of 30 is good enough for the CLT to kick-in, but occasionally you need larger sample sizes. A good way to test this is to use larger sample sizes and see if there's any trend in the data. 

General implications

The CLT is a double-edged sword: it enables us to use the same averaging processes regardless of the underlying distribution, but it also lulls us into a false sense of security and analysts have made blunders as a result.

Any data that's been through an averaging process will tend to follow a normal distribution. For example, if you were analyzing average school test scores you should expect them to follow a normal distribution, similarly for transaction values by retail stores, and so on. I've seen data scientists claim brilliant data insights by announcing their data is normally distributed, but they got it through an averaging process, so of course it was normally distributed. 

The CLT is one of the reasons why the normal distribution is so prevalent, but it's not the only reason and of course, not all data is normally distributed. I've seen junior analysts make mistakes because they've assumed their data is normally distributed when it wasn't. 

A little more rigor

I've been deliberately loose in my description of the CLT so far so I can explain the general idea. Let's get more rigorous so we can dig into this a bit more. Let's deal with some terminology first.

Central tendency

In statistics, there's something called a "central tendency" which is a measurement that summarizes a set of data by giving a middle or central value. This central value is often called the average. More formally, there are three common measures of central tendency:

• The mode. This is the value that occurs most often.
• The median. Rank order the data and this is the middle value.
• The mean. Sum up all the data and divide by the number of values.

These three measures of central tendency have different properties, different advantages, and different disadvantages. As an analyst, you should know what they are.

(Depending on where you were educated, there might be some language issues here. My American friends tell me that in the US, the term "average" is always a synonym for the mean, in Britain, the term "average" can be the mean, median, or mode but is most often the mean.)

For symmetrical distributions, like the normal distribution, the mean, median, and mode are the same, but that's not the case for non-symmetrical distributions. 

The term "central" in the central limit theorem is referring to the central or "average" value.

iid

If you were taught about the Central Limit Theorem, you were probably taught that it only applies to iid data, which means independent and identically distributed. Here's what iid means. 

• Each sample in the data is independent of the other samples. This means selecting or removing a sample does not affect the value of another sample.
• All the samples come from the same probability distribution.

Actually, this isn't true. The CLT applies even if the distributions are not the same. However, the independence requirement still holds,

When the CLT doesn't apply

Fortunately for us, the CLT applies to almost all distributions an analyst might come across, but there are exceptions. The underlying distribution must have a finite variance, which rules out using it with distributions like the Cauchy distribution. The samples must be iid as I said before.

A re-statement of the CLT

Given data that's distributed with a finite variance and is iid, if we take n samples, then:

• as \( n \to \infty \), the sample mean converges to the population mean
• as \( n \to \infty \), the distribution of the sample means approximates a normal distribution

Note this formulation is in terms of the mean. This version of the CLT also applies to sums because the mean is just the sum divided by a constant (the number of samples).

A different version of the CLT

There's another version of the CLT that's not well-known but does come up from time to time in more advanced analysis. The usual version of the CLT is expressed in terms of means (which is the sum divided by a constant). If instead of taking the sum of the samples, we take their product, then instead of the products tending to a normal distribution they tend to a log-normal distribution. In other words, where we have a quantity created from the product of samples then we should expect it to follow a log-normal distribution. 

What should I take away from all this?

Because of the CLT, the mean and standard deviation mostly work regardless of the underlying distribution. In other words, you don't have to know how your data is distributed to do basic analysis on it. BUT the CLT only kicks in above a certain sample size (which can vary with the underlying distribution but is usually around 30) and there are cases when it doesn't apply. 

You should know what to do when you have a small sample size and know what to watch out for when you're relying on the CLT.

You should also understand that any process that sums (or products) data will lead to a normal distribution (or log-normal).

W.E.B. Du Bois - data scientist

Changing the world through charts

Two of the key skills of a data scientist are informing and persuading others through data. I'm going to show you how one man, and his team, used novel visualizations to illustrate the lives of African-Americans in the United States at the start of the 20th century. Even though they created their visualizations by hand, these visualizations still have something to teach us over 100 years later. The team's lack of computers freed them to try different forms of data visualizations; sometimes their experimentation was successful, sometimes less so, but they all have something to say and there's a lesson here about communication for today's data scientists.

I'm going to talk about W.E.B. Du Bois and the astounding charts his team created for the 1900 Paris exhibition.

(W.E.B. Du Bois in 1904 and one of his 1900 data visualizations.)

Who was W.E.B. Du Bois?

My summary isn't going to do his amazing life justice so I urge you to read any of these short descriptions of who he was and what he did:

• Wikipedia
• NAACP
• Britannica
• History.com

To set the scene here's just a very brief list of some of the things he did. Frankly, summarizing his life in a few lines is ridiculous.

• Born 1868, Great Barrington, Massachusetts.
• Graduate of Fisk University and Harvard - the first African-American to gain a Ph.D. from Harvard.
• Conducted ground-breaking sociological work in Philadelphia, Virginia, Alabama, and Georgia.
• His son died in 1899 because no white doctor would treat him and black doctors were unavailable.
• Was the primary organizer of "The Exhibit of American Negroes" at the Exposition Universelle held in Paris between April and November 1900.
• NAACP director and editor of the NAACP magazine The Crisis.
• Debated Lothrop Stoddard, a "scientific racist" in 1929 and thoroughly bested him.
• Opposed US involvement in World War I and II.
• Life-long peace activist and campaigner, which led to the FBI investigating him in the 1950s as a suspected communist. They withheld his passport for 8 years.
• Died in Ghana in 1963.

Visualizing Black America at the start of the twentieth century

In 1897, Du Bois was a history professor at Atlanta University. His former classmate and friend, Thomas Junius Calloway, asked him to produce a study of African-Americans for the 1900 Paris world fair, the "Exposition Universelle". With the help of a large team of Atlanta University students and alumni, Du Bois gathered statistics on African-American life over the years and produced a series of infographics to bring the data to life. Most of the names of the people who worked on the project are unknown, and it's a mystery who originated the form of the plots, but the driving force behind the project was undoubtedly Du Bois. Here are some of my favorite infographics from the Paris exhibition.

The chart below shows where African-Americans lived in Georgia in 1890. There are four categories: 

• Red - country and villages
• Yellow - cities 2,500-5,000
• Blue - cities 5,000-10,000
• Green - cities over 10,000

the length of the lines is proportional to the population and obviously, the chart shows the huge majority of the population lived in the country and villages. I find the chart striking for three reasons: it doesn't follow any of the modern charting conventions, it clearly represents the data, and it's visually very striking. My criticism is that the design makes it hard to visually quantify the differences, for example, how many more people live in the country and villages compared to cities 5,000-10,000? If I were drawing a chart with the same data today, I might use an area chart to represent the same data; it would quantify things better, but it would be far less visually interesting.

The next infographic is two choropleth charts that show the African-American population of Georgia counties in 1870 and 1880. Remember that the US civil war ended in 1865, and with the Union victory came freedom for the slaves. As you might expect, there was a significant movement of the now-free people. Looking at the charts in detail raises several questions, for example, why did some areas see a growth in the African-American population while other areas did not? Why did the highest populated areas remain the highest populated? The role of any good visualization is to prompt meaningful questions.

This infographic shows the income and expenditure of 150 African-American families in Georgia. The income bands are on the left-hand side, and the bar chart breaks down the families' expenses by category:

• Black - rent
• Purple - food
• Pink - clothes
• Dark blue - direct taxes
• Light blue - other expenses and savings

There are several notable observations from this chart: the disappearance of rent above a certain income level, the rise in other expenses and savings with rising income, and the declining fraction spent on clothing. There's a lot on this chart and it's worthy of greater study; Du Bois' team crammed a great deal of meaning into a single page. For me, the way the key is configured at the top of the chart doesn't quite work, but I'm willing to give the team a pass on this because it was created in the 19th century. A chart like this wouldn't look out of place in a 2022 report - which of itself is startling.

My final example is a comparison of the occupations of African-Americans and the white population in Georgia. It's a sort-of pie chart, with the upper quadrant showing African Americans and the bottom quadrant showing the white population. Occupations are color-coded:

• Red - agriculture, fishery, and mining
• Yellow - domestic and personal service
• Blue - manufacturing and mechanical industries
• Grey - trade and transportation
• Brown - professions

The fraction of the population in these employment categories is written on the slices, though it's hard to read because the contrast isn't great. Notably, the order of the occupations is reversed from the top to the bottom quadrant, which has the effect of making the sizes of the slices easier to compare - this can't be an accident. I'm not a fan of pie charts, but I do like this presentation.

Influences on later European movements - or not?

Two things struck me about Du Bois' charts: how modern they looked and how similar they were to later art movements like the Italian Futurists and Bauhaus. 

At first glance, his charts look to me like they'd been made in the 1960s. The typography and coloring were obviously pre-computerization, but everything else about them suggests modernity, from the typography to the choice of colors to the layout. The experimentation with form is striking and is another reason why this looks very 1960s to me; perhaps the use of computers to visualize data has constrained us too much. Remember, Du Bois's mission was to explain and convince and he chose his charts and their layout to do so, hence the experimentation with form. It's quite astonishing how far ahead of his time he was.  

Italian Futurism started in 1909 and largely fizzled out at the end of the second world war due to its close association with fascism. The movement emphasized the abstract representation of dynamism and technology among other things. Many futurist paintings used a restricted color palette and have obvious similarities with Du Bois' charts, here are just a few examples (below). I couldn't find any reliable articles that examined the links between Du Bois' work and futurism.

Numbers In Love - Giacomo Balla Image from WikiArt	Music - Luigi Russolo Image from WikiArt

The Bauhaus design school (1919-1933) sought to bring modernity and artistry into mass production and had a profound and lasting effect on the design of everyday things, even into the present day. Bauhaus designs tend to be minimal ("less is more") and focus on functionality ("form follows function") but can look a little stark. I searched, but I couldn't find any scholarly study of the links between Du Bois and Bauhaus, however, the fact the Paris exposition charts and the Bauhaus work use a common visual language is striking. Here's just one example, a poster for the Bauhaus school from 1923.

(Joost Schmidt, Public domain, via Wikimedia Commons)

Du Bois' place in data visualization

I've read a number of books on data visualization. Most of them include Nightingale's coxcomb plots and Playfair's bar and pie charts, but none of them included Du Bois charts.  Du Bois didn't originate any new chart types, which is maybe why the books ignore him, but his charts are worth studying because of their experimentation with form, their use of color, and most important of all, their ability to communicate meaning clearly. Ultimately, of course, this is the only purpose of data visualization.

Reading more

W. E. B. Du Bois's Data Portraits: Visualizing Black America, Whitney Battle-Baptiste, Britt Rusert. This is the book that brought these superb visualizations to a broader audience. It includes a number of full-color plates showing the infographics in their full glory.

The Library of Congress has many more infographics from the Paris exhibition, it also has photos too. Take a look at it for yourself here https://www.loc.gov/collections/african-american-photographs-1900-paris-exposition/?c=150&sp=1&st=list - but note the charts are towards the end of the list. I took all my charts in this article from the Library of Congress site. 

"W.E.B. Du Bois’ Visionary Infographics Come Together for the First Time in Full Color" article in the Smithsonian magazine that reviews the Battle-Baptiste book (above).

"W. E. B. Du Bois' Hand-Drawn Infographics of African-American Life (1900)" article in Public Domain Review that reviews the Battle-Baptiste book (above).

Prediction, distinction, and interpretation: the three parts of data science

What does data science boil down to?

Data science is a relatively new discipline that means different things to different people (most notably, to different employers). Some organizations focus solely on machine learning, while other lean on interpretation, and yet others get close to data engineering. In my view, all of these are part of the data science role. 

I would argue data science generally is about three distinct areas:

• Prediction. The ability to accurately extrapolate from existing data sets to make forecasts about future behavior. This is the famous machine learning aspect and includes solutions like recommender systems.
• Distinction. The key question here is: "are these numbers different?". This includes the use of statistical techniques to decide if there's a difference or not, for example, specifying an A/B test and explaining its results. 
• Interpretation. What are the factors that are driving the system? This is obviously related to prediction but has similarities to distinction too.

(A similar view of data science to mine: Calvin.Andrus, CC BY-SA 3.0, via Wikimedia Commons)

I'm going to talk through these areas and list the skills I think a data scientist needs. In my view, to be effective, you need all three areas. The real skill is to understand what type of problem you face and to use the correct approach.

Distinction - are these numbers different?

This is perhaps the oldest area and the one you might disagree with me on. Distinction is firmly in the realm of statistics. It's not just about A/B tests or quasi-experimental tests, it's also about evaluating models too.

Here's what you need to know:

• Confidence intervals.
• Sample size calculations. This is crucial and often overlooked by experienced data scientists. If your data set is too small, you're going to get junk results so you need to know what too small is. In the real world. increasing the sample size is often not an option and you need to know why.
• Hypothesis testing. You should know the difference between a t-test and a z-test and when a z-test is appropriate (hint: sample size).
• α, β, and power. Many data scientists have no idea what statistical power is. If you're doing any kind of statistical testing, you need to have a firm grasp of power.
• The requirements for running a randomized control trial (RCT). Some experienced data scientists have told me they were analyzing results from an RCT, but their test just wasn't an RCT - they didn't really understand what an RCT was.
• Quasi-experimental methods. Sometimes, you just can't run an RCT, but there are other methods you can use including difference-in-difference, instrumental variables, and regression discontinuity.  You need to know which method is appropriate and when. 
• Regression to the mean. This is why you almost always need a control group. I've seen experienced data scientists present results that could almost entirely be explained by regression to the mean. Don't be caught out by one of the fundamentals of statistics.

Prediction - what will happen next?

This is the piece of data science that gets all the attention, so I won't go into too much detail.

Here's what you need to know:

• The basics of machine learning models, including:
• Generalized linear modeling
• Random forests (including knowing why they are often frowned upon)
• k-nearest neighbors/k-means clustering
• Support Vector Machines
• Gradient boosting.
• Cross-validation, regularization, and their limitations.
• Variable importance and principal component analysis.
• Loss functions, including RMSE.
• The confusion matrix, accuracy, sensitivity, specificity, precision-recall and ROC curves.

There's one topic that's not on any machine learning course or in any machine learning book that I've ever read, but it's crucially important: knowing when machine learning fails and when to stop a project.  Machine learning doesn't work all the time.

Interpretation - what's going on?

The main techniques here are often data visualization. Statistical summaries are great, but they can often mislead. Charts give a fuller picture. 

Here are some techniques all data scientists should know:

• Heatmaps
• Violin plots
• Scatter plots and curve fitting
• Bar charts
• Regression and curve fitting.

They should also know why pie charts in all their forms are bad. 

A good knowledge of how charts work is very helpful too (the psychology of visualization).

What about SQL and R and Python...?

You need to be able to manipulate data to do data science, which means SQL, Python, or R. But plenty of people use these languages without being data scientists. In my view, despite their importance, they're table stakes.

Book knowledge vs. street knowledge

People new to data science tend to focus almost exclusively on machine learning (prediction in my terminology) which leaves them very weak on data analysis and data exploration; even worse, their lack of statistical knowledge sometimes leads them to make blunders on sample size and loss functions. No amount of cross-validation, regularization, or computing power will save you from poor modeling choices. Even worse, not knowing statistics can lead people to produce excellent models of regression to the mean.

Practical experience is hugely important; way more important than courses. Obviously, a combination of both is best, which is why PhDs are highly sought after; they've learned from experience and have the theoretical firepower to back up their practical knowledge.

What is beta in statistical testing?

\(\beta\) is \(\alpha\) if there's an effect

In hypothesis testing, there are two kinds of errors:

• Type I - we say there's an effect when there isn't. The threshold here is \(\alpha\).
• Type II - we say there's no effect when there really is an effect. The threshold here is \(\beta\).

This blog post is all about explaining and calculating \(\beta\).

The null hypothesis

Let's say we do an A/B test to measure the effect of a change to a website. Our control branch is the A branch and the treatment branch is the B branch. We're going to measure the conversion rate \(C\) on both branches. Here are our null and alternative hypotheses:

• \(H_0: C_B - C_A = 0\) there is no difference between the branches
• \(H_1: C_B - C_A \neq 0\) there is a difference between the branches

Remember, we don't know if there really is an effect, we're using procedures to make our best guess about whether there is an effect or not, but we could be wrong. We can say there is an effect when there isn't (Type I error) or we can say there is no effect when there is (Type II error).

Mathematically, we're taking the mean of thousands of samples so the central limit theorem (CLT) applies and we expect the quantity \(C_B - C_A\) to be normally distributed. If there is no effect, then \(C_B - C_A = 0\), if there is an effect \(C_B - C_A \neq 0\).

\(\alpha\) in a picture

Let's assume there is no effect. We can plot out our expected probability distribution and define an acceptance region (blue, 95% of the distribution) and two rejection regions (red, 5% of the distribution). If our measured \(C_B - C_A\) result lands in the blue region, we will accept the null hypothesis and say there is no effect, If our result lands in the red region, we'll reject the null hypothesis and say there is an effect. The red region is defined by \(\alpha\).

One way of looking at the blue area is to think of it as a confidence interval around the mean \(x_0\):

\[\bar x_0 + z_\frac{\alpha}{2} s \; and \; \bar x_0 + z_{1-\frac{\alpha}{2}} s \]

In this equation, s is the standard error in our measurement. The probability of a measurement \(x\) lying in this range is:

\[0.95 = P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ] \]

If we transform our measurement \(x\) to the standard normal \(z\), and we're using a 95% acceptance region (boundaries given by \(z\) values of 1.96 and -1.96), then we have for the null hypothesis:

\[0.95 = P[-1.96 < z < 1.96]\]

\(\beta\) in a picture

Now let's assume there is an effect. How likely is it that we'll say there's no effect when there really is an effect? This is the threshold \(\beta\).

To draw this in pictures, I want to take a step back. We have two hypotheses:

• \(H_0: C_B - C_A = 0\) there is no difference between the branches
• \(H_1: C_B - C_A \neq 0\) there is a difference between the branches

We can draw a distribution for each of these hypotheses. Only one distribution will apply, but we don't know which one.

If the null hypothesis is true, the blue region is where our true negatives lie and the red region is where the false positives lie. The boundaries of the red/blue regions are set by \(\alpha\). The value of \(\alpha\) gives us the probability of a false positive.

If the alternate hypothesis is true, the true positives will be in the green region and the false negatives will be in the orange region. The boundary of the green/orange regions is set by \(\beta\). The value of \(\beta\) gives us the probability of a false negative.

Calculating \(\beta\)

Calculating \(\beta\) is calculating the orange area of the alternative hypothesis chart. The boundaries are set by \(\alpha\) from the null hypothesis. This is a bit twisty, so I'm going to say it again with more words to make it easier to understand.

\(\beta\) is about false negatives. A false negative occurs when there is an effect, but we say there isn't. When we say there isn't an effect, we're saying the null hypothesis is true. For us to say there isn't an effect, the measured result must lie in the blue region of the null hypothesis distribution.

To calculate \(\beta\), we need to know what fraction of the alternate hypothesis lies in the acceptance region of the null hypothesis distribution.

Let's take an example so I can show you the process step by step.

1. Assuming the null hypothesis, set up the boundaries of the acceptance and rejection region. Assuming a 95% acceptance region and an estimated mean of x, this gives the acceptance region as:
   \[P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ] \] which is the mean and 95% confidence interval for the null hypothesis. Our measurement \(x\) must lie between these bounds.
2. Now assume the alternate hypothesis is true. If the alternate hypothesis is true, then our mean is \(\bar x_1\).
3. We're still using this equation from before, but this time, our distribution is the alternate hypothesis.
   \[P \left [ \bar x_0 + z_\frac{\alpha}{2} s < x < \bar x_0 + z_{1-\frac{\alpha}{2}} s \right ] ] \]
4. Transforming to the standard normal distribution using the formula \(z = \frac{x - \bar x_1}{\sigma}\), we can write the probability \(\beta\) as:
   \[\beta = P \left [ \frac{\bar x_0 + z_\frac{\alpha}{2} s - \bar x_1}{s} < z < \frac{ \bar x_0 + z_{1-\frac{\alpha}{2}} s - \bar x_1}{s} \right ] \]

This time, let's put some numbers in. 

• \(n = 200,000\) (100,000 per branch)
• \(C_B = 0.062\)
• \(C_A =  0.06\)
• \(\bar x_0= 0\) - the null hypothesis
• \(\bar x_1 = 0.002\) - the alternate hypothesis
• \(s = 0.00107\)  - this comes from combining the standard errors of both branches, so \(s^2 = s_A^2 + s_B^2\), and I'm using the usual formula for the standard error of a proportion, for example, \(s_A = \sqrt{\frac{C_A(1-C_A)}{n} }\)

Plugging them all in, this gives:
\[\beta = P[ -3.829 < z < 0.090]\]
which gives \(\beta = 0.536\)

This is too hard

This process is complex and involves lots of steps. In my view, it's too complex. It feels to me that there must be an easier way of constructing tests. Bayesian statistics holds out the hope for a simpler approach, but widespread adoption of Bayesian statistics is probably a generation or two away. We're stuck with an overly complex process using very difficult language.

Reading more

• https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4347431/ 
• https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5635437.3/
• https://abtestguide.com/calc/