The Normal Distribution (or a Gaussian) shows up widely in statistics as a result of the Central Limit Theorem. Specifically, the Central Limit Theorem says that (in most common scenarios besides the stock market) anytime “a bunch of things are added up,” a normal distribution is going to result.
But why? Why that distribution? Why is it special? Why not some other distribution? Are there other statistical distributions where this happens?
Teaser: the answer is yes, there are other distributions that are special in the same way as the Normal distribution. The Normal distribution is still the most special because:
If you’re intrigued, read on! I’ll give an intuitive sketch of the Central Limit Theorem and a quick proof-sketch before diving into the Normal distribution’s oft-forgotten cousins.
Here is a quick official statement:
Formal statement of the Central Limit Theorem
Where Φ represents the normal distribution with mean and variance as given. (You may be used to seeing the equivalent standard deviation σ/√n instead). The “in distribution” is a technical bit about how the convergence works. We’ll ignore such technicalities from here on out.
The Central Limit Theorem shows up in all sorts of places in real-world situations. For example, it’s a pretty reasonable assumption that your height can be expressed as the sum of a bunch of factors related to, among others:
Take a whole bunch of factors, each of which makes a small difference in your final (adult) height, and, presto, you end up with a (roughly) normal distribution for human heights!
Note that I cheated slightly – the variables here aren’t i.i.d. But the independence assumption is a reasonable approximation, and there are stronger versions of the central limit theorem that relax the identical-distribution hypothesis. We did choose to leave out the cases of extreme genetic conditions that affect height however.
So in sum, any time something you measure is made up of a whole bunch of contributions from smaller parts being added up, you are likely to end up with a normal distribution.
This proof is necessarily a sketch because, well, if you want a full proof with all of the analysis and probability theory involved, go read a textbook. The main point I want to get across is that there is a reason Euler’s constant _e _shows up.
First of all, we will need one high-powered mathematical tool. To every reasonable random variable X there is a_ characteristic function φ _which is, in essence, the Fourier Transform of the Probability-Density Function (PDF) of X.
#mathematics #central-limit-theorem #statistics #data-science #data analysis
In a series of weekly articles, I will be covering some important topics of statistics with a twist.
The goal is to use Python to help us get intuition on complex concepts, empirically test theoretical proofs, or build algorithms from scratch. In this series, you will find articles covering topics such as random variables, sampling distributions, confidence intervals, significance tests, and more.
At the end of each article, you can find exercises to test your knowledge. The solutions will be shared in the article of the following week.
Articles published so far:
As usual, the code is available on my GitHub.
#statistics #distribution #python #machine-learning #sampling distributions with python #sampling distributions
What is Database Normalization
Database normalization is the step by step process of organizing data to minimize data redundancy i.e. Data duplication which in turn ensures data consistency
#sql server #1nf #2nf #3nf #4nf #5nf #6nf #data #database in sql server #normalization #normalization forms #normalization in database #what is data
I see a lot of data scientists using tests such as the Shapiro-Wilk test and the Kolmogorov–Smirnov to test for normality. Stop doing this. Just stop. If you’re not yet convinced (and I don’t blame you!), let me show you why these are a waste of your time.
We should care about normality. It’s an important assumption that underpins a wide variety of statistical procedures. We should always be sure of our assumptions and make efforts to check that they are correct. However, normality tests are not the way for us to do this.
However, in large samples (n > 30) which most of our work as data scientists is based upon the Central Limit Theorem usually applies and we need not worry about the normality of our data. But in cases where it does not apply let’s consider how we can check for normality in a range of different samples.
First let us consider a small sample. Say n=10. Let’s look at the histogram for this data.
Histogram of x (n=10). (Image by author)
Is this normally distributed? Doesn’t really look like it — does it? Hopefully you’re with me and accept that this isn’t normally distributed. Now let’s perform the Shapiro-Wilk test on this data.
Oh. p=0.53. No evidence to suggest that x is not normally distributed. Hmm. What do you conclude then. Well, of course, not being evidence that x is not normally distributed does not mean that x is normally distributed. What’s actually happening is that in small samples the tests are _underpowered _to detect deviations from normality.
Normal Q-Q Plot of x (n=10). (Image by author)
The best way to assess normality is through the use of a quantile-quantile plot — Q-Q plot for short. If the data is normally distributed we would expect to see a straight line. This data shows some deviation from normality, the line is not very straight. There appears to be some issues in the tail. Admittedly, without more data it is hard to say.
With this data, I would have concerns about assuming normality as there appears to be some deviation in the Q-Q plot and in the histogram. But, if we had just relied on our normality test, we wouldn’t have picked this up. This is because the test is underpowered in small samples.
Now let’s take a look at normality testing in a large sample (n=5000). Let’s take a look at a histogram.
Histogram of x (n=5000). (Image by author)
I hope you’d all agree that this looks to be normally distributed. Okay, so what does the Shapiro-Wilk test say. Bazinga! p=0.001. There’s very strong evidence that x is not normally distributed. Oh dear. Well, let’s take a quick look at our Q-Q plot. Just to double check.
Normal Q-Q plot for x (n=5000). (Image by author)
Wow. This looks to be normally distributed. In fact, there shouldn’t be any doubt that this is normally distributed. But, the Shapiro-Wilk test says it isn’t.
What’s going on here? Well the Shapiro-Wilk test (and other normality tests) are designed to test for theoretical normality (i.e. the perfect bell curve). In small samples these tests are underpowered to detect quite major deviations from normality which can be easily detected through graphical methods. In larger samples these tests will detect even extremely minor deviations from theoretical normality that are not of practical concern.
Hopefully, I have shown you that normality tests are not of practical utility for data scientists. Don’t use them. Forget about them. At best, they are useless; at worst, they are misleading. If you want to assess the normality of some data, use Q-Q plots and histograms. They’ll give you a much clearer picture about the normality of your data.
#normal-distribution #statistics #tests-of-normality #mathematics #data-science
This blog is an abridged version of the talk that I gave at the Apache Ignite community meetup. You can download the slides that I presented at the meetup here. In the talk, I explain how data in Apache Ignite is distributed.
Inevitably, the evolution of a system that requires data storage and processing reaches a threshold. Either too much data is accumulated, so the data simply does not fit into the storage device, or the load increases so rapidly that a single server cannot manage the number of queries. Both scenarios happen frequently.
Usually, in such situations, two solutions come in handy—sharding the data storage or migrating to a distributed database. The solutions have features in common. The most frequently used feature uses a set of nodes to manage data. Throughout this post, I will refer to the set of nodes as “topology.”
The problem of data distribution among the nodes of the topology can be described in regard to the set of requirements that the distribution must comply with:
#tutorial #big data #distributed systems #apache ignite #distributed storage #data distribution #consistent hashing
e-Distribución is an energy distribution company that covers most of South Spain area. If you live in this area, you probably are able to register into their website to get some information about your power demand, energy consumption, or even cycle billing (in terms of consumptions).
Although their application is great, this integration enables you to add a sensor to Home Assistant and getting updated automatically. However, it has some limitations yet, and no front-end support is being provided at the moment.
sensor: - platform: edistribucion username: !secret eds_user ## this key may exist in secrets.yaml! password: !secret eds_password ## this key may exist in secrets.yaml!
At this point, you got an unique default sensor for the integration, namely
sensor.edistribucion, linked to those credentials in the e-Distribución platform. This default sensor assumes the first CUPS that appears in the fetched list of CUPS, which frequently is the most recent contract, so this configuration may be valid for most users. If you need a more detailed configuration, please check the section below “What about customisation?”.
#machine learning #distribution #python #home assistant custom integration for e-distribution with python #home assistant #e-distribution with python