misuse of statistics pdf and cdf

Misuse Of Statistics Pdf And Cdf

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Springer Handbook of Engineering Statistics pp Cite as. This brief chapter presents some fundamental elements of engineering probability and statistics with which some readers are probably already familiar, but others may not be. Statistics is the study of how best one can describe and analyze the data and then draw conclusions or inferences based on the data available.

Statistical Issues in Particle Physics

P values are one of the most widely used concepts in statistical analysis. They are used by researchers, analysts and statisticians to draw insights from data and make informed decisions.

Along with statistical significance, they are also one of the most widely misused and misunderstood concepts in statistical analysis. Hypothesis testing is a standard approach to drawing insights from data. It is used in virtually every quantitative discipline, and has a rich history going back over one hundred years. The usual approach to hypothesis testing is to define a question in terms of the variables you are interested in.

Then, you can form two opposing hypotheses to answer it. For example, say you are testing whether caffeine affects programming productivity. There are two variables you are interested in - the dose of the caffeine, and the productivity of group of software developers. The word 'significant' has a very specific meaning here. It refers to a relationship between variables existing due to something more than chance alone.

Instead, the relationship exists at least in part due to 'real' differences or effects between the variables. The next step is to collect some data to test the hypotheses. This could be collected from an experiment or survey, or from a set of data you have access to.

The final step is to calculate a test statistic from the data. This is a single number that represents some characteristic of your data. Examples include the t-test, Chi-squared test, and the Kruskal-Wallis test - among many others.

Exactly which one to calculate will depend on the question you are asking, the structure of your data, and the distribution of your data. Here's a handy cheatsheet for your reference.

In the caffeine example, a suitable test might be a two-sample t-test. You will end up with a single test statistic from your data. All that is left to do is interpret this result to determine whether it supports or rejects the null hypothesis. Recall that you have calculated a test statistic, which represents some characteristic of your data. You want to understand whether it supports or rejects the null hypothesis. The approach taken is to assume the null hypothesis is true.

That is, assume there are no significant relationships between the variables you are interested in. Then, look at the data you have collected. How likely would your test statistic be if the null hypothesis really is true? But how 'extreme' does a result need to be before it is considered too unlikely to support the null hypothesis? This is what a P value lets you estimate.

It provides a numerical answer to the question: "if the null hypothesis is true, what is the probability of a result this extreme or more extreme? Usually, a threshold is chosen to determine statistical significance. If the P value is below the threshold , your results are ' statistically significant '. This means you can reject the null hypothesis and accept the alternative hypothesis.

There is no one-size-fits-all threshold suitable for all applications. Usually, an arbitrary threshold will be used that is appropriate for the context. For example, in fields such as ecology and evolution, it is difficult to control experimental conditions because many factors can affect the outcome.

It can also be difficult to collect very large sample sizes. In these fields, a threshold of 0. In other contexts such as physics and engineering, a threshold of 0.

In this example, there are two fictional variables: region, and political party membership. It uses the Chi-squared test to see if there's a relationship between region and political party membership. There are several mistakes that even experienced practitioners often make about the use of P values and hypothesis testing.

This section will aim to clear those up. This is not the same as "the probability of the null hypothesis being true, given the results". It does not tell you: "if these results are true, the null hypothesis is unlikely". It is the probability of observing a certain test statistic by chance alone.

There are correction methods that will let you calculate how much lower the threshold should be. This is one of the biggest weaknesses of hypothesis testing this way. It forces you to draw a line in the sand, even though no line can easily be drawn. Often, there are many causes for a given outcome. Some will be random, others less so. It is important not to mistake statistical significance with "effect size". If this article was helpful, tweet it.

Learn to code for free. Get started. Forum Donate. Peter Gleeson. This article will explain: how a P value is used for inferring statistical significance how P values are calculated and how to avoid some common misconceptions Recap: Hypothesis testing Hypothesis testing is a standard approach to drawing insights from data. The null hypothesis claims there is no statistically significant relationship between the variables The alternative hypothesis claims there is a statistically significant relationship between the variables For example, say you are testing whether caffeine affects programming productivity.

The null hypothesis would be: "Caffeine intake has no significant effect on programming productivity". The alternative hypothesis would be: "Caffeine intake does have a significant effect on productivity". This is where P values come into play. How unlikely is this statistic? Let's refer back to the caffeine intake example from before. Say that productivity levels were split about evenly between developers, regardless of whether they drank caffeine or not graph A.

This result would be likely to occur by chance if the null hypothesis were true. However, suppose that almost all of the highest productivity was seen in developers who drank caffeine graph B. This is a more 'extreme' result, and would be unlikely to occur just by chance if the null hypothesis were true.

A high P value indicates the observed results are likely to occur by chance under the null hypothesis. A low P value indicates that the results are less likely to occur by chance under the null hypothesis. Chi-squared example In this example, there are two fictional variables: region, and political party membership. You can change the number of members for each party.

Null hypothesis: "there is no significant relationship between region and political party membership" Alternative hypothesis: "there is a significant relationship between region and political party membership" Hit the "rerun" button to try different scenarios. Common misconceptions and how to avoid them There are several mistakes that even experienced practitioners often make about the use of P values and hypothesis testing.

Peter Gleeson Founder Associate at Revolut.

Basic Statistical Concepts

The Central Limit Theorem CLT is possibly the most famous theorem in all of statistics, being widely used in any field that wants to infer something or make predictions from gathered data. A first simple version of it was introduced in the eighteenth century, first by de Moivre and then later in a more refined way by Laplace, but it wasn't until around that the theorem as we know it today was published. The goal of these notes is to explain in broad terms what it says and, more importantly, what it doesn't. Informally, the theorem states that if we take random samples of a certain distribution and then average them, the result i. The most common example of the CLT in action is when considering a binomial distribution. In general, comparing cdfs is more accurate since histograms can differ wildly depending on the number of bins, and the convergence of the CLT is stated in terms of the cdfs.

The Central Limit Theorem and its misuse

In null hypothesis significance testing , the p -value [note 1] is the probability of obtaining test results at least as extreme as the results actually observed , under the assumption that the null hypothesis is correct. Reporting p -values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p -value is hard to grasp, misuse is widespread and has been a major topic in metascience. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis is tenable, but not, at the same time, to investigate other hypotheses, then such a test is called a significance test.

Springer Handbook of Engineering Statistics pp Cite as. This brief chapter presents some fundamental elements of engineering probability and statistics with which some readers are probably already familiar, but others may not be. Statistics is the study of how best one can describe and analyze the data and then draw conclusions or inferences based on the data available. The first section of this chapter begins with some basic definitions, including probability axioms, basic statistics and reliability measures. The third section describes statistical inference, including parameter estimation and confidence intervals.

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Camille L.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

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