Statistics By Jim

Making statistics intuitive

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Can High P-values Be Meaningful?

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Can high p-values be helpful? What do high p-values mean?

Typically, when you perform a hypothesis test, you want to obtain low p-values that are statistically significant. Low p-values are sexy. They represent exciting findings and can help you get articles published.

However, you might be surprised to learn that higher p-values, the ones that are not statistically significant, are also valuable. In this post, I’ll show you the potential value of a p-value that is greater than 0.05, or whatever significance level you’re using. [Read more…] about Can High P-values Be Meaningful?

Using Histograms to Understand Your Data

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Histograms are graphs that display the distribution of your continuous data. They are fantastic exploratory tools because they reveal properties about your sample data in ways that summary statistics cannot. For instance, while the mean and standard deviation can numerically summarize your data, histograms bring your sample data to life.

In this blog post, I’ll show you how histograms reveal the shape of the distribution, its central tendency, and the spread of values in your sample data. You’ll also learn how to identify outliers, how histograms relate to probability distribution functions, and why you might need to use hypothesis tests with them.
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Boxplots vs. Individual Value Plots: Comparing Groups

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Use boxplots and individual value plots when you have a categorical grouping variable and a continuous outcome variable. The levels of the categorical variables form the groups in your data, and the researchers measure the continuous variable. Both types of charts help you compare distributions of measurements between the groups. Boxplots are also known as box and whisker plots. [Read more…] about Boxplots vs. Individual Value Plots: Comparing Groups

Using Post Hoc Tests with ANOVA

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Post hoc tests are an integral part of ANOVA. When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all of the group means are equal. However, ANOVA results do not identify which particular differences between pairs of means are significant. Use post hoc tests to explore differences between multiple group means while controlling the experiment-wise error rate.

In this post, I’ll show you what post hoc analyses are, the critical benefits they provide, and help you choose the correct one for your study. Additionally, I’ll show why failure to control the experiment-wise error rate will cause you to have severe doubts about your results. [Read more…] about Using Post Hoc Tests with ANOVA

When Can I Use One-Tailed Hypothesis Tests?

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One-tailed hypothesis tests offer the promise of more statistical power compared to an equivalent two-tailed design. While there is some debate about when you can use a one-tailed test, the general consensus among statisticians is that you should use two-tailed tests unless you have concrete reasons for using a one-tailed test.

In this post, I discuss when you should and should not use one-tailed tests. I’ll cover the different schools of thought and offer my own opinion. [Read more…] about When Can I Use One-Tailed Hypothesis Tests?

One-Tailed and Two-Tailed Hypothesis Tests Explained

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Choosing whether to perform a one-tailed or a two-tailed hypothesis test is one of the methodology decisions you might need to make for your statistical analysis. This choice can have critical implications for the types of effects it can detect, the statistical power of the test, and potential errors.

In this post, you’ll learn about the differences between one-tailed and two-tailed hypothesis tests and their advantages and disadvantages. I include examples of both types of statistical tests. In my next post, I cover the decision between one and two-tailed tests in more detail.
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Central Limit Theorem Explained

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The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable’s distribution in the population.

Unpacking the meaning from that complex definition can be difficult. That’s the topic for this post! I’ll walk you through the various aspects of the central limit theorem (CLT) definition, and show you why it is vital in statistics. [Read more…] about Central Limit Theorem Explained

Introduction to Bootstrapping in Statistics with an Example

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Bootstrapping is a statistical procedure that resamples a single dataset to create many simulated samples. This process allows you to calculate standard errors, construct confidence intervals, and perform hypothesis testing for numerous types of sample statistics. Bootstrap methods are alternative approaches to traditional hypothesis testing and are notable for being easier to understand and valid for more conditions.

In this blog post, I explain bootstrapping basics, compare bootstrapping to conventional statistical methods, and explain when it can be the better method. Additionally, I’ll work through an example using real data to create bootstrapped confidence intervals. [Read more…] about Introduction to Bootstrapping in Statistics with an Example

Confounding Variables Can Bias Your Results

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In research studies, confounding variables influence both the cause and effect that the researchers are assessing. Consequently, if the analysts do not include these confounders in their statistical model, it can exaggerate or mask the real relationship between two other variables. By omitting confounding variables, the statistical procedure is forced to attribute their effects to variables in the model, which biases the estimated effects and confounds the genuine relationship. Statisticians refer to this distortion as omitted variable bias.
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Assessing Normality: Histograms vs. Normal Probability Plots

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Because histograms display the shape and spread of distributions, you might think they’re the best type of graph for determining whether your data are normally distributed. However, I’ll show you how histograms can trick you! Normal probability plots are a better choice for this task and they are easy to use. Normal probability plots are also known as quantile-quantile plots, or Q-Q Plots for short!
[Read more…] about Assessing Normality: Histograms vs. Normal Probability Plots

Luck and Statistics: Do You Feel Lucky, Punk?

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Clint Eastwood asking the punk if he was lucky.
Do you feel lucky, Punk?

Luck, statistics, and probabilities go together hand-in-hand. Clint Eastwood, playing Dirty Harry, famously asked a bad guy who was about to reach for his rifle whether he felt lucky. I’m quite sure that the crook carefully pondered the nature of luck, probabilities, and expected outcomes before deciding not to grab his rifle!

A while ago, I did something shocking . . . something that I hadn’t done for several decades. Just like the thief in the Dirty Harry movie, I started thinking about luck. Yes, you guessed it:  I bought a lottery ticket for the record-breaking Mega Millions Jackpot. This purchase is shocking for someone like me who knows statistics and is fully aware of how unlikely it is to win. Did I feel lucky? Or was I just a punk? [Read more…] about Luck and Statistics: Do You Feel Lucky, Punk?

Populations, Parameters, and Samples in Inferential Statistics

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Inferential statistics lets you draw conclusions about populations by using small samples. Consequently, inferential statistics provide enormous benefits because typically you can’t measure an entire population.

However, to gain these benefits, you must understand the relationship between populations, subpopulations, population parameters, samples, and sample statistics.

In this blog post, I discuss these concepts, and how to obtain representative samples using random sampling.

Related post: Difference between Descriptive and Inferential Statistics
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Types of Errors in Hypothesis Testing

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Hypothesis tests use sample data to make inferences about the properties of a population. You gain tremendous benefits by working with random samples because it is usually impossible to measure the entire population.

However, there are tradeoffs when you use samples. The samples we use are typically a minuscule percentage of the entire population. Consequently, they occasionally misrepresent the population severely enough to cause hypothesis tests to make errors.

In this blog post, you will learn about the two types of errors in hypothesis testing, their causes, and how to manage them. [Read more…] about Types of Errors in Hypothesis Testing

Practical vs. Statistical Significance

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Important ink stamp that relates to the concept of practical significance.You’ve just performed a hypothesis test and your results are statistically significant. Hurray! These results are important, right? Not so fast. Statistical significance does not necessarily mean that the results are practically significant in a real-world sense of importance.

In this blog post, I’ll talk about the differences between practical significance and statistical significance, and how to determine if your results are meaningful in the real world.
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7 Classical Assumptions of Ordinary Least Squares (OLS) Linear Regression

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Ordinary Least Squares (OLS) is the most common estimation method for linear models—and that’s true for a good reason. As long as your model satisfies the OLS assumptions for linear regression, you can rest easy knowing that you’re getting the best possible estimates. [Read more…] about 7 Classical Assumptions of Ordinary Least Squares (OLS) Linear Regression

Normal Distribution in Statistics

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The normal distribution is a continuous probability distribution that is symmetrical around its mean, most of the observations cluster around the central peak, and the probabilities for values further away from the mean taper off equally in both directions. Extreme values in both tails of the distribution are similarly unlikely.

As with any probability distribution, the normal distribution describes how the values of a variable are distributed. It is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and the bell curve.

In this blog post, learn how to use the normal distribution, about its parameters, the Empirical Rule, and how to calculate Z-scores to standardize your data and find probabilities. [Read more…] about Normal Distribution in Statistics

Understanding Probability Distributions

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Probability distributions are statistical functions that describe the likelihood of obtaining possible values that a random variable can take. In other words, the values of the variable vary based on the underlying probability distribution.

Suppose you draw a random sample and measure the heights of the subjects. As you measure heights, you create a distribution of heights. This type of distribution is useful when you need to know which outcomes are most likely, the spread of potential values, and the likelihood of different results.

In this blog post, you’ll learn about probability distributions for both discrete and continuous variables. I’ll show you how they work and examples of how to use them. [Read more…] about Understanding Probability Distributions

Interpreting Correlation Coefficients

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Correlation coefficients measure the strength of the relationship between two variables. A correlation between variables indicates that as one variable changes in value, the other variable tends to change in a specific direction.  Understanding that relationship is useful because we can use the value of one variable to predict the value of the other variable. For example, height and weight are correlated—as height increases, weight also tends to increase. Consequently, if we observe an individual who is unusually tall, we can predict that his weight is also above the average. [Read more…] about Interpreting Correlation Coefficients