Statistics By Jim

Making statistics intuitive

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Using Confidence Intervals to Compare Means

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To determine whether the difference between two means is statistically significant, analysts often compare the confidence intervals for those groups. If those intervals overlap, they conclude that the difference between groups is not statistically significant. If there is no overlap, the difference is significant.

While this visual method of assessing the overlap is easy to perform, regrettably it comes at the cost of reducing your ability to detect differences. Fortunately, there is a simple solution to this problem that allows you to perform a simple visual assessment and yet not diminish the power of your analysis.

In this post, I’ll start by showing you the problem in action and explain why it happens. Then, we’ll proceed to an easy alternative method that avoids this problem. [Read more…] about Using Confidence Intervals to Compare Means

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: Graphing Continuous Data by Groups

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Graphing your data before performing statistical analysis is a crucial step. Graphs bring your data to life in a way that statistical measures do not because they display the relationships and patterns. In this blog post, you’ll learn about using boxplots and individual value plots to compare distributions of continuous measurements between groups. You’ll also learn why you need to pair these plots with hypothesis tests when you want to make inferences about a population. [Read more…] about Boxplots vs. Individual Value Plots: Graphing Continuous Data by 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

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 so important in the field of 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

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.
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Normal Distribution in Statistics

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The normal distribution 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.

The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where 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.

In this blog post, you’ll learn how to use the normal distribution, its parameters, 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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A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. 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 can 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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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

Estimating a Good Sample Size for Your Study Using Power Analysis

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Determining a good sample size for a study is always an important issue. After all, using the wrong sample size can doom your study from the start. Fortunately, power analysis can find the answer for you. Power analysis combines statistical analysis, subject-area knowledge, and your requirements to help you derive the optimal sample size for your study.

Statistical power in a hypothesis test is the probability that the test will detect an effect that actually exists. As you’ll see in this post, both under-powered and over-powered studies are problematic. Let’s learn how to find a good sample size for your study! [Read more…] about Estimating a Good Sample Size for Your Study Using Power Analysis

Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation

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A measure of variability is a summary statistic that represents the amount of dispersion in a dataset. How spread out are the values? While a measure of central tendency describes the typical value, measures of variability define how far away the data points tend to fall from the center. We talk about variability in the context of a distribution of values. A low dispersion indicates that the data points tend to be clustered tightly around the center. High dispersion signifies that they tend to fall further away.

In statistics, variability, dispersion, and spread are synonyms that denote the width of the distribution. Just as there are multiple measures of central tendency, there are several measures of variability. In this blog post, you’ll learn why understanding the variability of your data is critical. Then, I explore the most common measures of variability—the range, interquartile range, variance, and standard deviation. I’ll help you determine which one is best for your data. [Read more…] about Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation

Measures of Central Tendency: Mean, Median, and Mode

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A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution. You can think of it as the tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendency are the mean, median, and mode. Each of these measures calculates the location of the central point using a different method.

Choosing the best measure of central tendency depends on the type of data you have. In this post, I explore these measures of central tendency, show you how to calculate them, and how to determine which one is best for your data. [Read more…] about Measures of Central Tendency: Mean, Median, and Mode

Guide to Data Types and How to Graph Them in Statistics

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In the field of statistics, data are vital. Data are the information that you collect to learn, draw conclusions, and test hypotheses. After all, statistics is the science of learning from data. However, there are different types of variables, and they record various kinds of information. Crucially, the type of information determines what you can learn from it, and, importantly, what you cannot learn from it. Consequently, it’s essential that you understand the different types of data. [Read more…] about Guide to Data Types and How to Graph Them in Statistics

Maximize the Value of Your Binary Data with the Binomial and Other Probability Distributions

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Binary data occur when you can place an observation into only two categories. It tells you that an event occurred or that an item has a particular characteristic. For instance, an inspection process produces binary pass/fail results. Or, when a customer enters a store, there are two possible outcomes—sale or no sale. In this post, I show you how to use the binomial, geometric, negative binomial, and the hypergeometric distributions to glean more information from your binary data. [Read more…] about Maximize the Value of Your Binary Data with the Binomial and Other Probability Distributions

Understanding Interaction Effects in Statistics

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Interaction effects occur when the effect of one variable depends on the value of another variable. Interaction effects are common in regression analysis, ANOVA, and designed experiments. In this blog post, I explain interaction effects, how to interpret them in statistical designs, and the problems you will face if you don’t include them in your model. [Read more…] about Understanding Interaction Effects in Statistics

Using Log-Log Plots to Determine Whether Size Matters

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Log-log plots display data in two dimensions where both axes use logarithmic scales. When one variable changes as a constant power of another, a log-log graph shows the relationship as a straight line. In this post, I’ll show you why these graphs are valuable and how to interpret them. [Read more…] about Using Log-Log Plots to Determine Whether Size Matters

Flu Shots, How Effective Are They?

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With the arrival of Fall in the Northern hemisphere, it’s flu season again.

Do you debate getting a flu shot every year? I do get flu shots every year. I realize that they’re not perfect, but I figure they’re a low-cost way to reduce my chances of a crummy week suffering from the flu.

The media report that flu shots have an effectiveness of approximately 68%. But, what does that mean exactly? What is the absolute reduction in risk? Are there long-term benefits?

In this blog post, I explore the effectiveness of flu shots from a statistical viewpoint. We’ll statistically analyze the data ourselves so we can go beyond the simplified accounts that the media presents. I’ll also model the long-term outcomes you can expect with regular flu vaccinations. By the time you finish this post, you’ll have a crystal clear picture of flu shot effectiveness. Some of the results surprised me! [Read more…] about Flu Shots, How Effective Are They?

Use Control Charts with Hypothesis Tests

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Typically, quality improvement analysts use control charts to assess business processes and don’t have hypothesis tests in mind. Do you know how control charts provide tremendous benefits in other settings and with hypothesis testing? Spoilers—control charts check an assumption that we often forget about for hypothesis tests! [Read more…] about Use Control Charts with Hypothesis Tests