Statistics By Jim

Making statistics intuitive

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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.
[Read more…] about One-Tailed and Two-Tailed Hypothesis Tests Explained

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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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
[Read more…] about Populations, Parameters, and Samples in Inferential Statistics

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 miniscule 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 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

Difference between Descriptive and Inferential Statistics

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Descriptive and inferential statistics are two broad categories in the field of statistics. In this blog post, I show you how both types of statistics are important for different purposes. Interestingly, some of the statistical measures are similar, but the goals and methodologies are very different. [Read more…] about Difference between Descriptive and Inferential Statistics