This chi-square table provides the critical values for chi-square (χ^{2}) hypothesis tests. The column and row intersections are the right-tail critical values for a given probability and degrees of freedom. [Read more…] about Chi-Square Table

# graphs

## Z-table

## Z-Score Table

A z-table, also known as the standard normal table, provides the area under the curve to the left of a z-score. This area represents the probability that z-values will fall within a region of the standard normal distribution. Use a z-table to find probabilities corresponding to ranges of z-scores and to find p-values for z-tests. [Read more…] about Z-table

## Lognormal Distribution: Uses, Parameters & Examples

## What is the Lognormal Distribution?

The lognormal distribution is a continuous probability distribution that models right-skewed data. The unimodal shape of the lognormal distribution is comparable to the Weibull and loglogistic distributions. [Read more…] about Lognormal Distribution: Uses, Parameters & Examples

## A Statistical Thanksgiving: Global Income Distributions

In the United States, our Thanksgiving holiday is fast approaching. On this day, we give thanks for the good things in our lives.

For this post, I wanted to quantify how thankful we should be. Ideally, I’d quantify something truly meaningful, like happiness. Unfortunately, most countries are not like Bhutan, which measures the gross national happiness and incorporates it into their five-year development plans.

Instead, I’ll focus on something that is more concrete and regularly measured around the world—income. By examining income distributions, I’ll show that you have much to be thankful for, and so does most of the world! [Read more…] about A Statistical Thanksgiving: Global Income Distributions

## Uniform Distribution: Definition & Examples

## What is a Uniform Distribution?

The uniform distribution is a symmetric probability distribution where all outcomes have an equal likelihood of occurring. All values in the distribution have a constant probability, making them uniformly distributed. This distribution is also known as the rectangular distribution because of its shape in probability distribution plots, as I’ll show you below. [Read more…] about Uniform Distribution: Definition & Examples

## Skewed Distribution: Definition & Examples

## What is a Skewed Distribution?

A skewed distribution occurs when one tail is longer than the other. Skewness defines the asymmetry of a distribution. Unlike the familiar normal distribution with its bell-shaped curve, these distributions are asymmetric. The two halves of the distribution are not mirror images because the data are not distributed equally on both sides of the distribution’s peak. [Read more…] about Skewed Distribution: Definition & Examples

## Heterogeneity in Data and Samples for Statistics

## What is Heterogeneity?

Heterogeneity is defined as a dissimilarity between elements that comprise a whole. When heterogeneity is present, there is diversity in the characteristic under study. The parts of the whole are different, not the same. It is an essential concept in science and statistics. Heterogeneous is the opposite of homogeneous. [Read more…] about Heterogeneity in Data and Samples for Statistics

## Range of a Data Set

The range of a data set is the difference between the maximum and the minimum values. It measures variability using the same units as the data. Larger values represent greater variability.

The range is the easiest measure of dispersion to calculate and interpret in statistics, but it has some limitations. In this post, I’ll show you how to find the range mathematically and graphically, interpret it, explain its limitations, and clarify when to use it. [Read more…] about Range of a Data Set

## Relative Frequencies and Their Distributions

A relative frequency indicates how often a specific kind of event occurs within the total number of observations. It is a type of frequency that uses percentages, proportions, and fractions.

In this post, learn about relative frequencies, the relative frequency distribution, and its cumulative counterpart. [Read more…] about Relative Frequencies and Their Distributions

## Empirical Rule: Definition & Formula

## What is the Empirical Rule?

The empirical rule in statistics, also known as the 68 95 99 rule, states that for normal distributions, 68% of observed data points will lie inside one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will occur within three standard deviations. [Read more…] about Empirical Rule: Definition & Formula

## Standard Deviation: Interpretations and Calculations

The standard deviation (SD) is a single number that summarizes the variability in a dataset. It represents the typical distance between each data point and the mean. Smaller values indicate that the data points cluster closer to the mean—the values in the dataset are relatively consistent. Conversely, higher values signify that the values spread out further from the mean. Data values become more dissimilar, and extreme values become more likely. [Read more…] about Standard Deviation: Interpretations and Calculations

## What is the Mean and How to Find It: Definition & Formula

## What is the Mean?

The mean in math and statistics summarizes an entire dataset with a single number representing the data’s center point or typical value. It is also known as the arithmetic mean, and it is the most common measure of central tendency. It is frequently called the “average.” [Read more…] about What is the Mean and How to Find It: Definition & Formula

## Gamma Distribution: Uses, Parameters & Examples

## What is the Gamma Distribution?

The gamma distribution is a continuous probability distribution that models right-skewed data. Statisticians have used this distribution to model cancer rates, insurance claims, and rainfall. Additionally, the gamma distribution is similar to the exponential distribution, and you can use it to model the same types of phenomena: failure times, wait times, service times, etc. [Read more…] about Gamma Distribution: Uses, Parameters & Examples

## Exponential Distribution: Uses, Parameters & Examples

## What is the Exponential Distribution?

The exponential distribution is a right-skewed continuous probability distribution that models variables in which small values occur more frequently than higher values. It is a unimodal distribution where small values have relatively high probabilities, which consistently decline as data values increase. Statisticians use the exponential distribution to model the amount of change in people’s pockets, the length of phone calls, and sales totals for customers. In all these cases, small values are more likely than larger values. [Read more…] about Exponential Distribution: Uses, Parameters & Examples

## Weibull Distribution: Uses, Parameters & Examples

## What is a Weibull Distribution?

The Weibull distribution is a continuous probability distribution that can fit an extensive range of distribution shapes. Like the normal distribution, the Weibull distribution is unimodal and describes probabilities associated with continuous data. However, unlike the normal distribution, it can also model skewed data. In fact, its extreme flexibility allows it to model both left- and right-skewed data. [Read more…] about Weibull Distribution: Uses, Parameters & Examples

## Poisson Distribution: Definition & Uses

## What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that describes probabilities for counts of events that occur in a specified observation space. It is named after Siméon Denis Poisson.

In statistics, count data represent the number of events or characteristics over a given length of time, area, volume, etc. For example, you can count the number of cigarettes smoked per day, meteors seen per hour, the number of defects in a batch, and the occurrence of a particular crime by county. [Read more…] about Poisson Distribution: Definition & Uses

## Using Excel to Calculate Correlation

Excel can calculate correlation coefficients and a variety of other statistical analyses. Even if you don’t use Excel regularly, this post is an excellent introduction to calculating and interpreting correlation.

In this post, I provide step-by-step instructions for having Excel calculate Pearson’s correlation coefficient, and I’ll show you how to interpret the results. Additionally, I include links to relevant statistical resources I’ve written that provide intuitive explanations. Together, we’ll analyze and interpret an example dataset! [Read more…] about Using Excel to Calculate Correlation

## Standard Error of the Mean (SEM)

The standard error of the mean (SEM) is a bit mysterious. You’ll frequently find it in your statistical output. Is it a measure of variability? How does the standard error of the mean compare to the standard deviation? How do you interpret it?

In this post, I answer all these questions about the standard error of the mean, show how it relates to sample size considerations and statistical significance, and explain the general concept of other types of standard errors. In fact, I view standard errors as the doorway from descriptive statistics to inferential statistics. You’ll see how that works! [Read more…] about Standard Error of the Mean (SEM)

## Autocorrelation and Partial Autocorrelation in Time Series Data

Autocorrelation is the correlation between two observations at different points in a time series. For example, values that are separated by an interval might have a strong positive or negative correlation. When these correlations are present, they indicate that past values influence the current value. Analysts use the autocorrelation and partial autocorrelation functions to understand the properties of time series data, fit the appropriate models, and make forecasts. [Read more…] about Autocorrelation and Partial Autocorrelation in Time Series Data

## Understanding Historians’ Rankings of U.S. Presidents using Regression Models

Historians rank the U.S. Presidents from best to worse using all the historical knowledge at their disposal. Frequently, groups, such as C-Span, ask these historians to rank the Presidents and average the results together to help reduce bias. The idea is to produce a set of rankings that incorporates a broad range of historians, a vast array of information, and a historical perspective. These rankings include informed assessments of each President’s effectiveness, leadership, moral authority, administrative skills, economic management, vision, and so on. [Read more…] about Understanding Historians’ Rankings of U.S. Presidents using Regression Models