Use scatterplots to show relationships between pairs of continuous variables. These graphs display symbols at the X, Y coordinates of the data points for the paired variables. Scatterplots are also known as scattergrams and scatter charts. [Read more…] about Scatterplots: Using, Examples, and Interpreting

# Blog

## Pie Charts: Using, Examples, and Interpreting

Use pie charts to compare the sizes of categories to the entire dataset. To create a pie chart, you must have a categorical variable that divides your data into groups. These graphs consist of a circle (i.e., the pie) with slices representing subgroups. The size of each slice is proportional to the relative size of each category out of the whole. [Read more…] about Pie Charts: Using, Examples, and Interpreting

## Bar Charts: Using, Examples, and Interpreting

Use bar charts to compare categories when you have at least one categorical or discrete variable. Each bar represents a summary value for one discrete level, where longer bars indicate higher values. Types of summary values include counts, sums, means, and standard deviations. Bar charts are also known as bar graphs. [Read more…] about Bar Charts: Using, Examples, and Interpreting

## Line Charts: Using, Examples, and Interpreting

Use line charts to display a series of data points that are connected by lines. Analysts use line charts to emphasize changes in a metric on the vertical Y-axis by another variable on the horizontal X-axis. Often, the X-axis reflects time, but not always. Line charts are also known as line plots. [Read more…] about Line Charts: Using, Examples, and Interpreting

## Dot Plots: Using, Examples, and Interpreting

Use dot plots to display the distribution of your sample data when you have continuous variables. These graphs stack dots along the horizontal X-axis to represent the frequencies of different values. More dots indicate greater frequency. Each dot represents a set number of observations. [Read more…] about Dot Plots: Using, Examples, and Interpreting

## Empirical Cumulative Distribution Function (CDF) Plots

Use an empirical cumulative distribution function plot to display the data points in your sample from lowest to highest against their percentiles. These graphs require continuous variables and allow you to derive percentiles and other distribution properties. This function is also known as the empirical CDF or ECDF. [Read more…] about Empirical Cumulative Distribution Function (CDF) Plots

## Contour Plots: Using, Examples, and Interpreting

Use contour plots to display the relationship between two independent variables and a dependent variable. The graph shows values of the Z variable for combinations of the X and Y variables. The X and Y values are displayed along the X and Y-axes, while contour lines and bands represent the Z value. The contour lines connect combinations of the X and Y variables that produce equal values of Z. [Read more…] about Contour Plots: Using, Examples, and Interpreting

## 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.

In this post, I cover both the autocorrelation function and partial autocorrelation function. You’ll learn about the differences between these functions and what they can tell you about your data. In later posts, I’ll show you how to incorporate this information in regression models of time series data and other time-series analyses.

## Autocorrelation and Partial Autocorrelation Basics

Autocorrelation is the correlation between two values in a time series. In other words, the time series data correlate with themselves—hence, the name. We talk about these correlations using the term “lags.” Analysts record time-series data by measuring a characteristic at evenly spaced intervals—such as daily, monthly, or yearly. The number of intervals between the two observations is the lag. For example, the lag between the current and previous observation is one. If you go back one more interval, the lag is two, and so on.

In mathematical terms, the observations at y_{t} and y_{t–k} are separated by k time units. K is the lag. This lag can be days, quarters, or years depending on the nature of the data. When k=1, you’re assessing adjacent observations. For each lag, there is a correlation.

The autocorrelation function (ACF) assesses the correlation between observations in a time series for a set of lags. The ACF for time series y is given by: Corr (y_{t},y_{t−k}), k=1,2,….

Analysts typically use graphs to display this function.

**Related posts**: Time Series Analysis Introduction and Interpreting Correlations

## Autocorrelation Function (ACF)

Use the autocorrelation function (ACF) to identify which lags have significant correlations, understand the patterns and properties of the time series, and then use that information to model the time series data. From the ACF, you can assess the randomness and stationarity of a time series. You can also determine whether trends and seasonal patterns are present.

In an ACF plot, each bar represents the size and direction of the correlation. Bars that extend across the red line are statistically significant.

### Randomness/White Noise

For random data, autocorrelations should be near zero for all lags. Analysts also refer to this condition as white noise. Non-random data have at least one significant lag. When the data are not random, it’s a good indication that you need to use a time series analysis or incorporate lags into a regression analysis to model the data appropriately.

This ACF plot indicates that these time series data are random.

### Stationarity

Stationarity means that the time series does not have a trend, has a constant variance, a constant autocorrelation pattern, and no seasonal pattern. The autocorrelation function declines to near zero rapidly for a stationary time series. In contrast, the ACF drops slowly for a non-stationary time series.

In this chart for a stationary time series, notice how the autocorrelations decline to non-significant levels quickly.

### Trends

When trends are present in a time series, shorter lags typically have large positive correlations because observations closer in time tend to have similar values. The correlations taper off slowly as the lags increase.

In this ACF plot for metal sales, the autocorrelations decline slowly. The first five lags are significant.

### Seasonality

When seasonal patterns are present, the autocorrelations are larger for lags at multiples of the seasonal frequency than for other lags.

When a time series has both a trend and seasonality, the ACF plot displays a mixture of both effects. That’s the case in the autocorrelation function plot for the carbon dioxide (CO2) dataset from NIST. This dataset contains monthly mean CO2 measurements at the Mauna Loa Observatory. Download the CO2_Data.

Notice how you can see the wavy correlations for the seasonal pattern and the slowly diminishing lags of a trend.

## Partial Autocorrelation Function (PACF)

The partial autocorrelation function is similar to the ACF except that it displays only the correlation between two observations that the shorter lags between those observations do not explain. For example, the partial autocorrelation for lag 3 is only the correlation that lags 1 and 2 do not explain. In other words, the partial correlation for each lag is the unique correlation between those two observations after partialling out the intervening correlations.

As you saw, the autocorrelation function helps assess the properties of a time series. In contrast, the partial autocorrelation function (PACF) is more useful during the specification process for an autoregressive model. Analysts use partial autocorrelation plots to specify regression models with time series data and Auto Regressive Integrated Moving Average (ARIMA) models. I’ll focus on that aspect in posts about those methods.

**Related post**: Using Moving Averages to Smooth Time Series Data

For this post, I’ll show you a quick example of a PACF plot. Typically, you will use the ACF to determine whether an autoregressive model is appropriate. If it is, you then use the PACF to help you choose the model terms.

This partial autocorrelation plot displays data from the southern oscillations dataset from NIST. The southern oscillations refer to changes in the barometric pressure near Tahiti that predicts El Niño. Download the southern_oscillations_data.

On the graph, the partial autocorrelations for lags 1 and 2 are statistically significant. The subsequent lags are nearly significant. Consequently, this PACF suggests fitting either a second or third-order autoregressive model.

By assessing the autocorrelation and partial autocorrelation patterns in your data, you can understand the nature of your time series and model it!

## Using Combinations to Calculate Probabilities

Combinations in probability theory and other areas of mathematics refer to a sequence of outcomes where the order does not matter. For example, when you’re ordering a pizza, it doesn’t matter whether you order it with ham, mushrooms, and olives or olives, mushrooms, and ham. You’re getting the same pizza! [Read more…] about Using Combinations to Calculate Probabilities

## Law of Large Numbers

The law of large numbers states that as the number of trials increases, sample values tend to converge on the expected result. The two forms of this law lay the foundation for both statistics and probability theory.

In this post, I explain both forms of the law, simulate them in action, and explain why they’re crucial for statistics and probability! [Read more…] about Law of Large Numbers

## Chebyshev’s Theorem in Statistics

Chebyshev’s Theorem estimates the minimum proportion of observations that fall within a specified number of standard deviations from the mean. This theorem applies to a broad range of probability distributions. Chebyshev’s Theorem is also known as Chebyshev’s Inequality. [Read more…] about Chebyshev’s Theorem in Statistics

## Using Permutations to Calculate Probabilities

Permutations in probability theory and other branches of mathematics refer to sequences of outcomes where the order matters. For example, 9-6-8-4 is a permutation of a four-digit PIN because the order of numbers is crucial. When calculating probabilities, it’s frequently necessary to calculate the number of possible permutations to determine an event’s probability.

In this post, I explain permutations and show how to calculate the number of permutations both with repetition and without repetition. Finally, we’ll work through a step-by-step example problem that uses permutations to calculate a probability. [Read more…] about Using Permutations to Calculate Probabilities

## 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

## Spearman’s Correlation Explained

Spearman’s correlation in statistics is a nonparametric alternative to Pearson’s correlation. Use Spearman’s correlation for data that follow curvilinear, monotonic relationships and for ordinal data. Statisticians also refer to Spearman’s rank order correlation coefficient as Spearman’s ρ (rho).

In this post, I’ll cover what all that means so you know when and why you should use Spearman’s correlation instead of the more common Pearson’s correlation. [Read more…] about Spearman’s Correlation Explained

## Effect Sizes in Statistics

Effect sizes in statistics quantify the differences between group means and the relationships between variables. While analysts often focus on statistical significance using p-values, effect sizes determine the practical importance of the findings. [Read more…] about Effect Sizes in Statistics

## Proxy Variables: The Good Twin of Confounding Variables

Proxy variables are easily measurable variables that analysts include in a model in place of a variable that cannot be measured or is difficult to measure. Proxy variables can be something that is not of any great interest itself, but has a close correlation with the variable of interest. [Read more…] about Proxy Variables: The Good Twin of Confounding Variables

## Multiplication Rule for Calculating Probabilities

The multiplication rule in probability allows you to calculate the probability of multiple events occurring together using known probabilities of those events individually. There are two forms of this rule, the specific and general multiplication rules.

In this post, learn about when and how to use both the specific and general multiplication rules. Additionally, I’ll use and explain the standard notation for probabilities throughout, helping you learn how to interpret it. We’ll work through several example problems so you can see them in action. There’s even a bonus problem at the end! [Read more…] about Multiplication Rule for Calculating Probabilities

## Exponential Smoothing for Time Series Forecasting

Exponential smoothing is a forecasting method for univariate time series data. This method produces forecasts that are weighted averages of past observations where the weights of older observations exponentially decrease. Forms of exponential smoothing extend the analysis to model data with trends and seasonal components. [Read more…] about Exponential Smoothing for Time Series Forecasting