The five-number summary is an exploratory data analysis tool that provides insight into the distribution of values for one variable. Collectively, this set of statistics describes where data values occur, their central tendency, variability, and the general shape of their distribution. [Read more…] about Five-Number Summary

# interpreting results

## Variance

Variance is a measure of variability in statistics. It assesses the average squared difference between data values and the mean. Unlike some other statistical measures of variability, it incorporates all data points in its calculations by contrasting each value to the mean. [Read more…] about Variance

## Mean Squared Error (MSE)

Mean squared error (MSE) measures the amount of error in statistical models. It assesses the average squared difference between the observed and predicted values. When a model has no error, the MSE equals zero. As model error increases, its value increases. The mean squared error is also known as the mean squared deviation (MSD). [Read more…] about Mean Squared Error (MSE)

## Paired T Test

Use a paired t-test when each subject has a pair of measurements, such as a before and after score. A paired t-test determines whether the mean change for these pairs is significantly different from zero. This test is an inferential statistics procedure because it uses samples to draw conclusions about populations.

Paired t tests are also known as dependent samples t tests. The two samples are dependent because they contain the same subjects. Conversely, an independent samples t test contains different subjects in the two samples. [Read more…] about Paired T Test

## Independent Samples T Test

Use an independent samples t test when you want to compare the means of precisely two groups—no more and no less! Typically, you perform this test to determine whether two population means are different. This procedure is an inferential statistical hypothesis test, meaning it uses samples to draw conclusions about populations. The independent samples t test is also known as the two sample t test. [Read more…] about Independent Samples T Test

## Stem and Leaf Plot

Stem and leaf plots display the shape and spread of a continuous data distribution. These graphs are similar to histograms, but instead of using bars, they show digits. It’s a particularly valuable tool during exploratory data analysis. They can help you identify the central tendency, variability, skewness of your distribution, and outliers. Stem and leaf plots are also known as stemplots. [Read more…] about Stem and Leaf Plot

## Pareto Charts

A Pareto chart is a specialized bar chart that displays categories in descending order and a line chart representing the cumulative amount. The chart effectively communicates the categories that contribute the most to the total. Frequently, quality analysts use Pareto charts to identify the most common types of defects or other problems.

Learn how to use and interpret these charts and understand the Pareto principle and the 80/20 rule that are behind it. I’ll also show you how to create them using Excel. [Read more…] about Pareto Charts

## 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

## Scatterplots: Using, Examples, and Interpreting

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

## 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)

## Understanding Historiansâ€™ 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

## 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

## Descriptive Statistics in Excel

Descriptive statistics summarize your dataset, painting a picture of its properties. These properties include various central tendency and variability measures, distribution properties, outlier detection, and other information. Unlike inferential statistics, descriptive statistics only describe your dataset’s characteristics and do not attempt to generalize from a sample to a population. [Read more…] about Descriptive Statistics in Excel