13: Distribution

13.1 Introduction to Distribution

Distribution refers to how the values of a dataset are spread or arranged. Understanding data distribution is crucial for interpreting the underlying patterns and relationships in the data. It provides insight into the data’s central tendency, variability, and shape, foundational for selecting appropriate statistical methods and making inferences. There are several key components when discussing distribution, including the shape of the distribution, skewness, kurtosis, percentiles, outliers, and the Central Limit Theorem (CLT). Each of these elements helps researchers understand the spread and patterns in the data and choose the right statistical tools for analysis.

13.2 Shape of the Distribution

The shape of a distribution refers to the overall pattern of the data when plotted. There are several common shapes that distributions may take. A normal distribution (Gaussian distribution) is symmetric and bell-shaped, with most of the data points clustered around the mean. It follows the well-known 68-95-99.7 rule, which states that 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The mean, median, and mode are all equal in a perfectly normal distribution. An example of a normal distribution could be the heights of adult humans.

On the other hand, skewed distributions are those where the data points are not evenly distributed. A positively skewed (right-skewed) distribution has a longer right tail, meaning most data is on the lower end, with a few larger values pulling the distribution to the right. The mean in a positively skewed distribution is greater than the median. An example of this could be household income, where a few high-income earners stretch the distribution’s right tail. Conversely, a negatively skewed (left-skewed) distribution has a longer left tail, with the mean being less than the median. An example of this is exam scores, where most students perform well, but a few perform poorly.

A uniform distribution occurs when all values in the dataset have the same probability of occurring, resulting in a flat, evenly distributed shape. An example of a uniform distribution could be the result of rolling a fair six-sided die. A bimodal distribution has two distinct peaks, which may occur when the data comes from two groups or sources. For example, the distribution of test scores from two different classrooms may exhibit two peaks representing the various performance levels of each group.

13.3 Skewness

Skewness refers to the degree of asymmetry or lopsidedness in a distribution. If a distribution is positively skewed, the right tail (larger values) is longer than the left, and most values are concentrated on the left. If the distribution is negatively skewed, the left tail (smaller values) is longer than the right, and most values are concentrated on the right. A normal distribution has a skewness of 0, indicating perfect symmetry. Positive skewness indicates a skew to the right, and negative skewness indicates a skew to the left. Skewness impacts the choice of statistical tests. For example, if the data is highly skewed, the mean may not accurately represent the central tendency, and the median would be a more informative measure.

13.4 Kurtosis

Kurtosis refers to the “tailedness” of the distribution, indicating how much of the data falls in the tails (extreme values) compared to a normal distribution. Leptokurtic distributions have high kurtosis, meaning they have more data in the tails and a sharper peak and are often prone to outliers. For example, stock market returns tend to have leptokurtic distributions, where extreme changes are frequent. Platykurtic distributions have low kurtosis, with fewer extreme values and a flatter peak. An example could be test scores that fall within a narrow range. Mesokurtic distributions have normal kurtosis, similar to the standard normal distribution. Understanding kurtosis helps assess the dataset’s likelihood of extreme values or outliers.

13.5 Percentiles

Percentiles divide a dataset into 100 equal parts and provide insight into the relative position of data points within a distribution. The 25th percentile (Q1) is the value below which 25% of the data falls. The 50th percentile (Q2 or median) is the middle value of the dataset, with 50% of the data falling below it. The 75th percentile (Q3) is the value below which 75% of the data falls. Percentiles are particularly useful when comparing individual data points to the overall distribution. For example, if a student’s score is in the 90th percentile, the student performed better than 90% of the other students.

13.6 Outliers

Outliers are data points that fall significantly outside the overall pattern of the distribution. These extreme values can arise from data entry errors or could represent rare but valid observations. Outliers can heavily influence statistical measures like the mean, variance, and standard deviation. They can distort the results of statistical tests if not appropriately handled. Boxplots are commonly used to visualize outliers, with any data points outside the whiskers of the boxplot typically considered outliers. Additionally, z-scores greater than 3 or less than -3 are often used to identify outliers. The IQR method considers values outside 1.5 times the IQR above Q3 or below Q1 as potential outliers.

13.7 Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a fundamental concept in statistics, stating that the distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution, provided the sample size is sufficiently large (typically n ≥ 30). The CLT justifies using normality assumptions in hypothesis testing and confidence interval estimation. It allows researchers to use parametric tests (like t-tests and ANOVA) even if the underlying population is not normally distributed. As the sample size increases, the sampling distribution of the sample mean becomes more concentrated around the population mean. The standard error of the mean decreases as the sample size increases, leading to more accurate estimates.

13.8 Distribution in Jamovi

Jamovi simplifies the process of analyzing distributions through its built-in Exploration and Descriptives features. To analyze a distribution in Jamovi, open your dataset and go to the Exploration menu under the Analyses tab. Select the variables you want to analyze, check the statistics you need (e.g., skewness, kurtosis, and percentiles), and click OK. The output will include summary statistics that help you understand the distribution of your data. You can also create visualizations, such as histograms or boxplots, to better understand the shape of the distribution.

Below is an example of the results generated when the steps are correctly followed.

IMAGE [INSERT NAME OF DATASET]

Interpretation