Learn how to use box plots to visualize and compare probability distributions, and how to create them in R. Box plots show the median, IQR, range, and outliers of a distribution.

Here is a simple guide on the interpretation: -The box's span (Q1 to Q3) represents the middle 50% of data, known as the interquartile range (IQR), indicating where most of your data is located. -The line inside the box shows the dataset's median value, a quick measure of its central tendency (use the median instead of the average as it minimizes the impact of outliers!). -Whiskers extend to the data's range (excluding outliers). -Outliers are the small singular points outside of the whisker range. -I've used a box plot that revealed an outlier in web form submissions, signaling bot/spam issues we'd seen before. After investigating, I found out that CAPTCHA had not been present on this specific form (but had been on our other forms).

Here is what else to consider: -Box plots are a starting point for your EDA. -Remember to dive deeper with tests like: the t-test, ANOVA, or chi-squared test. -Why? These will confirm whether you have statistically significant differences between means/distributions.

How can you use box plots to represent probability distributions?

Box plots are a type of graphical display that can help you visualize the shape, spread, and center of a probability distribution. They are also known as box-and-whisker plots, because they consist of a rectangular box that shows the interquartile range (IQR) and two whiskers that extend to the minimum and maximum values or to a specified distance from the box. In this article, you will learn how to use box plots to represent probability distributions and compare different groups of data.

1 What are probability distributions?

A probability distribution is a mathematical function that describes how likely it is that a random variable will take on different values. For example, the normal distribution is a common probability distribution that is symmetric and bell-shaped, and it is often used to model phenomena such as heights, weights, test scores, and errors. Probability distributions can be discrete or continuous, depending on whether the random variable can take on only a finite number of values or any value within an interval.

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Sirikonda Satish

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Box plots can represent probability distributions by visually displaying key statistical parameters such as the median, quartiles, and potential outliers. The box in the plot shows the interquartile range (IQR), where half the data lies, while the median line divides the box in half. The "whiskers" extend to the minimum and maximum values within a certain range, typically 1.5 times the IQR. Outliers beyond this range are plotted individually. This visualization offers a concise summary of the distribution's central tendency, spread, and presence of outliers, aiding in the comparison of different distributions or identifying unusual data points.

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2 How to construct a box plot?

Constructing a box plot requires calculating five summary statistics: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. The median is the middle value of the data, and it divides it into two halves. The first quartile is the median of the lower half, and the third quartile is the median of the upper half. The interquartile range (IQR) is the difference between Q3 and Q1, and it measures the spread of the middle 50% of the data. The minimum and maximum are the smallest and largest values of the data, respectively. To draw a box plot, you need to draw a horizontal or vertical axis that covers the range of the data, draw a box with its lower and upper edges at Q1 and Q3 respectively, draw a line inside the box at the median, draw whiskers from the box to the minimum and maximum or to a distance of 1.5 times IQR from it, and label it with appropriate titles and units. Any data points beyond this distance are considered outliers, which are marked with dots or stars.

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3 How to interpret a box plot?

A box plot is useful for understanding the shape, spread, and center of a probability distribution. It displays the median, which indicates the central tendency of the distribution. If the median is close to the middle of the box, the distribution is symmetric. If it’s closer to one end, it’s skewed. The IQR indicates variability in the middle 50% of the data, with a larger IQR indicating more variation. The whiskers indicate range in the lower or upper 25% of the data, with longer whiskers meaning more variation. Outliers indicate extreme values and can affect the mean and standard deviation, but not the median and IQR.

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Tomas Jancovic

Data @ Trustpilot (Microsoft-certified Power BI Data Analyst, Meta-certified Professional Data Scientist)
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Here is a simple guide on the interpretation: -The box's span (Q1 to Q3) represents the middle 50% of data, known as the interquartile range (IQR), indicating where most of your data is located. -The line inside the box shows the dataset's median value, a quick measure of its central tendency (use the median instead of the average as it minimizes the impact of outliers!). -Whiskers extend to the data's range (excluding outliers). -Outliers are the small singular points outside of the whisker range. -I've used a box plot that revealed an outlier in web form submissions, signaling bot/spam issues we'd seen before. After investigating, I found out that CAPTCHA had not been present on this specific form (but had been on our other forms).

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Akshit Pathak

Data Scientist @ NIQ | PGDM student at Adani Institute of Digital Technology Management | SIP @ Adani Energy Solutions Limited | Economics Graduate inquisitive about finance and Data Analytics | Python | SQL | Power BI
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Interpreting a box plot involves understanding key components that depict data distribution. The median line inside the box indicates the 50th percentile, dividing data into two halves. The box spans the interquartile range (IQR), showing where the middle 50% of data lies. Whiskers extend from the box to depict data beyond the IQR, often 1.5 times the IQR's length. Outliers, represented as individual points, fall outside the whiskers and signify extreme values. By analyzing these elements, one gains insights into data central tendency, spread, and outliers, crucial for informed decision-making.

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4 How to compare box plots?

Box plots are useful for comparing probability distributions of different groups of data, such as the heights of men and women, the test scores of different classes, or the incomes of different regions. To compare box plots, you need to analyze the location of the boxes (indicating the difference in the medians of the groups), the size of the boxes (indicating the difference in the IQRs of the groups), the length of the whiskers (indicating the difference in the ranges of the groups), and the number and position of outliers (indicating the difference in shapes of groups). If any two aspects are similar, then it is likely that their respective groups have similar central tendencies, variabilities, and shapes.

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5 How to create box plots in R?

R is a popular programming language for statistical analysis and visualization. You can create box plots in R using the boxplot() function, which takes a vector or a formula as an input and produces a box plot as an output. For example, you can create a box plot of the heights of men and women using the following code:

# Create a vector of heights of men
men <- c(175, 180, 178, 182, 174, 176, 179, 181, 183, 177)
# Create a vector of heights of women
women <- c(165, 168, 170, 162, 166, 167, 169, 163, 164, 171)
# Create a box plot of heights by gender
boxplot(men, women, names = c("Men", "Women"), ylab = "Height (cm)")

The result is a box plot that shows that men have a higher median height than women, and that both groups have similar variabilities and ranges. There are no outliers in either group.

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Akshit Pathak

Data Scientist @ NIQ | PGDM student at Adani Institute of Digital Technology Management | SIP @ Adani Energy Solutions Limited | Economics Graduate inquisitive about finance and Data Analytics | Python | SQL | Power BI
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# Create a box plot of Sepal Length for each Species boxplot(Sepal.Length ~ Species, data = iris, main = "Box Plot of Sepal Length by Species", xlab = "Species", ylab = "Sepal Length")

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Tomas Jancovic

Data @ Trustpilot (Microsoft-certified Power BI Data Analyst, Meta-certified Professional Data Scientist)
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Here is what else to consider: -Box plots are a starting point for your EDA. -Remember to dive deeper with tests like: the t-test, ANOVA, or chi-squared test. -Why? These will confirm whether you have statistically significant differences between means/distributions.

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Lucas Vicentini

Analista de Garantia de Qualidade Sr | Black Belt Lean Six-Sigma | Verificação Continuada de Processos | Análise Estatística de Processos
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Lembre-se que em algumas situações, gráficos box-plot podem esconder informações a respeito da distribuição do conjunto ou a quantidade amostral. Mantenha a mente aberta e considere também gráficos como dotplot (para distribuição e tamanho amostral) e o gráfico de violino (para avaliar a distribuição dos dados).

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How can you use box plots to represent probability distributions?

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3

4

5

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1 What are probability distributions?

2 How to construct a box plot?

3 How to interpret a box plot?

4 How to compare box plots?

5 How to create box plots in R?

6 Here’s what else to consider

Statistics

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How can you use box plots to represent probability distributions?

1

2

3

4

5

6

1 What are probability distributions?

2 How to construct a box plot?

3 How to interpret a box plot?

4 How to compare box plots?

5 How to create box plots in R?

6 Here’s what else to consider

Statistics

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