How to Analyze Box Plots

 

There are two ways that you can analyze box plots. Consider the above plot showing sales of a particular brand per day across all stores over the months from February (2) and June (6). Let's assume June has 30 days. 

1. Analyze a single Box Plot

2. Compare two or more box plots

We'll study them separately.

1. Analyze a single Box Plot

There are Six points that you need to focus on when you are analyzing a single box Plot

    a. Total Size of the Plot

The total size of the plot indicates the range of the values. For example, in June month, the sales per day varies from 0 to 20000. 

    b. Absolute Position of Median

Roughly 50% of the values fall below this value, and 50% of the values fall above. So in June month, the median sales per day is about 7500 Rs. With about 15 days below 7500 Rs. and 15 days above 7500 Rs.  Considering the range from the size of plot in point b,  it is lower, as it should be around 10000 Rs. Which means that there are more days with values less than 10K then there are days with values more than 10K.

    c. Position of the Median Relative to Box

Median is at the lower half of the box. It indicates that the distribution is right skewed, which means that more days have low values and some days have higher values. This is also supported by the point b. combined with a. as indicated above. 

    d. Size of the box compared to the range. 

The size of the box indicates the Inter-quartile range i.e. the values between 1st quartile and 3rd quartile. It simply indicates the middle 50% values of the data. It is relatively robust and free from the extreme values. So we can see for June data that values lie roughly between 5000 and 12000. Their average is 8500 whereas median is at 7500, lower than the ideal mid. The IQR is about 12000-5000 = 7000 which when comparing with the range of 0 to 20000, is relatively less. It indicates that there are more extreme values. 

7. Relative Lengths of two Whiskers

We can see that upper whisker is more than lower whisker. Whiskers indicate extreme values. So the data has more extreme values at the upper end than extreme values at the lower end. 

8. Relative Lengths of Whiskers compared to the Box

We can see that  size of upper whisker is less than 1.5 times that of size of box. It means that the value of the end of whisker is the maximum value ( apart from outlier)  at about 20000. Similarly the size of lower whisker is less than 1.5 times less than the size of box. It means that the value of the end of lower whisker is the min value at about 0. 

9. Outliers

These are values that more than 1.5 times the IQR. So there is no outlier here. 

SUMMARY

The box plot analysis of daily sales data for the brand in June reveals several key insights. The total range of sales per day varies from 0 to 20,000 Rs., indicating a wide range of sales values. The median sales per day stands at around 7,500 Rs., with approximately half of the days below this value and the other half above. Notably, there are more days with sales below 10,000 Rs. than above, indicating a skewed distribution skewed towards lower sales. The interquartile range (IQR), representing the middle 50% of the data, spans from 5,000 to 12,000 Rs., which is relatively small compared to the overall range. This suggests the presence of more extreme sales values, particularly at the upper end. The absence of outliers suggests a consistent dataset. The length of the upper whisker exceeds that of the lower whisker, indicating more extreme sales values at the higher end.

So the sales per day in the month shows a high variability with more extreme values towards the upper part of data, thus indicating a right skewed data. This could have happened because of some event, probably a discount sale. 

1. Compare two or more box Plots

To compare two box plots, you can visually analyze their key components and consider the following aspects:

Size : Size of the plot from whisker to whisker can be used for comparison . If the size is more, the data in the plot is more spread out. 

Overlapping: Check if the boxes and whiskers of the two box plots overlap. If the boxes or whiskers overlap significantly, it suggests that the distributions of the two datasets have similarities in terms of central tendency and spread. On the other hand, if the boxes and whiskers do not overlap or have minimal overlap, it indicates potential differences between the distributions.

for example comparing May and June, the boxes overlap significantly. 

Median Comparison: Compare the positions of the medians in the two box plots. If one median is higher than the other, it suggests a difference in the central tendency of the two datasets. A higher median in one box plot indicates higher values or sales compared to the other dataset.

for example comparing May and June, the medians are similar

Quartiles: Examine the quartiles (Q1 and Q3) of the two box plots. If the two datasets have similar quartiles, it suggests similarities in the lower and upper ranges of the data. If the quartiles differ, it indicates differences in the spread of the data or the range of sales.

The quartiles are also similar

Outliers: Pay attention to any outliers in the box plots. Compare the presence, position, and magnitude of outliers in each plot. Unusual outliers may indicate unique patterns or extreme values in one dataset compared to the other.

There is an outlier in May.

Overall Shape: Observe the overall shape of the box plots. If the boxes are similar in length, it suggests similar variability in the two datasets. If one box is longer than the other, it indicates a larger range or greater variability in the corresponding dataset.

Conclusion

In comparing box plots of May and June, there shapes are similar, however, there are more extreme values in June than in May

Post Notes

Relation between Box Plots and Normal Distribution

If we are looking at the box plot of a normal distribution, the relationship is as follows:

Thus the "box" is about 0.67 std deviation on both sides and the "whiskers" are about 2.69 std deviation on both sides. 

What is a Box Plot

What is a Box Plot 

A box plot, also known as a box-and-whisker plot, is a graphical representation of the distribution of a dataset. It displays a summary of the data's central tendency, spread, and potential outliers. A box plot provides a visual depiction of the quartiles, median, and range of the dataset.

The key components of a box plot include:

Box: The central rectangular shape in the plot represents the interquartile range (IQR), which contains the middle 50% of the data. The bottom of the box represents the first quartile (Q1), and the top represents the third quartile (Q3).

Median: Inside the box, there is a horizontal line that represents the median. The median is the value that separates the lower 50% of the data from the upper 50%.

Whiskers: The lines extending from the box, often with lines or horizontal bars at their ends, are known as whiskers. They represent the range of the data, excluding outliers. The length of the whiskers can vary depending on the method used to calculate them, such as 1.5 times the IQR or extending to the maximum and minimum values.

Outliers: Individual data points that fall significantly outside the whiskers are considered outliers. Outliers are often represented as individual points on the plot, indicated by dots or small circles.

Box plots provide several insights about a dataset:

Central Tendency: The position of the median within the box indicates the central tendency of the data.

Spread: The width of the box and the length of the whiskers provide information about the spread or variability of the data.

Skewness: The asymmetry of the box plot can indicate skewness in the distribution.

Outliers: The presence of outliers outside the whiskers suggests extreme or unusual values.

Box plots are useful for summarizing and comparing distributions across different groups or categories. They provide a concise visualization that helps in understanding the distributional characteristics of the data and identifying potential anomalies or patterns.

Why IQR is so important in a box plot

The interquartile range (IQR) is a crucial component of box plots because it provides valuable information about the spread or variability of the data. The IQR represents the range that contains the middle 50% of the dataset, which is a more robust measure than using the full range (i.e., maximum and minimum values) to describe the spread.

Here are some reasons why the IQR is important in box plots:

Robustness to Outliers: The IQR is less sensitive to outliers compared to the full range. By using the IQR, box plots focus on the central portion of the data and are less affected by extreme values. This makes box plots more resistant to the influence of outliers and provides a more representative measure of the typical spread of the majority of the data.

Summarizing Spread: The IQR summarizes the spread of the middle 50% of the dataset. It provides a compact measure that helps understand the variability of the data without considering each individual value. The width of the box in a box plot represents the IQR, giving a visual representation of the spread.

Comparison of Distributions: The IQR is useful for comparing the spread of different distributions or groups in box plots. By comparing the widths of the boxes, you can quickly assess the relative variability of the datasets being compared. A wider box indicates a larger spread or greater variability, while a narrower box suggests a more tightly clustered distribution.

Identifying Skewness: The IQR, along with the position of the median within the box, can help identify skewness in the data. If the IQR is asymmetrically distributed around the median, it suggests skewness in the dataset. This information helps in understanding the shape and characteristics of the distribution.

Outlier Detection: The IQR is instrumental in identifying potential outliers in the dataset. In many box plot constructions, outliers are defined as individual data points that fall outside a certain range, such as 1.5 times the IQR. By using the IQR as a threshold, box plots can effectively highlight potential extreme values that might require further investigation or analysis.

Overall, the IQR is important in box plots as it provides a robust and concise summary of the spread or variability of the data, allowing for easier comparison, outlier detection, and assessment of skewness. It helps in gaining insights into the distributional characteristics of the dataset while minimizing the influence of outliers.

What are the various possible shapes in a box plot and their interpretation

When analyzing the spread and symmetry of a box plot, you can encounter various shapes that provide insights into the distribution of the data. Here are some common shapes and their interpretations:

Symmetrical Distribution:

A symmetrical distribution is characterized by a box plot where the median is approximately centered within the box, and the whiskers are of similar length. The distribution is balanced, indicating that the data is evenly spread around the median. In such cases, the first quartile (Q1) and the third quartile (Q3) are equidistant from the median. A symmetrical distribution suggests that the dataset is well-behaved and lacks significant skewness.

Skewed Right (Positively Skewed) Distribution:

A skewed right distribution, also known as positively skewed or right-skewed, is indicated by a box plot where the median is closer to the bottom of the box, and the whisker on the upper side (above Q3) is longer than the lower whisker (below Q1). This means that the majority of the data is concentrated on the lower end of the distribution, while a few extreme values extend the upper tail. In this case, the mean is usually greater than the median.

Skewed Left (Negatively Skewed) Distribution:

A skewed left distribution, also known as negatively skewed or left-skewed, is the opposite of a skewed right distribution. The median is closer to the top of the box, and the whisker on the lower side (below Q1) is longer than the upper whisker (above Q3). This indicates that the majority of the data is concentrated on the higher end of the distribution, with a few extreme values in the lower tail. In a negatively skewed distribution, the mean is usually less than the median.

Bimodal Distribution:

A bimodal distribution appears when there are two distinct peaks or modes in the data. In a box plot, this is represented by two separate boxes, each with its own median, whiskers, and outliers. This suggests that the dataset consists of two separate groups or categories, and there may be different underlying factors influencing each group.

Outliers and Extreme Values:

In any distribution, outliers are individual data points that fall significantly outside the whiskers. They are represented as individual points on the plot. Outliers can occur in any distribution shape and may indicate anomalies, errors, or unusual observations. They can have a significant impact on the overall interpretation of the data, so it's important to carefully consider their presence and possible explanations.

By examining the shape of the box plot, including the width of the box, length of the whiskers, and the position of the median, you can gain insights into the spread, symmetry, and potential underlying characteristics of the distribution being represented by the data.

What are the limitations of Box Plots

While box plots are a useful visualization tool, they do have some limitations. It's important to be aware of these limitations when interpreting and using box plots:

Limited Descriptive Statistics: Box plots provide a summary of the data's central tendency, spread, and potential outliers. However, they do not provide detailed information about the shape of the distribution, such as the presence of multiple modes, skewness, or kurtosis. Other statistical measures or additional visualizations may be required to obtain a more comprehensive understanding of the data.

Loss of Information: Box plots provide a simplified representation of the data and can result in the loss of some information. They only show summary statistics, such as quartiles and medians, and do not display the individual data points. Consequently, specific patterns or variations within the data may be obscured.

Unequal Sample Sizes: When comparing box plots with different sample sizes, it's essential to consider that the box sizes may not be directly comparable. A box plot with a larger sample size will typically have a smaller box compared to one with a smaller sample size, even if the spread of the data is similar.

Insensitivity to Distributional Shape: Box plots do not provide detailed information about the shape of the distribution, such as whether it is symmetric, skewed, or bimodal. They cannot differentiate between different types of distributions with similar box plot characteristics. Depending on the context, additional visualizations or statistical tests may be necessary to explore the shape of the distribution.

Handling of Outliers: Box plots can help identify potential outliers, but they do not provide a precise definition or account for the impact of outliers on the distribution. The choice of the method used to define and display outliers, such as the whisker length or threshold, can affect the interpretation of the plot.

Limited to Univariate Analysis: Box plots are primarily designed for univariate analysis, where only one variable is represented. They may not be suitable for exploring relationships or comparisons involving multiple variables simultaneously. In such cases, other types of plots or multivariate techniques might be more appropriate.

Subjective Interpretation: The interpretation of box plots can be subjective to some extent. Different viewers may interpret the same plot differently, especially when assessing the presence or significance of outliers or the symmetry of the distribution. It's crucial to provide context and consider the specific characteristics of the dataset being analyzed.

Despite these limitations, box plots remain a valuable tool for summarizing and comparing distributions, providing a quick visual overview of essential statistical measures. They can serve as a starting point for data exploration and hypothesis generation, but additional analyses and visualizations may be necessary for a comprehensive understanding of the data.