Question

Examine issues of location and spread for boxplots. In each case, draw sideby- side boxplots of the datasets on the same scale. There are many possible answers. One dataset has median 50, interquartile range 20 , and range 40 . A second dataset has median 50, interquartile range 50 , and range 100 . A third dataset has median 50 , interquartile range 50 , and range 60 .

Short answer

All three datasets are centred around the same value (Median = 50), but they differ in their spread. The interquartile ranges for the second and third datasets are larger than for the first dataset, indicating a greater spread of the middle 50% of data. The range is the largest for the second dataset, suggesting overall greater data dispersion. The third dataset, despite having an IQR equivalent to dataset 2, has less overall spread due to a smaller range.

Step-by-step solution

Understand the Data

Firstly, identify the median which is the middle value for each data set. Also, acknowledge the interquartile range (IQR) which is the range of the middle 50% of the data, and is calculated as Q3 - Q1. The 'range' is the difference between the maximum and minimum values. Dataset 1: median 50, IQR 20, range 40; Dataset 2: median 50, IQR 50, range 100; Dataset 3: median 50, IQR 50, range 60.

Create the Boxplots

Plot three boxplots side-by-side. For boxplot of each data set, draw a box from the Q1 to Q3 value. To find Q1 and Q3, add or subtract half of the IQR to the median. The line inside the box represents the median. The whiskers extend from Q1 and Q3 to the minimum and maximum value respectively. To find maximum and minimum, add or subtract half of the range to the median.

Examine Location and Spread

Examine issues of location and spread for the boxplots. All three datasets have the same median (50), so they have the same location. However, the spread (IQR and range) differ for each dataset. Dataset 2 and 3 have larger IQR showing greater middle 50% data dispersion than Dataset 1. Dataset 2 has the largest spread (range) indicating that its data is the most dispersed overall.

Key concepts

Interquartile Range (IQR)

One of the fundamental components of a boxplot is the interquartile range (IQR), which measures the spread of the middle 50% of a dataset. Essentially, the IQR reflects the range between the first quartile (Q1) and the third quartile (Q3) values of a dataset.

Calculating the IQR is straightforward: you subtract Q1 from Q3, and the result shows how much variability there is in the central portion of your data. In our exercise, for instance, Dataset 1 has an IQR of 20, which means there is less variance in the middle 50% compared to Dataset 2 and 3, which both have an IQR of 50. Understanding IQR is essential as it helps identify the compactness of the data and is less affected by outliers and extreme values than the full range.

Dataset Comparison

Comparing datasets is an integral part of statistical analysis, often to assess differences and similarities in their central tendency and variability. Boxplots are particularly useful for this purpose as they summarize the data through five-number summaries (minimum, Q1, median, Q3, and maximum) and showcase the data’s spread.

When comparing datasets with boxplots, you should check for the IQR, which indicates the concentration of the middle 50% of values. In our example, although all datasets share the same median of 50, indicating a similar central location, their IQRs differ significantly. Moreover, comparing the ranges (total spread) alongside the IQR provides insights into how spread out the entire set of values is, not just the middle 50%.

Data Spread

Data spread, or variability, is a key concept in statistics, exhibiting how much the data points differ from each other. A higher data spread means the values are more spread out from the center, and a lower spread means the values are closer to the center. The range, IQR, and boxplot whiskers are all indicators of data spread.

The boxplot visually communicates the spread of data: the wider the box (representing the IQR), the greater the variability within the central portion of the data. Whiskers on boxplots extend to the minimum and maximum values, demonstrating the total spread. For example, in the exercise, Dataset 2 has a much larger range than the others, signaling that its values are more widely dispersed.