Question

Construct a box plot for these data and identify any outliers: $$ 25,22,26,23,27,26,28,18,25,24,12 $$.

Short answer

Answer: The IQR is 4, and there is one potential outlier - the value 12.

Step-by-step solution

Arrange the data in ascending order

First, arrange the given data in ascending order: $$ 12, 18, 22, 23, 24, 25, 25, 26, 26, 27, 28 $$

Find the median (Q2)

Since there are 11 data points, the median is the middle value, which is the 6th value in the ordered list: Median = 25

Find the first quartile (Q1)

To find the first quartile, we take the median of the lower half of the data (excluding the median itself, since there are an odd number of data points). This would be the middle value of the first 5 data points: First Quartile (Q1) = 22

Find the third quartile (Q3)

Similarly, find the median of the upper half of the data (excluding the median itself). This would be the middle value of the last 5 data points: Third Quartile (Q3) = 26

Calculate the interquartile range (IQR)

The interquartile range (IQR) measures the spread of the central 50% of the data and is calculated as the difference between Q3 and Q1: IQR = Q3 - Q1 = 26 - 22 = 4

Determine the range for potential outliers

Potential outliers are values that fall outside of the range defined by: Lower bound = Q1 - 1.5 * IQR Upper bound = Q3 + 1.5 * IQR Calculate the lower and upper bounds: Lower bound = 22 - 1.5 * 4 = 16 Upper bound = 26 + 1.5 * 4 = 32

Identify any outliers

Check if any data points lie outside of the defined lower and upper bounds: The value 12 is less than the lower bound of 16, so it is a potential outlier. All other values fall within the acceptable range.

Construct the box plot

The box plot can now be constructed using the determined values: 1. Draw a number line and indicate the Q1, median, and Q3 values with vertical lines. 2. Connect the vertical lines for Q1, median, and Q3 to form a box. 3. Extend whiskers from the box edges to the smallest and largest data points that are not outliers. 4. Add a symbol (such as a circle or an asterisk) to represent the identified outlier (12). The resulting box plot will display the data distribution and the outlier (12) clearly.

Key concepts

Outliers

Outliers are data points that stand out significantly from the rest of the dataset. They can heavily influence statistical calculations and, if not addressed properly, might lead to incorrect conclusions. In a box plot, outliers are often displayed as individual points that are distinctly separated outside the "whiskers" of the plot.
To identify outliers, we establish a rule using the interquartile range (IQR). Any data point that lies more than 1.5 times the IQR below the first quartile (Q1), or above the third quartile (Q3), is considered an outlier. Let's apply this principle to our data:

• Calculate the IQR, which is the difference between Q3 and Q1. In this case, it's 4.
• Determine the lower boundary: Q1 minus 1.5 times the IQR. Here, it's calculated as 22 - 1.5 * 4 = 16.
• Determine the upper boundary: Q3 plus 1.5 times the IQR. It's calculated as 26 + 1.5 * 4 = 32.
• Now, check the data points. Any value outside the range of 16 to 32 is an outlier. For our dataset, the value 12 is below 16, marking it as an outlier.

In summary, outliers are extreme values that need special attention in data analysis to preserve the integrity of the dataset's interpretation.

Quartiles

Quartiles divide a dataset into four equal parts, helping to understand the data's distribution. They are crucial for creating a box plot and identifying outliers.
There are three key quartiles to remember:

• The first quartile, Q1, marks the 25th percentile and represents the value below which 25% of the data falls. Given our ordered data, Q1 is 22.
• The second quartile, Q2, also known as the median, corresponds to the 50th percentile. For our data, the median is the center value, 25.
• The third quartile, Q3, indicates the 75th percentile, below which 75% of data points lie. In our example, Q3 is 26.

Estimating quartiles involves sorting the data in ascending order and selecting key points that help divide the data into quarters. Quartiles are used extensively in statistics to highlight various spread and density details within datasets.

Interquartile Range

The interquartile range (IQR) is a measure of statistical dispersion, describing the middle 50% of the dataset. It provides a view of the variability by determining how spread out the central values are.
To calculate the IQR, subtract Q1 from Q3, giving you the range within the first and third quartiles:

• Using our data: IQR = Q3 - Q1 = 26 - 22 = 4.

The IQR is resistant to outliers because it only focuses on the middle portion of the dataset, making it a very reliable measure of spread. It's also a key component in constructing a box plot, as it determines the "box," the central part of the plot. This measurement is used to calculate the threshold for identifying outliers, establishing a range within which most of the data should reside. Understanding the IQR allows you to make informed decisions about the variability of your dataset.