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

**Question:** Based on the given dataset (12, 18, 22, 23, 24, 25, 26, 26, 27, 28), construct a box plot and identify any potential outliers. **Answer:** Following the steps for constructing a box plot and identifying outliers, we found one outlier in the dataset: the number 12. The constructed box plot displays the data, including the quartiles (Q1 = 22; Q2 = 24.5; Q3 = 26.5) and the outlier marked appropriately.

Step-by-step solution

Arrange the numbers in ascending order

First, we need to list the given data in ascending order: $$ 12,18,22,23,24,25,26,26,27,28 $$

Calculate the median, lower quartile (Q1), and upper quartile (Q3)

To find the median, we look for the middle value in the ordered data set. With an even number of \(10\) data points, we will take the average of the middle two values: Median \((Q2) = \frac{24+25}{2} = 24.5\) Next, we need to calculate the lower quartile (Q1) and upper quartile (Q3). Q1 is the median value of the lower half of the dataset: \(Q1 = 22\) Q3 is the median value of the upper half of the dataset: \(Q3 = 26.5\)

Determine the Interquartile Range (IQR)

Calculate the Interquartile Range (IQR) by subtracting Q1 from Q3: $$ IQR = Q3 - Q1 = 26.5 - 22 = 4.5 $$

Check for outliers

To detect possible outliers, we will use the 1.5 x IQR rule. Any value below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR will be considered an outlier. Lower bound for outliers: \(Q1 - 1.5*IQR = 22 - 1.5*(4.5) = 14.25\) Upper bound for outliers: \(Q3 + 1.5*IQR = 26.5 + 1.5*(4.5) = 34.25\) By looking at the dataset, we see that the number \(12\) is below the lower bound for outliers, so it is considered an outlier.

Construct the box plot

To construct the box plot, follow these steps: 1. Draw a number line with the minimum value (12) and maximum value (28). 2. Using the values of Q1, Q2 (median), and Q3, draw a box and whiskers plot. 3. For the outlier(s), use a different mark (e.g., a circle or asterisk) beyond the whiskers. Doing this, we have constructed a box plot for the given dataset with the outlier marked appropriately.

Key concepts

Outliers

Outliers are data points that significantly differ from other observations in your dataset. In simpler terms, they are numbers that don't seem to fit with the rest. These outliers can skew the data's overall trend and affect statistical calculations, such as the mean and standard deviation. Identifying outliers is crucial for drawing more accurate conclusions from data.

There are various methods to detect outliers, but a common approach is the 1.5 x Interquartile Range (IQR) rule. By using this rule, you can determine if a number is too far from other data points:

• Calculate the bounds for potential outliers by using the IQR. The lower bound is given by subtracting 1.5 times the IQR from the first quartile (Q1), while the upper bound is found by adding 1.5 times the IQR to the third quartile (Q3).
• Any data point that lies outside these bounds (either below the lower bound or above the upper bound) is considered an outlier.

In our dataset, the number 12 is identified as an outlier because it falls below the calculated lower bound, emphasizing its unusual position relative to other numbers in the dataset.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion or spread within a dataset. It represents the range within which the middle 50% of the data lies, effectively focusing on the crux of the data while ignoring extremes.

To compute the IQR, subtract the first quartile (Q1) from the third quartile (Q3). This calculation gives us an understanding of where the bulk of the dataset exists:

• Q1 (first quartile) divides the lowest 25% of data from the rest. It's the middle of the first half of the data.
• Q3 (third quartile) divides the lowest 75% of data from the highest 25%. It's the median of the second half of the data.
• IQR = Q3 - Q1: This tells us about the spread of the central part of the data, regardless of any potential outliers.

A larger IQR indicates more variability within the central 50% of the dataset, while a smaller IQR suggests less variability. In our example, the IQR is 4.5, which helps assess the spread of the data points.

Quartiles

Quartiles are a set of values that divide your data into four equal parts, each comprising 25% of the data points. They are fundamental in understanding the spread and distribution of your data.

When establishing quartiles, you generally calculate three different points:

• Q1 (First Quartile): This is the median of the lower half of the dataset. It indicates the 25th percentile, meaning 25% of data points are less than or equal to Q1.
• Q2 (Second Quartile or Median): This is the median of the entire dataset, showing the cut-off point where half of the data lies below and half above.
• Q3 (Third Quartile): This is the median of the upper half of the dataset. It represents the 75th percentile, with 75% of data below it.

These quartiles are crucial in constructing a box plot, where Q1, median (Q2), and Q3 create the boundaries of the box, giving you a concise visual representation of your data's distribution. For our set of numbers, these quartiles help us visualize the layout and identify if any numbers deviate significantly from the main group.