Question

Find the five-number summary and the IQR. Use this information to construct a box plot and identify any outliers. A count of the number of bacteria (per 100 milliliters) in 10 samples of seawater gave these readings: $$ \begin{array}{llllllllll} 49, & 70, & 54, & 67, & 59, & 40, & 61, & 69, & 71, & 52 \end{array} $$

Short answer

The five-number summary for the given data set is {40, 52, 60, 67, 71}. The IQR is 15, calculated by subtracting Q1 (52) from Q3 (67). The outlier boundaries are 30.5 for the lower limit and 88.5 for the upper limit. In this case, there are no outliers within the data set.

Step-by-step solution

Sort the data in ascending order

Arrange the given data in ascending order: $$ \begin{array}{llllllllll} 40, & 49, & 52, & 54, & 59, & 61, & 67, & 69, & 70, & 71 \end{array} $$ Now the data is sorted, we can proceed with finding the five-number summary.

Find the five-number summary

The five-number summary consists of the Minimum, First Quartile (Q1), Median (Q2), Third Quartile (Q3), and Maximum values. Here's how to find each of them: 1. Minimum (smallest value): 40 2. Maximum (largest value): 71 3. Median (Q2): Since there are 10 data points, the median is the average of the 5th and 6th values: $$\frac{59+61}{2} = 60$$ 4. First Quartile (Q1): The median of the lower half of the data (not including the median if the number of data points is odd). Since there are 5 data points in the lower half, Q1 is the median of this set, which is 52. 5. Third Quartile (Q3): The median of the upper half of the data (not including the median if the number of data points is odd). Since there are 5 data points in the upper half, Q3 is the median of this set, which is 67. The five-number summary is: {40, 52, 60, 67, 71}

Calculate Interquartile Range (IQR)

To find the IQR, subtract the value of Q1 from the value of Q3: IQR = Q3 - Q1 = 67 - 52 = 15

Check for outliers

Any data points that lie below $$Q1 - 1.5 \times IQR$$ or above $$Q3 + 1.5 \times IQR$$ can be considered outliers. Lower outlier boundary = Q1 - 1.5 × IQR = 52 - 1.5 × 15 = 30.5 Upper outlier boundary = Q3 + 1.5 × IQR = 67 + 1.5 × 15 = 88.5 By checking the sorted data, we can see that there are no values below 30.5 or above 88.5. Therefore, there are no outliers.

Construct the box plot

To create a box plot, follow these steps: 1. Draw a number line that covers the range of the data. 2. Above the number line, draw a box from Q1 (52) to Q3 (67). 3. Draw a line within the box at the median (60). 4. Draw a line extending from Q1 (52) to the minimum value (40). 5. Draw a line extending from Q3 (67) to the maximum value (71). And that is the box plot representing the given data.

Key concepts

Five-Number Summary

Understanding the five-number summary is crucial for statistical analysis, especially in creating box plots. This summary efficiently describes the spread and center of a dataset using five key data points:

• Minimum: The smallest number in your data set. For our example, it's 40.
• Maximum: The largest number, here it is 71.
• Median (Q2): The middle value when the data is ordered. In a set with an even number of values, such as ours, the median is the average of the two central numbers. Here, it is \( \frac{59 + 61}{2} = 60 \).
• First Quartile (Q1): This is the median of the lower half of the data set. For these numbers, it's 52.
• Third Quartile (Q3): This represents the median of the upper half, which in this case is 67.

These five numbers provide a concise summary of the dataset, showing its range and where most data points lie. It's a great way to quickly understand the distribution of the data.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion, or how spread out the values in a dataset are. It focuses on the middle 50% of the data, offering a clearer picture of data distribution by excluding outliers or extreme values.

The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1), capturing the central spread of the data set. In the given example, we calculate it as follows:

• Formula: \( \text{IQR} = Q3 - Q1 \)
• Calculation: \( 67 - 52 = 15 \)

The result tells us that the middle 50% of the data points fall within a range of 15 units. This helps us understand the consistency and concentration of the dataset. Unlike the range, which can be swayed by extreme values, the IQR offers a more robust measure of the central tendency.

Outliers

Outliers are data points that significantly deviate from the rest of the dataset. They can skew the data analysis, making it important to identify and understand how they affect the results.

To detect outliers using the IQR, we calculate what is known as the outlier boundaries. Any points outside these boundaries may be considered outliers:

• Lower Outlier Boundary: \( Q1 - 1.5 \times \text{IQR} \)
• Upper Outlier Boundary: \( Q3 + 1.5 \times \text{IQR} \)

Applying these calculations to our dataset:

• Lower Boundary: \( 52 - 1.5 \times 15 = 30.5 \)
• Upper Boundary: \( 67 + 1.5 \times 15 = 88.5 \)

In the example provided, no data points fall below 30.5 or above 88.5. Therefore, our dataset does not contain any outliers. Identifying outliers is crucial as they can provide insights into anomalies within the data or point towards measurement errors or exceptional cases.