Question

Construct a box plot for these data and identify any outliers: $$ 3,9,10,2,6,7,5,8,6,6,4,9,22 $$

Short answer

Question: Based on the data set (2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 22), create a box plot and identify any outliers. Answer: In the given data set, after constructing the box plot, we identified the number 22 as an outlier.

Step-by-step solution

Sort the data

First, we must sort the given data in ascending order: $$ 2,3,4,5,6,6,6,7,8,9,9,10,22 $$

Calculate the Quartiles

Now, we will find the first quartile (Q1), median (Q2), and third quartile (Q3). - Since there are 13 data points, the median is the middle value, which is the 7th value in the sorted data set: $$\text{Q2 (Median)}= 6$$ - The first quartile (Q1) is the median of the lower half of the data set (excluding Q2). There are 6 values in the lower half, so Q1 is the average of the 3rd and 4th values: $$\text{Q1}= \frac{4+5}{2} = 4.5$$ - The third quartile (Q3) is the median of the upper half of the data set (excluding Q2). There are 6 values in the upper half, so Q3 is the average of the 9th and 10th values: $$\text{Q3}= \frac{8+9}{2} = 8.5$$

Calculate Interquartile Range (IQR) and Outlier Boundaries

Now, we calculate the Interquartile Range (IQR) and boundaries for outliers. - IQR = Q3 - Q1 = $$8.5 - 4.5 = 4$$ - Lower outlier boundary = Q1 - 1.5 × IQR = $$4.5 - 1.5 × 4 = -1.5$$ - Upper outlier boundary = Q3 + 1.5 × IQR = $$8.5 + 1.5 × 4 = 14.5$$

Identify Outliers

We will identify any data points that are below the lower outlier boundary or above the upper outlier boundary. - The only data point that is outside of these boundaries is 22.

Construct the Box Plot

Now, we can construct the box plot using the data from the previous steps. 1. Draw a number line. 2. Mark the minimum value (2), Q1 (4.5), Q2 (6), Q3 (8.5), and maximum value (10) on the number line. 3. Create a box with the left edge at Q1 (4.5) and the right edge at Q3 (8.5). Place a line inside the box at the median (Q2 = 6). 4. Draw a line (whisker) from the left edge of the box to the minimum value (2) and from the right edge of the box to the maximum value (10). 5. Plot the outlier (22) as a separate point above the box plot. With the box plot constructed, we have identified 22 as an outlier in the data set.

Key concepts

Outliers Detection

When visualizing data, detecting outliers is crucial, as they can significantly influence statistical analyses and visual impressions. Outliers are values in a data set that are significantly higher or lower than the rest of the data.

In the given problem, we calculated outlier boundaries using the 1.5×IQR rule, which is a commonly used method. To detect outliers, subtract 1.5 times the Interquartile Range (IQR) from the first quartile, Q1, for the lower boundary, and add 1.5 times the IQR to the third quartile, Q3, for the upper boundary. Any data points outside these boundaries are considered outliers. In our case, the number 22 was identified as an outlier, being above the upper boundary of 14.5.

Quartiles Calculation

Quartiles divide a sorted data set into four equal parts and are essential in describing the spread and center of the data. In the exercise, we calculated three quartiles: Q1, the median (Q2), and Q3.

The median, Q2, divides the data set into two equal halves. For an odd number of data points, it is the middle value; for an even number, the average of the two middle values. In our data, the median was 6.

Q1 is found by taking the median of the first half of data, excluding Q2 if necessary, which was 4.5 for our data. Q3 is similarly the median of the second half; here, it was 8.5. The location of these quartiles provides insight into the distribution of our data and is a key part of constructing a box plot.

Interquartile Range

The Interquartile Range, or IQR, is the range within which the middle 50% of the data falls. It is calculated by subtracting Q1 from Q3, and it provides a measure of the data's dispersion that is resistant to outliers. The IQR can be particularly useful for comparing differences between the data sets and identifying where the bulk of data points lie.

In the problem, the IQR was 4, derived from Q3 (8.5) minus Q1 (4.5). This measure told us that the central 50% of the data was spread across a range of 4 units. This information, coupled with the outlier detection step, gives a clear picture of the data's variability.

Data Visualization

Data visualization is a powerful tool to communicate complex information clearly and effectively. A box plot, or box-and-whisker plot, is a standardized way of displaying the distribution of a data set based on a five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

In constructing the box plot for this exercise, we used these five key statistics to draw a clear and informative picture of the data distribution. The box plot's edges represent Q1 and Q3, the line inside represents the median, and whiskers extend to the minimum and maximum values that are not outliers. Outliers, such as the value 22 in our problem, are plotted as individual points. This visual representation helps immediately identify key aspects like symmetry, skewness, and where the data is concentrated.