Problem 11 - Algebra 2

Question

Identify the outlier of each set of values. $$\begin{array}{llllllllll}17 & 21 & 19 & 10 & 15 & 19 & 14 & 0 & 11 & 16\end{array}$$

Step-by-Step Solution

Answer

The outlier in this dataset is 0.

Step 1: Arrange the data in ascending order

First, the dataset should be arranged in ascending order: \(0, 10, 11, 14, 15, 16, 17, 19, 19, 21\).

Step 2: Find the first (Q1) and third quartiles (Q3)

Q1 is the median of the first half of the dataset, and Q3 is the median of the second half. In this dataset, Q1 is \((10 + 14) / 2 = 12\) and Q3 is \((19 + 19) / 2 = 19\).

Step 3: Calculate the interquartile range (IQR)

The IQR is the difference between Q3 and Q1, so IQR= \(19 - 12 = 7\).

Step 4: Determine outliers

Outliers are values below \(Q1 - 1.5 * IQR\) or above \(Q3 + 1.5 * IQR\). In this case, they are values below \(12 - 1.5 * 7 = -4.5\) or above \(19 + 1.5 * 7 = 29.5\). In this dataset, only 0 is below -4.5.

Key Concepts

Quartiles

When analyzing a set of numerical data, understanding quartiles is crucial. Quartiles divide the data into four equal parts. The first quartile, or \(Q1\), is the median of the lower half of the data. This means it separates the lowest 25% of the data from the rest. On the other hand, the third quartile, or \(Q3\), is the median of the upper half, separating the top 25% of the data. This division helps in identifying the spread and tendencies within the dataset.
To find the quartiles, the data must first be sorted in ascending order. In our example, the dataset is \(0, 10, 11, 14, 15, 16, 17, 19, 19, 21\). Next, \(Q1\) is calculated by finding the median of the first five numbers: \(0, 10, 11, 14\), and \(15\), which gives us \(Q1 = 12\). Similarly, \(Q3\) is found by taking the median of the last five numbers: \(16, 17, 19, 19, 21\), producing \(Q3 = 19\). These quartiles then help in further analysis by defining the boundaries for calculating IQR and identifying outliers.

Interquartile Range (IQR)

The interquartile range, abbreviated as IQR, is a measure of statistical dispersion. It is especially useful for understanding the spread and variability in a dataset. The IQR is found by subtracting the first quartile \(Q1\) from the third quartile \(Q3\): \[ IQR = Q3 - Q1 \]A higher IQR implies greater variability in the data.
For our example, with \(Q1 = 12\) and \(Q3 = 19\), the IQR is calculated as \(19 - 12 = 7\). This IQR informs us about the spread of the central 50% of the data. Importantly, the IQR is resistant to outliers, which makes it a robust indicator of variability compared to other measures like the range. Understanding the IQR is key to identifying outliers in the dataset, as it sets thresholds beyond which values are considered suspect or atypical.

Data Analysis

Data analysis involves extracting meaningful insights from raw data. In our example, after arranging the data in order and calculating quartiles and the IQR, you're equipped to identify outliers. Outliers are data points significantly different from others, potentially indicating errors or unique cases.
To find outliers, calculate thresholds using the IQR. An outlier is any number below \[ Q1 - 1.5 \times IQR \] or above \[ Q3 + 1.5 \times IQR \]. Calculating these limits for our data gives us \[ 12 - 1.5 \times 7 = -4.5 \] and \[ 19 + 1.5 \times 7 = 29.5 \]. Any data point outside these bounds is considered an outlier. For this dataset, only the number 0 appears suspicious, as it lies below the lower threshold. Recognizing such outliers can help make informed decisions and possibly refine data collection methods in future analyses.