Problem 14 - Algebra 2 and Trigonometry

Question

The table shows the number of pounds lost during the first month by people enrolled in a weight-loss program. a. Find the range. b. Find the interquartile range. c. Which of the data values is an outlier? $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline \text { Pounds Lost } & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {11} & {15} \\ \hline \text { Frequency } & {1} & {1} & {2} & {2} & {6} & {10} & {7} & {7} & {2} & {1} & {1} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

a. Range: 14 b. IQR: 3 c. Outlier: 15

Step 1: Organize the Data

First, list out all the data points based on the frequency given: 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 11, 15.

Step 2: Calculate the Range

The range is the difference between the largest and the smallest number in the dataset. Here, the smallest number is 1 and the largest number is 15. So, the range is: \[ 15 - 1 = 14 \]

Step 3: Calculate the Quartiles

To find quartiles, first determine the median. There are 40 data points, so the median is the average of the 20th and the 21st values. Both are 6, so the median is 6. The first quartile (Q1) is the median of the first half of the data (1 to 20), and is the average of the 10th and 11th values: 5 and 5, so Q1 is 5. The third quartile (Q3) is the median of the second half (21 to 40), and is the average of the 30th and 31st values: 8 and 8, so Q3 is 8.

Step 4: Calculate the Interquartile Range

The interquartile range (IQR) is the difference between the third quartile and the first quartile. \[ IQR = Q3 - Q1 = 8 - 5 = 3 \]

Step 5: Identify Outliers

An outlier is defined as a value that is below \( Q1 - 1.5 \times IQR \) or above \( Q3 + 1.5 \times IQR \). First, calculate \( 1.5 \times IQR \):\[ 1.5 \times 3 = 4.5 \]So, outliers are below \( 5 - 4.5 = 0.5 \) and above \( 8 + 4.5 = 12.5 \). The only data point larger than 12.5 is 15, making it the outlier.

Key Concepts

Range Calculation in Statistics

The range in statistics is a simple measure of variability that tells us the spread of a dataset. It is calculated by subtracting the smallest value in the dataset from the largest value.

This gives us a quick overview of how far apart the data values are from each other. For instance, if the smallest number in a data set is 1 and the largest is 15, we find the range by calculating: \[15 - 1 = 14 \]

• The range here is 14, which indicates the dataset's overall spread.
• Knowing the range can help identify extreme points that might skew data analyses.

Understanding the Interquartile Range

The interquartile range (IQR) is a measure of where the "middle fifty" is in a dataset. It shows where the bulk of the data points lie by calculating the difference between the first quartile (Q1) and the third quartile (Q3).

First, determine Q1, the median of the first half of the data, and Q3, the median of the second half. Then compute the IQR: \[IQR = Q3 - Q1 \]

• The IQR is less affected by outliers as it focuses on the central data points.
• In our example: \[IQR = 8 - 5 = 3 \]
• This tells us that the middle 50% of our data spans 3 units.

Identifying Outliers in Data

Outliers are unusual data points that stand apart from the rest. Identifying them is crucial as they can affect statistical analyses and conclusions.

To find outliers, calculate the boundaries outside which a data point is considered an outlier. This is done using the IQR:

• Calculate \[1.5 imes IQR \]
• Determine lower bound: \[Q1 - 1.5 \times IQR \]
• Determine upper bound: \[Q3 + 1.5 \times IQR \]

For example, with an IQR of 3:

• Lower: \[5 - 4.5 = 0.5 \]
• Upper: \[8 + 4.5 = 12.5 \]
• Anything below 0.5 or above 12.5 is an outlier. Here, 15 is the only outlier.

Organizing Your Data

Data organization is a fundamental first step before performing any statistical analysis. It involves arranging the collected data points in a logical manner, often from the smallest to the largest value.

This helps in accurately identifying the range, quartiles, and any potential outliers.

• Sort your data to see trends and patterns more clearly.
• Data is typically organized in lists, tables, or plots.
• A systematic approach helps in efficient calculation and error reduction.

How to Compute Quartiles

Quartiles are values that divide your data into quarters, making it easier to understand the distribution. In a dataset, there are three main quartiles:

• Q1 (first quartile) marks the 25th percentile.
• The median (second quartile or Q2) indicates the 50th percentile.
• Q3 (third quartile) represents the 75th percentile.

To calculate:

• Arrange data in ascending order.
• Locate median (for even number, average middle two values).
• For Q1, find the median of the lower half (below median).
• For Q3, find the median of the upper half (above median).

Understanding quartiles is crucial for computing narrower ranges, such as the IQR, and spotting outliers.