Problem 13 - Algebra 2 and Trigonometry

Question

The table shows the number of hours that 40 third graders reported studying a week. Find the range and the interquartile range. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Hours } & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} & {11} & {12} \\ \hline \text { Frequency } & {2} & {1} & {3} & {3} & {5} & {8} & {8} & {5} & {4} & {1} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The range is 9 and the interquartile range is 3.

Step 1: List the Data

Start by listing all the data points. Each frequency corresponds to how many times a specific number of hours was reported. The data set is 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12.

Step 2: Determine the Range

The range is the difference between the maximum and minimum values in the data set. In this case, the minimum value is 3 and the maximum value is 12. The range is calculated as: \(12 - 3 = 9\).

Step 3: Find the First Quartile (Q1)

To find Q1, we must determine the data point that lies at the 25th percentile. With 40 data points, the position for Q1 is \(\frac{40+1}{4} = 10.25\). Since it is not a whole number, Q1 is the value at the 10th position plus 0.25 times the difference between the 11th position and 10th position values. In the ordered data list, the 10th position is 7, and the 11th position is also 7. Therefore, Q1 = 7.

Step 4: Find the Third Quartile (Q3)

To find Q3, calculate the position of the 7th percentile. For 40 data points, Q3 is calculated as \(\frac{(40+1)\times3}{4} = 30.75\). This means Q3 is the value at the 30th position plus 0.75 times the difference between the values of the 31st and 30th positions. Both the 30th and 31st positions are 10, so Q3 = 10.

Step 5: Calculate the Interquartile Range (IQR)

The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1). So, \(\text{IQR} = 10 - 7 = 3\).

Key Concepts

Understanding the Range

The range is a simple measure of variability or spread in a data set. It provides basic insight into how spread out the values are.
This is useful when you want to understand the difference between the largest and smallest values in a set.

• To calculate the range, you subtract the smallest value from the largest value.
• For example, in the data given with a minimum of 3 and a maximum of 12, the range is calculated as \( 12 - 3 = 9 \).

However, while the range gives a quick sense of spread, it does not account for how data is distributed between these extremes.
For more nuanced insights, we turn to measures like the interquartile range.

Decoding the Interquartile Range

The interquartile range (IQR) provides a glimpse into the spread of the middle 50% of the data.
This helps in understanding data variability with more depth than the simple range.

• The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
• For the given exercise, Q1 is found at the 10.25th position and calculated as 7, while Q3 is at the 30.75th position, also resulting in a value of 10.
• Thus, the IQR is \( 10 - 7 = 3 \).

The IQR is more robust as it is less affected by extreme values or outliers compared to the range.
It provides a clearer picture of where the bulk of the data points lie.

Exploring Quartiles

Quartiles divide a data set into four equal parts, providing key insights into distribution.

• The first quartile (Q1) is the value below which 25% of the data falls.
• The second quartile (Q2), also known as the median, splits the data into two equal halves.
• The third quartile (Q3) marks the boundary below which 75% of the data resides.

For the exercise, Q1 is 7 and Q3 is 10. These quartiles supply more context than the median alone by showcasing where data clusters or spreads out.
Analyzing quartiles helps in identifying outliers and understanding the symmetrical or skewed nature of the data distribution.