Problem 12 - Algebra 2 and Trigonometry

Question

The ages of students in a Spanish class are shown in the table. Find the range and the interquartile range. $$ \begin{array}{|c|c|}\hline \text { Age } & {\text { Frequency }} \\ \hline 19 & {1} \\ {18} & {8} \\ {17} & {8} \\ {16} & {6} \\ {15} & {2} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

Range: 4, Interquartile Range: 2.

Step 1: List the Ages

First, list all the ages in order from the table using their frequencies. The ages, based on frequency, are: 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19.

Step 2: Calculate the Range

Find the range by subtracting the smallest age from the largest age. The smallest age is 15 and the largest age is 19, so: \( 19 - 15 = 4 \). Thus, the range is 4.

Step 3: Determine the Median

Count the total number of students to find the median. There are 25 students. The median age will be the 13th number because \( \frac{25+1}{2} = 13 \). In the ordered list, the 13th age is 17.

Step 4: Find Quartiles

Now find the lower quartile (Q1) and the upper quartile (Q3). For Q1, consider the first 12 numbers: 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17. The middle number is the average of the 6th and 7th numbers, both 16, so Q1 is 16. For Q3, consider the last 12 numbers: 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19. The middle number is the average of the 6th and 7th numbers, both 18, so Q3 is 18.

Step 5: Calculate the Interquartile Range (IQR)

The IQR is the difference between Q3 and Q1. Thus, \( IQR = Q3 - Q1 = 18 - 16 = 2 \). The interquartile range is 2.

Key Concepts

Interquartile Range

The interquartile range (IQR) is a measure of variability, which describes the spread of the middle 50% of the data. It's particularly helpful to identify the range in which the bulk of your data points lie, offering insights into the data's consistency. To compute the IQR, you take the difference between the upper quartile (Q3) and the lower quartile (Q1). In simpler terms, this is calculated as:

• Identify Q1, the 25th percentile
• Identify Q3, the 75th percentile
• Subtract Q1 from Q3 to get IQR

From our example, Q1 is 16 and Q3 is 18, so the IQR is calculated as:\[ IQR = 18 - 16 = 2 \]A smaller IQR means that the data is more compact around the median, while a larger IQR indicates more variability. It acts as a robust and resistant measure of statistical dispersion when compared to other measures like the range.

Range in Statistics

Range is one of the simplest measures of dispersion in statistics. It provides a quick understanding of the data spread by identifying the difference between the maximum and minimum values. Here is how you can calculate it:

• Identify the smallest value in your data set (min)
• Identify the largest value in your data set (max)
• Subtract the smallest value from the largest value

Expressed in a formula, \[ \text{Range} = \text{max} - \text{min} \]For the ages given, the smallest (min) age is 15 and the largest (max) age is 19, thus:\[ \text{Range} = 19 - 15 = 4 \]The range is a straightforward way to see the extremes in your data, but it can be sensitive to outliers. Unlike the IQR, it does include all data points in its calculation.

Quartiles

Quartiles break down an ordered data set into four equal parts, each representing a quarter of the distribution. This is how you locate each quartile:

• The first quartile (Q1) is the median of the first half of the data set and signifies the 25th percentile.
• The second quartile (Q2) is the overall median and represents the 50th percentile.
• The third quartile (Q3) is the median of the second half of the data set, indicating the 75th percentile.

In our exercise with ages, you split the ordered list of ages in half to separately find these quartiles:

• For Q1, calculate the median of the first half of data. Given as 16.
• For Q3, calculate the median of the last half of data. This is 18.

Each quartile provides a valuable understanding of the distribution, identifying how your data is spread out.

Median in Data Sets

The median is the middle value of an ordered data set, providing a critical measure of central tendency that isn't skewed by outliers. To find the median, follow these steps:

• Order your data set from smallest to largest
• If the total number of data points is odd, the median is the middle value
• If the total number of data points is even, the median is the average of the two middle numbers

In the example provided, the data set consisted of 25 ages, an odd number:
In the ordered list, the middle position is the 13th, which corresponds to the value 17. Therefore, the median age is 17, splitting the data into two equal halves. The median offers a measure of the center of your data, especially useful for skewed data sets where the mean could mislead.