Problem 9 - Algebra 2 and Trigonometry

Question

In \(7-9,\) find the mean, median, range, and interquartile range for each set of data to the nearest tenth. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 11 & {5} \\ {16} & {8} \\\ {19} & {9} \\ {31} & {6} \\ {37} & {5} \\ {32} & {5} \\ {35} & {6} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

Mean: 30.4, Median: 19, Range: 26, IQR: 16

Step 1: List all data points

To solve the exercise, we first need to list all values using the given frequencies. The table shows that the data points and their frequencies are: - The value 11 appears 5 times. - The value 16 appears 8 times. - The value 19 appears 9 times. - The value 31 appears 6 times. - The value 37 appears 5 times. - The value 32 appears 5 times. - The value 35 appears 6 times. This gives us the dataset: \[ [11, 11, 11, 11, 11, 16, 16, 16, 16, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19, 19, 19, 19, 31, 31, 31, 31, 31, 31, 32, 32, 32, 32, 32, 35, 35, 35, 35, 35, 35, 37, 37, 37, 37, 37] \]

Step 2: Calculate the mean

To find the mean, sum all data points and divide by the total number of points. The sum is: \[ 11 \times 5 + 16 \times 8 + 19 \times 9 + 31 \times 6 + 37 \times 5 + 32 \times 5 + 35 \times 6 = 3525 \]The total number of data points is \[ 5+8+9+6+5+5+6 = 44 \] Therefore, the mean is:\[ \text{Mean} = \frac{3525}{44} \approx 30.4 \]

Step 3: Calculate the median

To find the median, order all points and find the middle value. With 44 data points, the median is the average of the 22nd and 23rd values. When ordered, the 22nd and 23rd values are both 19:\[ \text{Median} = \frac{19 + 19}{2} = 19 \]

Step 4: Calculate the range

The range is the difference between the maximum and minimum values:- Maximum value = 37- Minimum value = 11Thus, the range is:\[ \text{Range} = 37 - 11 = 26 \]

Step 5: Calculate the interquartile range

The interquartile range (IQR) is the difference between the upper quartile (Q3) and lower quartile (Q1):1. **Find Q1 (lower quartile):** It's the 11th data value in the ordered set, which is 16.2. **Find Q3 (upper quartile):** It's the 33rd data value, which is 32.So, the IQR is:\[ \text{IQR} = Q3 - Q1 = 32 - 16 = 16 \]

Key Concepts

Mean

The mean is a way of finding the "average" of a dataset. It gives us a sense of the central value around which all other values revolve.
Here's how we calculate it:

• Add up all the numbers—this tells us the sum of all data points.
• Divide that sum by the number of data points present.

For example, in the exercise above, the sum of all numbers is 3525, and there are 44 numbers in total. So, the mean is \[ \frac{3525}{44} \approx 30.4 \].
Thus, the mean tells us that if each value in our data set was equal, they all would approximately be 30.4. A helpful way to think of mean is as the balancing point of our data.

Median

The median is another measure that helps us understand the center of our dataset. Unlike the mean, the median is less affected by extremely high or low values.
To find the median:

• First, list all data values in order from the smallest to the largest.
• For an even number of data points, find the two middle numbers and calculate their average.
• If there is an odd number of data points, the median is the middle number.

In the provided exercise, with 44 data points, the median is the average of the 22nd and 23rd values, both of which are 19. As a result, the median of this dataset is 19. This tells us that half of the data values are below 19, and half are above it.

Range

The range is a simple measure of how spread out the data points are. It tells us the difference between the highest and the lowest values in the dataset.
The range can be found by subtracting the smallest number from the largest number in the dataset:

• Find the maximum value.
• Find the minimum value.
• Subtract the min from the max.

In the dataset given, the maximum value is 37, and the minimum is 11. Therefore, the range is \[ 37 - 11 = 26 \].
Knowing the range helps us understand the span of our dataset, indicating the overall variability.

Interquartile Range

The interquartile range (IQR) provides a more robust measure of variability compared to the range. It focuses on the middle 50% of the data, which helps ignore outliers and extreme values.
Here’s how to calculate the IQR:

• Order the dataset.
• Identify the lower quartile (Q1), the 25th percentile point of the data.
• Identify the upper quartile (Q3), the 75th percentile point.
• Subtract Q1 from Q3 to get the IQR.

In the example, Q1 (lower quartile) is 16 and Q3 (upper quartile) is 32.The IQR is \[ 32 - 16 = 16 \].
This tells us that the middle 50% of our data values fall within this range, providing a clearer snapshot of data variability.