Problem 19 - Algebra 2 and Trigonometry

Question

The weights in pounds of the members of the football team are shown below: $$\begin{array}{cccccccccc}{181} & {199} & {178} & {203} & {211} & {208} & {209} & {202} & {212} & {194} \\ {185} & {208} & {223} & {206} & {202} & {213} & {202} & {186} & {189} & {203}\end{array}$$ a. Find the mean. b. Find the median. c. Find the mode or modes. d. Find the first and third quartiles. e. Draw a box-and-whisker plot.

Step-by-Step Solution

Answer

Mean: 204.7, Median: 202.5, Mode: 202, Q1: 191.5, Q3: 208.5.

Step 1: Calculate the Mean

To find the mean, add together all the weights and then divide by the number of weights. The weights are: 181, 199, 178, 203, 211, 208, 209, 202, 212, 194, 185, 208, 223, 206, 202, 213, 202, 186, 189, 203.First, sum the weights: \[181 + 199 + 178 + 203 + 211 + 208 + 209 + 202 + 212 + 194 + 185 + 208 + 223 + 206 + 202 + 213 + 202 + 186 + 189 + 203 = 4094\]Then, divide the sum by the number of data points (20):\[\text{Mean} = \frac{4094}{20} = 204.7\]The mean weight is 204.7 pounds.

Step 2: Identify the Median

To find the median, organize the weights in ascending order and identify the middle value. If there is an even number of values, find the average of the two middle numbers.The weights arranged in ascending order are: 178, 181, 185, 186, 189, 194, 199, 202, 202, 202, 203, 203, 206, 208, 208, 209, 211, 212, 213, 223.There are 20 values, so the median will be the average of the 10th and 11th values:\[ \text{Median} = \frac{202 + 203}{2} = 202.5 \]The median weight is 202.5 pounds.

Step 3: Determine the Mode

The mode is the number that appears most frequently in the data set. List the frequency of each weight: - 178: 1 time - 181: 1 time - 185: 1 time - 186: 1 time - 189: 1 time - 194: 1 time - 199: 1 time - 202: 3 times - 203: 2 times - 206: 1 time - 208: 2 times - 209: 1 time - 211: 1 time - 212: 1 time - 213: 1 time - 223: 1 time The mode is 202 pounds as it appears 3 times.

Step 4: Find the Quartiles

Quartiles divide data into four equal parts.First Quartile (Q1): Calculate the median of the first half of the data (below the median). The first half is 178, 181, 185, 186, 189, 194, 199, 202, 202, 202.Median of the first half is the average of the 5th and 6th values:\[ Q1 = \frac{189 + 194}{2} = 191.5 \]Third Quartile (Q3): Do the same for the second half of the data (above the median). The second half is 203, 203, 206, 208, 208, 209, 211, 212, 213, 223.Median of the second half is the average of the 5th and 6th values:\[ Q3 = \frac{208 + 209}{2} = 208.5 \]The first quartile is 191.5 pounds and the third quartile is 208.5 pounds.

Step 5: Create the Box-and-Whisker Plot

To draw a box-and-whisker plot: 1. Draw a number line that includes the range of our data set (from 178 to 223). 2. Plot points for the minimum value (178), maximum value (223), first quartile (191.5), median (202.5), and third quartile (208.5). 3. Draw a box from Q1 to Q3 with a line at the median. 4. Draw whiskers from the minimum to Q1 and the maximum to Q3. The diagram visually represents the data set, its center, and spread.

Key Concepts

Mean

The mean, often referred to as the average, is a key measurement in descriptive statistics. It represents the central tendency of a dataset. To find the mean, you sum up all the available data points and then divide by the total number of points.
For the football team's weights, the calculation was:

• Total sum of weights: 4094
• Number of members: 20
• Mean weight: \( \frac{4094}{20} = 204.7 \) pounds

A mean value can provide an at-a-glance number representing the typical member weight, but keep in mind that it is sensitive to extremely high or low values (outliers), which can skew the mean.

Median

The median offers a different glimpse into the central tendency of a dataset. It is found by arranging all data points in ascending order and selecting the middle value. If the dataset has an even number of items, like in our example, the median is calculated by averaging the two middlemost numbers.
For the team's weights organized from 178 to 223, the middle lies between the 10th and 11th values:

• 10th value: 202
• 11th value: 203
• Median: \( \frac{202+203}{2} = 202.5 \) pounds

The median can be particularly useful because it is not influenced by outliers, making it a robust measure of central tendency.

Mode

The mode is the number that appears most frequently in a dataset. In some cases, a dataset might have one mode, multiple modes, or no mode at all when no number repeats more than once. For the football team's weights, we observed:

• The most frequent weight: 202 pounds (appears 3 times)

In real-world data, the mode offers insight into the most common characteristics, like frequently occurring values in categorical data, and can be useful in understanding the most typical value in a dataset.

Box-and-Whisker Plot

A Box-and-Whisker Plot is a graphical representation that summarizes a dataset by displaying its median, quartiles, and outliers. It highlights how data is distributed and is especially useful for identifying variability and skewness. To create the plot:

• Draw a horizontal line covering your data range: 178 to 223 pounds.
• Mark minimum, first quartile (Q1), median, third quartile (Q3), and maximum values on the line: 178, 191.5, 202.5, 208.5, and 223 respectively.
• Draw a box from Q1 to Q3 with a line at the median.
• Extend whiskers from Q1 to the minimum and from Q3 to the maximum.

This plot provides a quick visual summary and is a powerful tool for visualizing the data distribution without making any assumptions about the underlying statistical distribution.

Quartiles

Quartiles divide a dataset into four equal parts to provide detailed insights into data distribution. Each quartile represents 25% of the dataset.

• First Quartile (Q1) is the median of the first half, indicating the 25th percentile: \( Q1 = 191.5 \) pounds
• Third Quartile (Q3) is the median of the second half, showing the 75th percentile: \( Q3 = 208.5 \) pounds
• Interquartile Range (IQR) is the difference between Q3 and Q1, representing the middle 50% of the data: \( IQR = Q3 - Q1 = 208.5 - 191.5 = 17 \) pounds

By understanding quartiles, you can determine the spread and variance within your data, and they are helpful for identifying outliers beyond the typical spread.