Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Hourly wages: \(\$ 6.90, \$ 7.10, \$ 7.50, \$ 7.50, \$ 8.25, \$ 9.30, \$ 9.50, \$ 10.00\)

Step-by-Step Solution

Answer

Mean: \$8.13, Median: \$7.88, Mode: \$7.50.

Step 1: Organize the Data

The given hourly wages are already in an increasing order: \(6.90, 7.10, 7.50, 7.50, 8.25, 9.30, 9.50, 10.00\). This will help us in calculating the median and identifying the mode.

Step 2: Find the Mean

To find the mean, add all the wages together and then divide by the number of wages: \[\frac{6.90 + 7.10 + 7.50 + 7.50 + 8.25 + 9.30 + 9.50 + 10.00}{8} = \frac{65.05}{8} = 8.13\] So, the mean hourly wage is \(\$8.13\).

Step 3: Find the Median

The median is the middle value of a dataset. Since there are 8 values, the median will be the average of the 4th and 5th values in the dataset: \[\frac{7.50 + 8.25}{2} = \frac{15.75}{2} = 7.875\] Thus, the median hourly wage is \(\$7.88\) when rounded to two decimal places.

Step 4: Find the Mode

The mode is the value that appears most frequently. In our dataset, the value \(7.50\) occurs twice, more frequently than any other value. Thus, the mode is \(\$7.50\).

Key Concepts

Data Organization

Before diving into calculations, organizing your data is crucial. For the set of hourly wages, we start with ensuring the numbers are in ascending order. Organized data allows us to easily locate the median and verify any repeated values for the mode. - The data given in increasing order is: - $6.90, 7.10, 7.50, 7.50, 8.25, 9.30, 9.50, 10.00$. - In this step, we ensure that every data point aligns correctly on the number line. This sorted format will simplify all subsequent calculations.

Calculating Mean

The mean, often called the "average," gives a single value representing the center of the data. To find the mean of our dataset:- Add together all the numbers: - \(6.90 + 7.10 + 7.50 + 7.50 + 8.25 + 9.30 + 9.50 + 10.00 = 65.05\).- Divide this sum by the number of values present in the dataset: \[\frac{65.05}{8} = 8.13125\]Rounding this number to two decimal places, we find the mean hourly wage is \(\\)8.13$. The mean provides a general understanding of what a typical data point in the dataset might be, and it can be especially useful when comparing different datasets.

Finding Median

The median tells us the middle value when all data points are listed in order. For an even count of data points, the median is found by averaging the two central numbers:- In our organized list: - the 4th and 5th numbers are \(7.50\) and \(8.25\). - Calculate their average: \[\frac{7.50 + 8.25}{2} = 7.875\]Rounding to two decimal places gives \(\\)7.88$. The median is less affected by outliers and skewed data, which makes it a reliable measure of central tendency when the dataset has extreme values.

Identifying Mode

The mode represents the most frequently appearing value in a dataset. For the wages we analyzed, observe which number appears more than once: - The value $7.50$ is present twice, while all other numbers appear only once. Thus, the mode is $7.50$. The mode provides insight into the most common characteristic within a dataset. This measure is particularly useful in situations where we need to focus on popularity or frequency, rather than trends or averages.