Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Tips: \(\$ 1.00, \$ 1.50, \$ 2.25, \$ 3.00, \$ 3.30, \$ 3.50, \$ 4.00, \$ 4.75, \$ 5.00, \$ 5.00, \$ 5.00\)

Step-by-Step Solution

Answer

Mean: \$3.48, Median: \$3.30, Mode: \$5.00.

Step 1: Arrange Data

List the values in increasing order: \[\\(1.00, \\)1.50, \\(2.25, \\)3.00, \\(3.30, \\)3.50, \\(4.00, \\)4.75, \\(5.00, \\)5.00, \$5.00\] These values are already sorted.

Step 2: Calculate the Mean

To find the mean, sum all of the numbers, then divide by the count of numbers. Sum: \(1.00 + 1.50 + 2.25 + 3.00 + 3.30 + 3.50 + 4.00 + 4.75 + 5.00 + 5.00 + 5.00 = 38.30\)Count of numbers: 11Mean: \(\frac{38.30}{11} \approx 3.48\)

Step 3: Find the Median

The median is the middle value in a data set arranged in order. Since there are 11 data points (an odd number), the median is the value at position \((11+1)/2 = 6\).Thus, the median is \$3.30.

Step 4: Determine the Mode

The mode is the number that appears most frequently. Here, \\(5.00 appears three times, more than any other number in the set.Therefore, the mode is \\)5.00.

Key Concepts

mean calculation

Calculating the mean, also known as the average, is a useful way to summarize a set of numbers into a single value that represents the group. In this exercise, we learned how to calculate the mean of a set of prices. The first step is to add all the numbers.

• For our exercise, the sum of all given prices is 38.30.
• Next, count how many numbers are in the set - here, it's 11.

To find the mean, divide the sum by the count of numbers, which is: \[\text{Mean} = \frac{38.30}{11} \approx 3.48\]This means that, on average, the value of our data set is approximately $3.48. The mean can be very helpful to understand the general cost level in this context.

median determination

The median is the middle value of a data set and gives insight into the center of your data when it is sorted in ascending order. Understanding how the median works can help when you want to know more about the distribution of your data.

• First, make sure your data is sorted. In this case, it already is: "\(1.00, \)1.50, \(2.25... \)5.00, \(5.00".
• For an odd number of values, such as our set of 11 numbers, the median is directly the middle number.

To find which position the median is, calculate:\[\text{Median position} = \frac{11+1}{2} = 6\]
This means the median is the value at the 6th position in our sorted list, which is \)3.30. Median determination is crucial in statistics as it is less affected by extremely high or low values compared to the mean.

mode identification

The mode represents the most frequently occurring number in a data set and can be particularly useful for understanding which values are common or popular.

• Look through the data to count how many times each number appears.
• The most frequent number is your mode.

In our example, $5.00 appears three times, while all other numbers appear less frequently. Therefore, $5.00 is the mode. Often, datasets can have no mode if all numbers appear with the same frequency, or more than one mode if multiple numbers appear frequently. Knowing the mode can help identify the most typical value in your data, providing useful insights into trends.