Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Grades: \(74,78,78,80,80,80,82,88,90\)

Step-by-Step Solution

Answer

Mean: 81.11, Median: 80, Mode: 80.

Step 1: Understanding the Data Set

The grades given are: 74, 78, 78, 80, 80, 80, 82, 88, and 90.

Step 2: Calculate the Mean

To find the mean, sum all the grades and divide by the number of grades. Sum: \(74 + 78 + 78 + 80 + 80 + 80 + 82 + 88 + 90 = 730\)Count of grades: 9Mean:\( \frac{730}{9} \approx 81.11 \) (rounded to two decimal places).

Step 3: Find the Median

The median is the middle value of an ordered data set. The grades in numerical order are already given. Since there are 9 values, the median is the 5th value: Median: 80.

Step 4: Determine the Mode

The mode is the most frequently occurring value in the data set. Here, the grade 80 appears three times, more frequent than any other value: Mode: 80.

Key Concepts

Understanding Mean Calculation

The mean, often referred to as the "average," is a central concept that represents the typical value in a dataset. Calculating the mean involves a couple of straightforward steps:

• First, you sum all the values in your dataset together. For our exercise, this means adding up the grades: 74, 78, 78, 80, 80, 80, 82, 88, and 90, which gives us a total of 730.
• Next, you count how many values are in the dataset. In this instance, we have 9 grades.
• The final step is to divide the total sum by the count of values. So, the calculation for the mean would be \( \frac{730}{9} \), which results in approximately 81.11 when rounded to two decimal places.

This value, 81.11, gives an idea of the overall performance in terms of grades. It shows us an "average" score around which most grades might be expected to cluster. Knowing how to calculate the mean is essential because it allows you to summarize a group of numbers in a single value.

Exploring Median Finding

Finding the median involves identifying the middle value in a dataset. It's a simple yet powerful measure of central tendency:

• First, make sure your dataset is in numerical order. Fortunately, the grades are already sorted: 74, 78, 78, 80, 80, 80, 82, 88, 90.
• If the dataset has an odd number of values, like here with 9 grades, the median is simply the middle number. In this case, it's the 5th grade when the grades are ordered, which gives us a median of 80.
• If there had been an even number of values, you'd take the average of the two middle numbers.

The median is a very useful measure, especially when your data contains outliers or is not symmetrically distributed, as it gives you a better measure of a central tendency than the mean in those cases.

Grasping Mode Determination

The mode is another form of central tendency, focusing on the most frequent value in your dataset. Here's how you determine it:

• Review your dataset to see which number appears most frequently.
• In our grades example, the number 80 appears three times, more than any other number. Thus, the mode is 80.
• It is possible for a dataset to have more than one mode (bimodal or multimodal) if multiple numbers appear with the same highest frequency.

The mode is particularly useful in categorical data where we wish to know which is the most common category, but it is also applicable in numerical data, helping us understand the most frequent scores or observations.