Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Number of student absences: \(0,0,0,1,1,2,2,2,3,4,5,9\)

Step-by-Step Solution

Answer

Mean: 2.42, Median: 2, Mode: 2.

Step 1: Organize the data set

The data set is already in ascending order: \(0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 5, 9\). This makes it easy to find the measures of central tendency.

Step 2: Find the mean

To find the mean, sum all the data points and divide by the number of data points. Calculate the sum: \(0 + 0 + 0 + 1 + 1 + 2 + 2 + 2 + 3 + 4 + 5 + 9 = 29\)Count the data points: There are 12 data points.Calculate the mean: \(\text{Mean} = \frac{29}{12} \approx 2.42\).

Step 3: Find the median

The median is the middle value in a data set. Since there are 12 data points (an even number), the median is the average of the 6th and 7th values.The 6th data point is 2 and the 7th data point is also 2.Calculate the median: \(\text{Median} = \frac{2 + 2}{2} = 2\).

Step 4: Find the mode

The mode is the number that appears most frequently in the data set. Look for the number with the highest frequency: In this set, 2 appears three times. Thus, the mode is 2.

Key Concepts

Mean Calculation

Calculating the mean, often called the average, involves two simple steps. First, sum up all the numbers in the data set. For our data on student absences, this involves adding each occurrence:

• 0 + 0 + 0 + 1 + 1 + 2 + 2 + 2 + 3 + 4 + 5 + 9, which gives a total of 29.

Next, divide this total by the number of data points you have. In this case, there are 12 data points. So, the mean calculation becomes:

• \[ ext{Mean} = \frac{29}{12} \approx 2.42 \].

The mean gives us a central value that represents the entire data set. It's useful for understanding the overall trend in data, though it can be skewed by very high or low values.

Median Determination

The median of a data set is the value that lies in the middle when the data is ordered. The key step is to find this midpoint. Since we have an even number of data points in our set (12 absences listed), we take the average of the 6th and 7th numbers.

• Here, both the 6th and 7th values are 2.
• Thus, the median calculation is: \[ ext{Median} = \frac{2 + 2}{2} = 2 \].

The median is particularly useful as a measure of central tendency when you have outliers, like our number 9, which could skew the mean.

Mode Identification

When identifying the mode, you're looking for the number that appears most frequently in the data set. This gives insight into the most common occurrence or trend within the data. In our student absences data set, review the frequency of each number:

• The number 2 appears three times, more than any other number.

So, the mode of this data set is 2. Knowing the mode can be helpful in understanding the most typical case or frequent trend in a group of numbers.

Data Set Analysis

Analyzing a data set means understanding its central tendencies—mean, median, and mode—and its distribution. Our data set of student absences provides a great example. To begin, ensure your data is organized, usually in ascending order, as it was here:

• 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 5, 9.

This makes calculating the measures of central tendency straightforward. Each measure offers unique insights:

• The mean (average) indicates a general central value.
• The median gives a midpoint that isn't skewed by extreme values.
• The mode highlights the most frequent value.

Together, these measures provide a comprehensive look at the data, allowing you to spot trends, identify outliers, and better understand the situation at hand.