Problem 62 - Algebra 2

Question

Find the mean, median, and mode for each set of values. $$ \begin{array}{lllllllllll}{7.1} & {8.5} & {7.0} & {7.6} & {8.5} & {8.1} & {7.9} & {8.2} & {7.3} & {9.1} & {8.7} & {7.9}\end{array} $$

Step-by-Step Solution

Answer

Mean: 8.24, Median: 8.0, Modes: 8.5 and 7.9.

Step 1: Calculate the mean

First, add all the values together: \(7.1 + 8.5 + 7.0 + 7.6 + 8.5 + 8.1 + 7.9 + 8.2 + 7.3 + 9.1 + 8.7 + 7.9 = 98.9\). Then, divide the sum by the total count of values, which is 12: \(\frac{98.9}{12} \approx 8.24\)

Step 2: Calculate the median

Arrange all the values in ascending order: \(7.0, 7.1, 7.3, 7.6, 7.9, 7.9, 8.1, 8.2, 8.5, 8.5, 8.7, 9.1\). Because there is an even number of values, take the average of the two middle numbers, which are 7.9 and 8.1: \(\frac{7.9 + 8.1}{2} = 8.0\)

Step 3: Calculate the mode

Look at the ordered list and find which value appears most frequently. The values 8.5 and 7.9 each appear twice, meaning that there are two modes: 8.5 and 7.9

Key Concepts

Mean

Calculating the mean of a data set involves finding the average of all the numbers in the list. It's a measure of "central tendency," indicating the central point of the numbers through the arithmetic average. To calculate the mean:

• Add up all the numbers in the set: Consider each value and sum them up. In our exercise, you're adding numbers like 7.1, 8.5, 7.0, and so forth.
• Divide the total by how many numbers there are: After adding, take the total and divide it by the count of numbers included in your set. Here, our total sum was 98.9, which divided by 12 data points gives us an approximate mean of 8.24.

The mean offers a quick numerical insight into how the data behaves as a whole. It helps you comprehend what an average point looks like within a numerical series.

Median

The median is another way to measure central tendency. It gives you the middle point of your data when it's arranged in order. It's particularly useful in understanding the data's center when dealing with skewed data and outliers.

• First, order the numbers from lowest to highest: This step is crucial because the median is the middle number. If not ordered, you cannot accurately determine the central value.
• Find the middle value: If your set has an odd number of values, the median is simply the middle one. However, if there is an even number of values, you need to calculate the average of the two central numbers. Here, we had an even set of 12 values, so the median is calculated from the 6th and 7th values in order, which are 7.9 and 8.1, leading to a median of 8.0.

The median is often more informative than the mean in datasets with extremities because it does not get influenced by very high or low values.

Mode

Mode is the simplest measure of central tendency to find. It points out the most frequently occurring number in your dataset. If a number repeats more than others, it is identified as the mode.

• Identify the frequency of each number: Check the frequency of each number in your data set. This means looking for the number that appears the most.
• Find any repeating values more than once: In our example, both 8.5 and 7.9 appear twice, more frequently than any other number, so the dataset has two modes.

Having multiple modes can indicate a dataset with more than one peak, which might suggest different subgroups or factors within the data. Multiple modes are described as bimodal or multimodal depending on the count. Understanding the mode can be particularly helpful in recognizing patterns or preferences in categories of data.