Problem 3 - Algebra 2

Question

Find the mean, median, and mode of each set of values. 475 \(\quad\) 722 \(\quad\) 499\(\quad 572 \quad 402 \quad 809 \quad 499 \quad 828 \quad 405 \quad 499 \quad 800 \quad 422 \quad\) 672\(\quad\) 800

Step-by-Step Solution

Answer

The mean is 670.57, the median is 499, and the mode is 499

Step 1: Sort the Numbers

First, sort the numbers from smallest to largest: 402, 405, 422, 475, 499, 499, 499, 572, 672, 722, 800, 800, 809, 828

Step 2: Find the Mean

Add all the numbers together and then divide by the total count. The sum of all the numbers is 9388 and the total count is 14. So, the mean is \( \frac{9388}{14} = 670.57 )

Step 3: Find the Median

Since there are 14 numbers in total (an even number), the median is the average of the 7th and 8th number. Therefore, the median is \( \frac{499 + 499}{2} = 499 )

Step 4: Find the Mode

The Mode is the number that appears most frequently. So, the mode is 499 since it appears 3 times

Key Concepts

Mean

The mean is a fundamental concept in statistics and algebra, often referred to as the "average". To calculate it, you sum up all the values in a dataset and then divide by the number of values. In this exercise, after sorting and adding the numbers, we get a total of 9388. With 14 values in our set, the mean is computed as follows: \[\text{Mean} = \frac{9388}{14} = 670.57\]Keep in mind that the mean can be impacted heavily by extremely high or low values in your dataset, known as outliers. It provides a useful measure of central tendency by showing the center of the data distribution.The mean is helpful in various real-world contexts, such as determining average scores, incomes, or measurements. Always ensure you're interpreting the mean within the right context of your data to draw accurate conclusions.

Median

The median offers another perspective on the central tendency of a dataset, which can be quite different from the mean. To find the median, you must first arrange the data in order. This ensures that the median value correctly represents the middle of the dataset.In our exercise, we have an even number of values, 14, hence:

• Identify the middle two numbers, which are the 7th and 8th values after sorting.
• Here, both numbers are 499.

The median is then calculated by averaging these two numbers:\[\text{Median} = \frac{499 + 499}{2} = 499\]The median is especially useful when the data set contains outliers because it is not as affected by extreme values as the mean. It's a reliable measure for understanding the typical value in skewed distributions, providing a more resilient indicator of where the center really lies.

Mode

The mode is the simplest measure of central tendency, highlighting the most frequently occurring value in a dataset. In some cases, a dataset may have more than one mode or no mode at all if there are no repeating numbers.For this exercise:- The list of numbers is sorted.- We observe that the number 499 appears three times, more than any other number.Therefore, the mode of the dataset is:\[\text{Mode} = 499\]The mode provides valuable insights, especially in categorical data where numbers are used to represent categories or labels, like identifying the most common survey response. While it may not always represent the central location of data, it's still a significant aspect when the frequency of occurrence is important. Understanding how the mode relates to the other measures (mean and median) gives a more comprehensive view of the dataset's distribution.