Problem 81 - Algebra 1

Question

Find the mean, the median, and the mode of the collection of numbers. $$1,5,2,4,3,6,1$$

Step-by-Step Solution

Answer

The mean of the given set of numbers is approximately 3.14, the median is 3 and the mode is 1.

Step 1: Calculate the Mean

Firstly, the mean or average needs to be calculated. For this, add all the numbers together and then divide the sum by the number of numbers. For this set \(1,5,2,4,3,6,1\), the sum is \(1+5+2+4+3+6+1=22\) and the total count is 7. So, the mean is calculated as \(\frac{22}{7}=3.14\)

Step 2: Find the Median

The median is the middle value of a number set, ordered in ascending order. The given set arranged in ascending order is \(1,1,2,3,4,5,6\). Since there are 7 numbers, the median is the fourth number, which is 3.

Step 3: Determine the Mode

The mode is the number that appears most frequently in the given set. In this set \(1,1,2,3,4,5,6\), the number 1 appears twice while all others appear only once, so the mode is 1.

Key Concepts

Understanding the Mean

The mean, often referred to as the average, represents the central value of a numerical set. It is calculated by adding all the numbers in a dataset and then dividing the sum by the total count of numbers. For instance, consider the dataset: \(1, 5, 2, 4, 3, 6, 1\).

In this set, summing the numbers gives us:

• Sum = \(1 + 5 + 2 + 4 + 3 + 6 + 1 = 22\)

To find the mean, divide the sum by the number of elements in the set, which here is 7:

• Mean = \(\frac{22}{7} \approx 3.14\)

This value indicates the average of the dataset, providing a central point around which the data points are spread. It is important to note that while the mean provides a central value, it can be influenced by very high or very low numbers, which may skew the result.

Deciphering the Median

The median is the middle value in an ordered data set, which splits the dataset into two equal halves. To find it, the numbers must first be arranged in ascending order. Taking our example dataset: \(1, 5, 2, 4, 3, 6, 1\), we reorder it to get:

• Ordered set: \(1, 1, 2, 3, 4, 5, 6\)

For this set, which has 7 numbers, the median is the fourth number. This is because the data is now symmetrical around this central value. Therefore, the median is:

• Median = 3

The median is especially useful in determining the central tendency when the dataset includes outliers or skewed data. Unlike the mean, it is not affected by extreme values, making it a reliable measure in such situations.

Identifying the Mode

The mode represents the value that occurs most frequently within a dataset. It highlights the most common item in a sequence. In the dataset \(1, 5, 2, 4, 3, 6, 1\), you look for numbers that appear more than once. Reordering the dataset, you have:

• Ordered set: \(1, 1, 2, 3, 4, 5, 6\)

Here, the number 1 appears twice, while all other numbers only appear once:

• Mode = 1

The mode is particularly helpful in categorical data analysis or when one is interested in the popularity of a certain outcome or feature. It is possible for a dataset to have more than one mode (bimodal or multimodal) if multiple values appear with the same highest frequency.