Problem 1 - Basic Engineering Mathematics

Question

Determine the mean, median and mode for the set: $$ \\{2,3,7,5,5,13,1,7,4,8,3,4,3\\} $$

Step-by-Step Solution

Answer

Mean is 5, median is 4, and mode is 3.

Step 1: Understand the Problem

We need to find the mean, median, and mode of the given set of numbers: \(\{2,3,7,5,5,13,1,7,4,8,3,4,3\}\).

Step 2: Organize and Sort the Set

To simplify calculation, first sort the numbers in ascending order: \(\{1,2,3,3,3,4,4,5,5,7,7,8,13\}\).

Step 3: Calculate the Mean

Sum all the numbers: \(1 + 2 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 7 + 7 + 8 + 13 = 65\).Divide the sum by the number of elements: \(65 \div 13 = 5\). The mean is 5.

Step 4: Determine the Median

Since there are 13 numbers, the median is the 7th number in the ordered set: \(4\). Thus, the median is 4.

Step 5: Identify the Mode

Find the number that appears most frequently. In the sorted set, the number 3 appears three times, more than any other number. Thus, the mode is 3.

Key Concepts

Mean

The mean is often referred to as the arithmetic average. It is one of the most common measures of central tendency used to summarize a data set. To find the mean of a set of numbers, follow these steps:

• Add up all the numbers in the data set.
• Divide the total sum by the number of data points.

In our example, we're given the numbers \( \{2,3,7,5,5,13,1,7,4,8,3,4,3\} \). First, we need to organize the data by sorting it in ascending order: \( \{1,2,3,3,3,4,4,5,5,7,7,8,13\} \). Then, add all the numbers together: \(1 + 2 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 7 + 7 + 8 + 13 = 65\).
Next, count the numbers in the set. There are 13 numbers in total. Finally, divide the sum by the number of values to find the mean: \(65 \div 13 = 5\).
Therefore, the mean, or average, of this data set is 5. This value represents a balance point or typical value within the data set, providing a general idea of where the numbers tend to cluster.

Median

The median is the middle value in a data set when the numbers are arranged in order. Unlike the mean, the median is less affected by extreme values or outliers.To determine the median:

• Organize the data in ascending order.
• Find the middle number. If there is an odd number of observations, the median is the center number. If there is an even number, average the two central numbers.

Looking at our sorted data set \( \{1,2,3,3,3,4,4,5,5,7,7,8,13\} \) with 13 numbers, you find the 7th value to determine the median, as 13 is an odd number. The 7th number, well-positioned in the middle of the sequence, is 4.
Thus, the median of this set is 4. This value effectively divides the data set into two equal parts, helping show the central point of a data series.

Mode

The mode is the value that appears most frequently in a data set. It's possible for a set of data to have one mode, more than one mode, or no mode at all if no number repeats.To identify the mode:

• Examine the frequency of each number in the data set.
• Determine which number(s) appear most often.

In the given set \( \{1,2,3,3,3,4,4,5,5,7,7,8,13\} \), observe that the number 3 appears three times. This is more frequent than any other number in the sequence, making 3 the mode.
The mode provides insight into the most common or popular value within a data set, helping to identify trends or patterns in frequency.