Problem 2 - Basic Engineering Mathematics

Question

The following set of data refers to the amount, of money in \(£\) s taken by a news vendor for 6 days. Determine the mean, median and modal values of the set: $$ \\{27.90,34.70,54.40,18.92,47.60,39.68\\} $$

Step-by-Step Solution

Answer

Mean: 37.20, Median: 37.19, Mode: None

Step 1: Order the Data Set

First, arrange the given data set in ascending order. This helps in calculating both the median and mode. The ordered set is \( \{18.92, 27.90, 34.70, 39.68, 47.60, 54.40\} \).

Step 2: Calculate the Mean

To find the mean, sum all the values in the data set, and divide by the number of values. Sum: \( 18.92 + 27.90 + 34.70 + 39.68 + 47.60 + 54.40 = 223.20 \)Divide by the number of data points (6):Mean \( = \frac{223.20}{6} = 37.20\).

Step 3: Calculate the Median

The median is the middle value in an ordered list. Since there are 6 values (an even number), the median is the average of the 3rd and 4th values.Average of 34.70 and 39.68: Median \( = \frac{34.70 + 39.68}{2} = 37.19\).

Step 4: Identify the Mode

The mode is the value that appears most frequently in a data set. Here, each value appears exactly once. Thus, there is no mode in this data set.

Key Concepts

Mean Calculation

To understand the concept of mean calculation, think of it as the average of a set of numbers. Calculating mean is a straightforward method in descriptive statistics. You follow these steps:

• Add together all the numbers in your data set.
• Then divide this sum by the number of data points.

Let's break it down using our example with vendor earnings. You have six values: 27.90, 34.70, 54.40, 18.92, 47.60, and 39.68. Start by adding them up, which totals to 223.20. Next, count the numbers; in this case, we have six. To find the mean, divide 223.20 by 6, resulting in a mean value of 37.20. The mean gives you an idea of the average earnings across the days, providing a central value that represents the data set.

Median Calculation

The median is another way to find a central value in your data set. It is slightly different from the mean because it is about the middle value rather than the average.First, you need to arrange your data in numerical order. This step is crucial. Our ordered set for the vendor's earnings is 18.92, 27.90, 34.70, 39.68, 47.60, and 54.40.When the number of values in the data set is even, like here with six numbers, find the two middle numbers, which are the 3rd and 4th in the ordered list.

• The 3rd value is 34.70.
• The 4th value is 39.68.

To calculate the median, take the average of these two numbers: \( \text{Median} = \frac{34.70 + 39.68}{2} = 37.19 \).The median gives a better picture of the dataset's center, especially if any data points are very high or low compared to others.

Mode Identification

Identifying the mode in a data set helps you find the value that occurs most frequently. However, not all data sets have a mode.

• If one number appears more than any other, that number is the mode.
• If no number repeats, as in our example (where each value appears once), there is no mode.
• If multiple numbers repeat with the same frequency, all these numbers are modes.

In the provided vendor earnings set, 18.92, 27.90, 34.70, 39.68, 47.60, and 54.40 all appear once. Thus, there is no mode. This means that none of the amounts stand out as being more common than the others, indicating uniform variance among values.