Question

RMS Average. The root-mean-square (rms) average is another way of calculating a mean for a set of numbers. The ms average of a series of numbers is the square root of the arithmetic mean of the squares of the numbers $$ \text { rms average }=\sqrt{\frac{1}{N} \sum_{\eta=1}^{N} x_{\eta}{ }^{2}} $$ Write a MATLAB program that will accept an arbitrary number of positive input values and calculate the rms average of the numbers. Prompt the user for the number of values to be entered, and use a for loop to read in the numbers. Test your program by calculating the rms average of the four numbers \(10,5,2\), and 5 .

Short answer

To calculate the root-mean-square (rms) average of a set of numbers in MATLAB, follow these steps: 1. Prompt the user for the total number of values to be entered: `num_values = input('Enter the number of values to calculate the rms average for: ');`. 2. Initialize the sum of the squares of the numbers: `sum_of_squares = 0;`. 3. Use a for loop to read in the numbers and sum their squares: ```MATLAB for i = 1:num_values curr_number = input(['Enter positive number ' num2str(i) ': ']); sum_of_squares = sum_of_squares + curr_number^2; end ``` 4. Compute the rms average of the numbers: `rms_average = sqrt(sum_of_squares / num_values);`. 5. Display the rms average: `disp(['The rms average of the given numbers is: ', num2str(rms_average)]);`. Upon testing the program with the given numbers (10, 5, 2, and 5), the rms average is found to be approximately \(6.2048\).

Step-by-step solution

Prompt user for the total number of values to be entered

In the MATLAB script, start by using input() function to ask the user how many values they want to enter. This could be written: `num_values = input('Enter the number of values to calculate the rms average for: ');`.

Initialize the sum of the squares of the numbers

Before the for loop, initialize a variable to store the sum of the squares of the input numbers. This variable should start at 0: `sum_of_squares = 0;`.

Use a for loop to read in the numbers and sum their squares

Create a for loop, which iterates for the number of values given by the user. Inside the loop, ask the user for a positive number and then add its square to the sum_of_squares. This could be coded: ```MATLAB for i = 1:num_values curr_number = input(['Enter positive number ' num2str(i) ': ']); sum_of_squares = sum_of_squares + curr_number^2; end ```

Compute the rms average of the numbers

After getting the sum of squares of all the numbers, compute the rms average using the given formula. The result will be the square root of the sum_of_squares divided by num_values. This could be written: `rms_average = sqrt(sum_of_squares / num_values);`.

Display the rms average

Finally, display the value of the rms average computed. Use the following line to display the result: `disp(['The rms average of the given numbers is: ', num2str(rms_average)]);`.

Test the program with the given numbers

Now run the program and input the given numbers (10, 5, 2, and 5) to verify that it calculates the correct rms average. After entering these numbers, the program should output the rms average as 6.2048.

Key concepts

RMS Average Explained

The root-mean-square (rms) average is a valuable concept in mathematics, particularly useful in engineering and statistics. It isn't just an average; instead, it focuses on the size of the numbers, providing a measure of the magnitude of a list of numbers. To compute the rms average, you follow a specific process:

• First, square each number in your series.
• Then, calculate the arithmetic mean by summing these squared values and dividing by the number of elements.
• Finally, take the square root of this mean.

Mathematically, it is expressed as:\[\text{rms average} = \sqrt{\frac{1}{N} \sum_{\eta=1}^{N} x_{\eta}{ }^{2}}\]This formula ensures that larger values have a more substantial effect, reflecting their true contribution to the average. The rms average is particularly beneficial when dealing with both positive and negative values, as squaring eliminates any negative signs.

Understanding For Loop in MATLAB

In MATLAB, a for loop is an essential programming structure that allows you to execute a block of code multiple times. It is particularly suited for operations that require repetitive calculations, like summing a series of numbers. Using a for loop involves the following steps:

• Initialize the loop with a starting value.
• Define the upper bound, indicating how many times the loop will iterate.
• Within the loop, perform whatever tasks are needed during each cycle.

In our rms average example, the for loop iterates over the number of values specified by the user. This loop requests each number, squares it, and adds it to a cumulative total. This way, you can easily handle many inputs efficiently without writing repetitive lines of code. This loop ends when all numbers are processed, making it both efficient and easy to use.

The Versatility of Input Function

The input function in MATLAB is a flexible feature that allows you to introduce user interactivity within your script. It prompts the user for input, which can be used throughout your program. Here’s how it works:

• You use the input function to display a message or question to the user.
• The user then enters a response, which the script captures and can process further.
• This input can be stored as a variable for calculations or decision-making processes.

For instance, in calculating the rms average, the `input()` function is used twice: first, to gather how many numbers will be inputted, and second, during each loop iteration, to collect each number itself. This empowers users to calculate rms for any number of values without altering the script.

Calculating the Arithmetic Mean

The arithmetic mean, commonly considered the simple average, is a fundamental measure of central tendency in a dataset. To find the arithmetic mean, you:

• Add up all the numbers in your dataset.
• Divide this sum by the total number of values.

Represented mathematically, if you have a series of numbers \(x_1, x_2, ..., x_N\), the arithmetic mean \(\mu\) is:\[\mu = \frac{1}{N} \sum_{i=1}^{N} x_i\]While the arithmetic mean is straightforward and widely used, it can sometimes misrepresent data with outliers or skewed distributions. In the context of rms average calculations, individual squares are averaged to ensure that the emphasis is on the magnitude of numbers, not just their total or simple average.