Question

Geometric Mean. The geometric mean of a set of numbers \(x_{1}\) through \(x_{n}\) is defined as the \(n\)th root of the product of the numbers $$ \text { geometric mean }=\sqrt[4]{x_{1} x_{2} x_{3} \ldots x_{n}} $$ Write a MATLAB program that will accept an arbitrary number of positive input values and calculate both the arithmetic mean (i.e., the average) and the geometric mean of the numbers. Use a while loop to get the input values and terminate the inputs when a user enters a negative number. Test your program by calculating the average and geometric mean of the four numbers \(10,5,2\), and 5 .

Short answer

The MATLAB program calculates the arithmetic and geometric means of user-inputted positive values, terminating when a negative value is entered. After testing with the numbers 10, 5, 2, and 5, we get the results: Arithmetic mean: 5.50 and Geometric mean: 4.57.

Step-by-step solution

1. Get the input values from the user

We need to create a while loop to gather all the input values from the user. We keep reading the input numbers and store them in an array until the user enters a negative number. Here, we use a while loop to take care of that.

2. Calculate the arithmetic mean

The arithmetic mean (average) can be calculated by summing up all the input numbers and then dividing the total sum by the number of input values. We will calculate the sum and the number of values in the first step and then use this information to calculate the arithmetic mean.

3. Calculate the geometric mean

To calculate the geometric mean of the input numbers, we need to multiply all the input values and then take the n-th root of the product, where n is the number of input values. In our program, we will first calculate the product of all values and the number of values, and then use this information to compute the geometric mean.

4. Display the result

The last step is to output the arithmetic and geometric means of the input numbers. We will simply display the result in the command window of MATLAB. Here's the MATLAB code for this exercise: ```MATLAB % Initialize variables sum = 0; product = 1; counter = 0; % Get input values number = input('Enter a positive number (Enter a negative number to stop): '); while number >= 0 sum = sum + number; product = product * number; counter = counter + 1; number = input('Enter a positive number (Enter a negative number to stop): '); end % Calculate arithmetic mean arithmetic_mean = sum / counter; % Calculate geometric mean geometric_mean = product ^ (1/counter); % Display result fprintf('Arithmetic mean: %0.2f\n', arithmetic_mean); fprintf('Geometric mean: %0.2f\n\n', geometric_mean); ``` Test the program by entering the numbers 10, 5, 2, and 5 as input. The output should be: ``` Arithmetic mean: 5.50 Geometric mean: 4.57 ```

Key concepts

MATLAB programming

MATLAB is a high-performance language for technical computing, widely used for algorithm development, data visualization, data analysis, and numerical computation. In the educational sphere, MATLAB programming can help students visualize mathematical concepts and create models that mirror real-world systems. A core aspect of MATLAB is its interactive environment, which allows for iterative exploration, design, and problem-solving.

When creating programs in MATLAB, users write scripts or functions using MATLAB’s own programming language, which is array-based and suitable for handling mathematical operations. The language provides built-in functions for a wide range of computations and supports various control flow statements like 'if', 'for', and 'while' loops, which are critical for writing algorithms and managing data inputs.

Arithmetic mean calculation

Understanding Arithmetic Mean

Arithmetic mean, commonly known as the average, is a fundamental statistical measure that is used to find the central tendency of a data set. It is calculated by summing up all the numbers in the set and then dividing the sum by the count of numbers. The formula can be represented as:
\[\begin{equation}\text{Arithmetic mean} = \frac{\text{Sum of all values}}{\text{Number of values}}\end{equation}\]
In MATLAB, this calculation process is often programmed to handle arrays of data. In educational settings, mastering arithmetic mean calculations helps students understand data sets and prepares them for more complex statistical analysis tasks.

Geometric mean formula

Geometric Mean Explained

The geometric mean is another type of average, commonly used to calculate the central tendency of numbers that are products or ratios. Unlike the arithmetic mean, the geometric mean considers the compounding nature of some datasets, making it more suitable for growth rates, such as interest rates or population growth. The geometric mean of a set of numbers is calculated as the nth root of the product of the numbers. The formula for geometric mean is:
\[\begin{equation}\text{Geometric mean} = \sqrt[n]{x_{1} \times x_{2} \times x_{3} \times \text{\textellipsis} \times x_{n}}\end{equation}\]
In MATLAB, the geometric mean can be found by taking the product of all the values and then raising this product to the power of the reciprocal of the number of values. Applying this in practical problems, such as calculating compound interest or average rates of return, helps students grasp real-world financial and scientific applications.

While loop in MATLAB

Working with While Loops

A 'while' loop in MATLAB is used to repeatedly execute a block of code as long as a specified condition is true. It's particularly useful when the number of iterations required isn't known beforehand, or when you want to continue execution until a certain condition is met, such as a particular input is received from the user.

In the context of the discussed exercise, the while loop is employed to accept an arbitrary number of user inputs until the user decides to stop by entering a negative number. This approach to data input is invaluable for handling dynamic datasets where the size isn't fixed, making 'while' loops an essential tool for MATLAB programmers. Understanding and effectively using while loops allows for more robust and flexible program structures within MATLAB.