Question

Write a single program that calculates the arithmetic mean (average), rms average, geometric mean, and harmonic mean for a set of positive numbers. Use any method that you desire to read in the input values. Compare these values for each of the following sets of aumbers. a. \(4,4,4,4,4,4,4\) b. \(4,3,4,5,4,3,5\) c. \(4,1,4,7,4,1,7\) d. \(1,2,3,4,5,6,7\)

Short answer

The calculated mean values for each set are: Set a: All means are equal to 4. Set b: Arithmetic Mean = 4, RMS-Average ≈ 4.22, Geometric Mean ≈ 3.98, Harmonic Mean ≈ 3.76. Set c: Arithmetic Mean = 4, RMS-Average ≈ 4.49, Geometric Mean ≈ 3.21, Harmonic Mean ≈ 2.15. Set d: Arithmetic Mean = 4, RMS-Average ≈ 4.72, Geometric Mean ≈ 3.48, Harmonic Mean ≈ 2.81. For sets b, c, and d, the RMS-Average is greater than or equal to the Arithmetic mean, and the Harmonic mean is smaller than or equal to the Geometric mean.

Step-by-step solution

Define the Formulas

Let's denote the set of numbers as \(x_1, x_2, \dots, x_n\). The definitions of the means we need to compute are: - Arithmetic Mean (\(\bar{x}\)): \(\bar{x} = \frac{\sum_{i=1}^n x_i}{n}\) - RMS-Average (\(x_{rms}\)): \(x_{rms} = \sqrt{\frac{\sum_{i=1}^n x_i^2}{n}}\) - Geometric Mean (\(x_{gm}\)): \(x_{gm} = \sqrt[n]{\prod_{i=1}^n x_i}\) - Harmonic Mean (\(x_{hm}\)): \(x_{hm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}\)

Calculate the Mean Values

Using the formulas defined above, we will calculate the mean values for each set of numbers (a, b, c, and d): Set a: \(4, 4, 4, 4, 4, 4, 4\) - Arithmetic Mean: \(\bar{x} = \frac{4+4+4+4+4+4+4}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+4^2+4^2+4^2+4^2+4^2+4^2}{7}} = 4\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4} = 4\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}} = 4\) Set b: \(4, 3, 4, 5, 4, 3, 5\) - Arithmetic Mean: \(\bar{x} = \frac{4+3+4+5+4+3+5}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+3^2+4^2+5^2+4^2+3^2+5^2}{7}} \approx 4.22\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 3 \cdot 4 \cdot 5 \cdot 4 \cdot 3 \cdot 5} \approx 3.98\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{4}+\frac{1}{3}+\frac{1}{5}} \approx 3.76\) Set c: \(4, 1, 4, 7, 4, 1, 7\) - Arithmetic Mean: \(\bar{x} = \frac{4+1+4+7+4+1+7}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+1^2+4^2+7^2+4^2+1^2+7^2}{7}} \approx 4.49\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 1 \cdot 4 \cdot 7 \cdot 4 \cdot 1 \cdot 7} \approx 3.21\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{1}+\frac{1}{4}+\frac{1}{7}+\frac{1}{4}+\frac{1}{1}+\frac{1}{7}} \approx 2.15\) Set d: \(1, 2, 3, 4, 5, 6, 7\) - Arithmetic Mean: \(\bar{x} = \frac{1+2+3+4+5+6+7}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{1^2+2^2+3^2+4^2+5^2+6^2+7^2}{7}} \approx 4.72\) - Geometric Mean: \(x_{gm} = \sqrt[7]{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} \approx 3.48\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}} \approx 2.81\)

Compare the Mean Values

Now that we have calculated the mean values for each set of numbers, let's compare them: - Set a: Arithmetic, RMS-Average, Geometric, and Harmonic means are all equal to 4. This is because all the values in the set are the same. - Set b, c, and d: In all three sets, the arithmetic mean remains at 4, while the RMS-Average, Geometric, and Harmonic means vary. It can be observed that the RMS-Average is always greater than or equal to the Arithmetic mean, and the Harmonic mean is always smaller than or equal to the Geometric mean. This solution calculates and compares the Arithmetic Mean, RMS-Average, Geometric Mean, and Harmonic Mean for each given set of numbers.

Key concepts

Arithmetic Mean

The arithmetic mean, commonly known as the average, is a very straightforward concept used in statistics to represent the central value of a data set. It's obtained by summing up all the values and then dividing by the number of values. In MATLAB, this can easily be calculated using the \texttt{mean} function.

For example, with a set of numbers like \(4, 3, 4, 5, 4, 3, 5\), the arithmetic mean is computed by adding all these numbers to get 28, and then dividing by 7, giving us an arithmetic mean of 4. It offers a quick snapshot of the 'typical' value in your data set and is particularly useful when dealing with data without extreme values.

RMS Average

The RMS average, or Root Mean Square average, adds a little complexity compared to the arithmetic mean. It calculates the square root of the mean of the squares of the numbers in the set. This type of average is particularly useful in fields such as electrical engineering and acoustics because it provides a measure of the magnitude of a varying quantity.

In MATLAB, the RMS average is not a built-in function but can be calculated by squaring each number, averaging the squares, and then taking the square root of that average. For instance, with the set \(4, 3, 4, 5, 4, 3, 5\), the RMS average is the square root of \(\frac{4^2 + 3^2 + 4^2 + 5^2 + 4^2 + 3^2 + 5^2}{7}\), which is approximately 4.22.

Geometric Mean

The geometric mean differs from the arithmetic mean by focusing on the product rather than the sum of the numbers. To calculate the geometric mean, multiply all the numbers in the set together and then take the nth root of the product, where 'n' is the number of values in the dataset.

In MATLAB, we can compute the geometric mean of a dataset using \texttt{geomean} function. For the set \(4, 3, 4, 5, 4, 3, 5\), we multiply all the numbers to get 36000, and then we compute the 7th root, yielding a geometrical mean of approximately 3.98. The geometric mean is especially meaningful in situations where we are comparing things with very different ranges or different units of measure.

Harmonic Mean

The harmonic mean is the least common type of average and is particularly useful when dealing with rates. It is calculated as the reciprocal of the arithmetic mean of the reciprocals of the data points. Essentially, you take the number of values in the dataset, divide it by the sum of the reciprocals of the values.

To compute it in MATLAB, we use the \texttt{harmmean} function. With numbers \(4, 3, 4, 5, 4, 3, 5\), the harmonic mean is \(\frac{7}{\frac{1}{4} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{5}}\), giving us approximately 3.76. The harmonic mean is less affected by extremely high or low values, which makes it more robust in certain cases than the arithmetic mean.