Let's break down the steps for computing the arithmetic mean, geometric mean, and harmonic mean for the given data set:

Data set:
37, 72, 33, 54, 66, 28, 80, 68, 54, 37, 92, 47

1. Arithmetic Mean (AM)

The arithmetic mean is the sum of all values divided by the number of values in the data set.

$AM = \frac{\sum_{i=1}^{n} x_i}{n}$

Where:

• $\sum_{i=1}^{n} x_i$ is the sum of all the values.
• $n$ is the number of values.

First, let's calculate the sum of all the values:

$\text{Sum} = 37 + 72 + 33 + 54 + 66 + 28 + 80 + 68 + 54 + 37 + 92 + 47 = 610$

There are 12 data points. So, the arithmetic mean is:

$AM = \frac{610}{12} = 50.83$

2. Geometric Mean (GM)

The geometric mean is the $n$-th root of the product of all the values:

$GM = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}$

To compute this, we first find the product of all the values and then take the 12th root:

$\text{Product} = 37 \times 72 \times 33 \times 54 \times 66 \times 28 \times 80 \times 68 \times 54 \times 37 \times 92 \times 47$

Now, taking the 12th root of this product will give us the geometric mean.

Let's compute the geometric mean:

$GM \approx \left( 2.254 \times 10^{16} \right)^{\frac{1}{12}} \approx 54.43$

3. Harmonic Mean (HM)

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the values:

$HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$

We first compute the reciprocals of each data point and then sum them:

$\frac{1}{37} + \frac{1}{72} + \frac{1}{33} + \frac{1}{54} + \frac{1}{66} + \frac{1}{28} + \frac{1}{80} + \frac{1}{68} + \frac{1}{54} + \frac{1}{37} + \frac{1}{92} + \frac{1}{47}$

Calculating the sum of the reciprocals:

$\sum = 0.027 + 0.014 + 0.030 + 0.019 + 0.015 + 0.036 + 0.013 + 0.015 + 0.019 + 0.027 + 0.011 + 0.021 = 0.239$

Now, the harmonic mean is:

$HM = \frac{12}{0.239} \approx 50.21$

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Summary of Results:

• Arithmetic Mean (AM): $50.83$
• Geometric Mean (GM): $54.43$
• Harmonic Mean (HM): $50.21$

Relationship Between the Means:

• The arithmetic mean is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean.
• In this case, $AM > GM > HM$, which holds true for most data sets. This is known as the inequality of means.

5 Related Questions:

1. How do the means change if we include a higher value in the data set, like 100?
2. What would happen if all the data points were equal? How would the means compare then?
3. How do the harmonic mean and geometric mean behave with very large or small numbers?
4. Can the geometric mean be used for non-positive data points?
5. How does the relationship between the means change when dealing with negatively skewed data?

Tip: When dealing with a set of numbers that have a wide range (like in this case), the geometric and harmonic means are less sensitive to extreme values than the arithmetic mean.