3. Describing Data Numerically
Mean
2:43 minutesProblem 3.1.39
Textbook Question
Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 0.58%, 0.29%, 0.13%, 0.14%, 0.15%, and 0.19%. Identify the single percentage growth rate that is the same as the six consecutive growth rates by computing the geometric mean of 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019.
Verified step by step guidance1
Step 1: Understand the problem. The geometric mean is used to find a single growth rate that represents the average of multiple growth rates. The formula for the geometric mean is: \( \text{Geometric Mean} = \sqrt[n]{x_1 \cdot x_2 \cdot x_3 \cdot \ldots \cdot x_n} \), where \( n \) is the number of values and \( x_1, x_2, \ldots, x_n \) are the values.
Step 2: Identify the values to use in the formula. The given growth rates are 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019. These represent the annual growth factors for the 6-year period.
Step 3: Multiply all the growth factors together. This means calculating \( 1.0058 \cdot 1.0029 \cdot 1.0013 \cdot 1.0014 \cdot 1.0015 \cdot 1.0019 \).
Step 4: Take the 6th root of the product obtained in Step 3. The 6th root can be expressed as raising the product to the power of \( \frac{1}{6} \), i.e., \( \text{Geometric Mean} = (\text{Product})^{1/6} \).
Step 5: Subtract 1 from the geometric mean obtained in Step 4 and multiply by 100 to convert it back to a percentage growth rate. This gives the single percentage growth rate equivalent to the six consecutive growth rates.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Mean
The geometric mean is a measure of central tendency that is particularly useful for sets of positive numbers, especially when dealing with rates of change or growth. It is calculated by multiplying all the values together and then taking the nth root of the product, where n is the number of values. This mean is less affected by extreme values compared to the arithmetic mean, making it ideal for financial and economic data.
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Rates of Change
Rates of change represent how a quantity changes over time, often expressed as a percentage. In finance, they are crucial for understanding growth rates, such as interest rates or investment returns. The geometric mean is particularly suited for calculating average rates of change over multiple periods, as it accounts for compounding effects, providing a more accurate representation of growth.
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Compounding
Compounding refers to the process where the value of an investment increases because the earnings on an investment earn interest as time passes. This concept is fundamental in finance, as it affects how growth rates are calculated over time. The geometric mean effectively captures the impact of compounding by averaging growth rates, allowing for a single growth rate that reflects the cumulative effect of multiple periods.
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