A certian company enjoyed 3.5% profits in year 1 over its revenue in Year 0. In Year 2, it enjoyed 5.75% profits over Year 1. In Year 3, it posted 1.25% profits over Year 2. What single, constant profit over this period of years would have produced the same profit? The teacher stated to use either harmonic or geometric mean, which ever is appropriate. I am not sure which one to use or how to set it up. My stats book is sadly lacking with examples of these.
Any direction would be helpful and much appreciated.
To find a single, constant profit over the three-year period, we can use the geometric mean. The geometric mean is appropriate in this case because we are looking for a constant growth rate over multiple periods.
The formula for the geometric mean is:
Geometric Mean = ((1 + r1) * (1 + r2) * (1 + r3)) ^ (1/3) - 1
where r1, r2, and r3 are the growth rates for each year.
In this case, we need to find the geometric mean of the growth rates for the three years: 3.5%, 5.75%, and 1.25%.
To calculate the geometric mean, follow these steps:
Step 1: Convert the growth rates to decimal form. Divide each percentage by 100.
- The growth rates become: 0.035, 0.0575, and 0.0125
Step 2: Apply the formula for the geometric mean:
- Geometric Mean = ((1 + 0.035) * (1 + 0.0575) * (1 + 0.0125)) ^ (1/3) - 1
- Simplify the expression within parentheses:
Geometric Mean = (1.035 * 1.0575 * 1.0125) ^ (1/3) - 1
- Calculate the product within parentheses:
Geometric Mean ≈ 1.10532
Step 3: Take the cube root of the product:
- Geometric Mean ≈ 1.10532 ^ (1/3)
- Geometric Mean ≈ 1.0348
Step 4: Subtract 1 from the result to get the constant growth rate:
- Constant Growth Rate ≈ 1.0348 - 1
- Constant Growth Rate ≈ 0.0348 (or 3.48%)
Therefore, a constant profit of approximately 3.48% over the three-year period would produce the same profit as the given variable profits.