Problem 28 - Basic Statistics for Business and Economics with Student

Question

Compute the geometric mean of the following percent increases: \(2,8,6,4,10,6,8,\) and 4

Step-by-Step Solution

Answer

The geometric mean of the percent increases is approximately 1.058, or 5.8%.

Step 1: Convert Percent Increases to Decimal Multipliers

To compute the geometric mean of percent increases, we first need to express each percentage as a multiplier in decimal form. This is done by dividing each percent by 100 and adding 1. For example, a 2% increase becomes 1.02, an 8% increase becomes 1.08, and so forth. Therefore, the list of decimal multipliers will be: 1.02, 1.08, 1.06, 1.04, 1.10, 1.06, 1.08, and 1.04.

Step 2: Multiply All Decimal Multipliers

Next, multiply all the decimal multipliers together to obtain the product. The calculation will be as follows:\(1.02 \times 1.08 \times 1.06 \times 1.04 \times 1.10 \times 1.06 \times 1.08 \times 1.04\).

Step 3: Calculate the Geometric Mean

To find the geometric mean, take the eighth root (since there are 8 values) of the product obtained in the previous step. Let's denote the product from Step 2 as P. Then, the geometric mean GM is given by:\(GM = P^{\frac{1}{8}}\).

Step 4: Compute the Final Result

After performing the calculations (multiplying and taking the root), we find the value of \(GM\). This value represents the average multiplier.

Key Concepts

Percent Increase

Understanding percent increase is vital to statistical and business calculations. A percent increase signifies how much a quantity grows in relation to its original size, expressed as a percentage. For instance, if a product originally priced at $100 is now $110, it has experienced a 10% increase. It's calculated by dividing the increase by the original value and then multiplying by 100.
To find the percent increase:

• Determine the absolute increase by subtracting the initial value from the new value.
• Divide the increase by the original value.
• Multiply the result by 100 to get a percentage.

This concept is key in various fields such as finance and economics, where it helps in forecasting trends and measuring growth rates.

Decimal Multipliers

Decimal multipliers are used to convert percentages to a format suitable for mathematical operations, like finding the geometric mean. This involves converting a percent change directly into a multiplier by dividing the percentage by 100 and adding 1.
For example, a 5% increase converts to a decimal multiplier of 1.05 (i.e., 5/100 + 1). This transformation is essential because it simplifies subsequent calculations such as multiplying several changes together or raising these changes to a power.
Decimal multipliers are especially useful in compound interest calculations and scenarios where a series of percentage changes need to be combined to find cumulative effects.

Statistical Calculations

Statistical calculations often involve finding means or averages, which provide insights into data sets. The geometric mean is one such calculation, especially used when dealing with percentage changes, rates, or proportional growth.
Unlike the arithmetic mean, which sums numbers and divides by the count, the geometric mean multiplies numbers and takes the nth root, where n is the count of numbers. This makes the geometric mean ideal for comparing different datasets or understanding compounded growth over time.
In our exercise, the geometric mean offers a balanced measure of average growth or rate change. It is more accurate for datasets involving products or varying percentage increases as it minimizes the impact of very high or low values compared to arithmetic means.

Business Mathematics

Business mathematics involves using mathematical concepts to solve problems in business contexts. This includes calculations like the geometric mean, essential in analyzing growth trends, financial ratios, and investment returns.
In this context, understanding how to convert percentages to decimal multipliers and apply them for statistical calculations becomes crucial. Businesses use these methods to gauge performance, make financial projections, and evaluate market conditions.
Applying geometric mean and other statistical measures help companies make informed decisions. For instance, assessing the average rate of return on investment portfolios or understanding product profitability are scenarios where business mathematics, within this framework, is applied effectively.