3. Describing Data Numerically

Mean

3. Describing Data Numerically

Mean

2:43 minutes

Problem 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 guidance

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.

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.

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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Textbook Question

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

c. What is the benefit of using a trimmed mean versus using a mean found using all data entries? Explain your reasoning.

Textbook Question

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.

Final Grade The scores and their percents of the final grade for a statistics student are shown below. What is the student’s mean score?

Textbook Question

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.

Credit Card Balance For the month of October, a credit card has a balance of $115.63 for 12 days, $637.19 for 6 days, $1225.06 for 7 days, $0 for 2 days, and $34.88 for 4 days. What is the account’s mean daily balance for October?

Multiple Choice

Find the mean of the sample data below.

Textbook Question

Weighted Mean A student of the author earned grades of 63, 91, 88, 84, and 79 on her five regular statistics tests. She earned grades of 86 on the final exam and 90 on her class projects. Her combined homework grade was 70. The five regular tests count for 60% of the final grade, the final exam counts for 10%, the project counts for 15%, and homework counts for 15%. What is her weighted mean grade? What letter grade did she earn (A, B, C, D, or F)? Assume that a mean of 90 or above is an A, a mean of 80 to 89 is a B, and so on.

Textbook Question

In Exercises 29–32, compute the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (29) 31.4 minutes; (Exercise 30) 140.6 minutes; (Exercise 31) 55.2 years; (Exercise 32) 240.2 seconds.

Textbook Question

What’s Wrong? Education Week magazine published a list consisting of the mean teacher salary in each of the 50 states for a recent year. If we add the 50 means and then divide by 50, we get $56,479. Is the value of $56,479 the mean teacher salary for the population of all teachers in the 50 United States? Why or why not?

Textbook Question

Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47.

a. Find the missing value.

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