Question

A set of numbers is transformed by taking the log base 10 of each number. The mean of the transformed data is \(1.65 .\) What is the geometric mean of the untransformed data?

Short answer

The geometric mean of the original data is approximately 44.67.

Step-by-step solution

Understanding the Problem

The problem involves a set of data that has been transformed using logarithms (base 10). The mean of the transformed data is given, and we need to find the geometric mean of the original data set.

Recall the Relationship Between Logarithms and Geometric Mean

The mean of the log-transformed data is the logarithm of the geometric mean of the original data, i.e., if the mean of log-transformed data is \( M \), then the geometric mean of the original data is \( 10^M \).

Apply the Given Mean Value

We know the mean of the log-transformed data is \( 1.65 \). Therefore, the geometric mean of the original data is the antilogarithm (base 10) of this mean value.

Calculate the Geometric Mean

Compute \( 10^{1.65} \) to find the geometric mean of the untransformed data:\[ 10^{1.65} = 44.6683592150963 \approx 44.67 \]

Key concepts

Logarithmic Transformation

Logarithmic transformation involves taking the logarithm of a dataset, in this case with base 10. This transformation is particularly useful for datasets where the data spans several orders of magnitude. It helps to scale down larger values and make data more manageable and interpretable.

By using the log transformation, you can:

• Reduce skewness in data.
• Handle multiplicative relationships between variables.
• Simplify complex calculations by turning multiplication into addition.

In our problem, each number in the dataset is transformed by applying the log base 10. This makes it easier to compute average properties like the mean, allowing us to later access important characteristics like the geometric mean of original data.

Mean of Logarithms

When you compute the mean of logarithms, you're essentially finding the average logarithmic value of a data set. In this case, the mean of these logarithmic values was given to be 1.65.

Calculating the mean of logarithms is a stepping stone to understanding the structure and magnitude of the original data.

• It reflects the average behavior of the log-transformed dataset.
• The values are typically smaller and easier to compute than working with very large original values.

The significance here is that this mean of logarithms corresponds directly, in a base 10 sense, to the logarithm of the geometric mean of the original data set.

Antilogarithm

The antilogarithm is the reverse of taking the logarithm. It is used to "undo" a logarithm, effectively allowing us to retrieve the original number from its logarithmic form.

To compute the geometric mean from the mean of the logarithms, we use the antilogarithm of the mean value. Here, given that the mean of the transformed data is 1.65, the geometric mean of the original numbers is found using the antilogarithm:

• This involves calculating:
  \( 10^{1.65} \).
• This results in approximately 44.67.

The process of taking the antilogarithm is crucial as it converts the log-scaled data back to its original scale, which in this problem, is vital for finding the geometric mean.

Statistical Concepts

Statistical concepts such as the geometric mean are essential for analyzing data that involve products or exponential growth over time. The geometric mean is particularly useful for datasets that include values that grow exponentially or sequentially over time.

Here's why the geometric mean is important:

• It is less affected by extreme values or outliers in the data.
• Provides a more accurate representation of average growth rates, particularly with financial and population data.
• Helps in comparing different datasets by reducing dimensional variance.

The understanding and calculation of geometric mean through statistical concepts like logarithms and antilogarithms highlight the interconnectivity of these mathematical tools in practical problem-solving.