Question

A biologist made a certain pH measurement in each of 24 frogs; typical values were \(^{45}\) $$\begin{array}{lll}7.43, & 7.16, & 7.51, \ldots\end{array}$$ She calculated a mean of 7.373 and a \(\mathrm{SD}\) of 0.129 for these original pH measurements. Next, she transformed the data by subtracting 7 from each observation and then multiplying by 100 . For example, 7.43 was transformed to 43.The transformed data are $$43, \quad 16, \quad 51, \ldots$$ What are the mean and SD of the transformed data?

Short answer

The mean of the transformed data is 37.3 and the standard deviation (SD) is 12.9.

Step-by-step solution

Understanding the Transformation

The biologist transformed the original pH measurements by firstly subtracting 7 from each observation and then multiplying the result by 100. Mathematically, if 'x' is an original pH measurement, the transformed value 'y' would be given by the formula: y = (x - 7) * 100.

Finding the Mean of the Transformed Data

To find the mean of the transformed data, we apply the transformation to the mean of the original data. This can be expressed as: transformed mean = (original mean - 7) * 100. Substituting the given original mean (7.373) into the formula gives us the transformed mean = (7.373 - 7) * 100 = 0.373 * 100 = 37.3.

Finding the Standard Deviation of the Transformed Data

When all values in a dataset are multiplied by a constant, the standard deviation is also multiplied by the same constant. Since the original standard deviation (SD) is 0.129 and the multiplication constant is 100, the transformed SD is 0.129 * 100 = 12.9.

Key concepts

Mean Calculation

Understanding mean calculation is crucial in statistics, as it gives us a measure of the central tendency or average of a set of data. In simpler terms, the mean provides a single value that summarizes the essence of a collection of numbers. It's calculated by adding up all the values within a set and then dividing that total by the number of values.

For example, if we have five values: 2, 3, 5, 7, and 11, the mean would be: \( \frac{2 + 3 + 5 + 7 + 11}{5} = \frac{28}{5} = 5.6 \). The biologists' use of mean calculation in our exercise demonstrates an essential step in understanding the typical pH level across a sample of frogs. By knowing the mean pH, we can get a sense of 'normal' pH levels for this group, which aids in further studies or comparisons.

Moreover, the transformation of data by subtracting a constant and multiplying doesn't change the 'shape' of the data. It simply shifts the mean accordingly. After the biologist transformed the data by deducting 7 and multiplying by 100, she applied the same operation to the mean, which is a direct and efficient method to obtain the transformed mean.

Standard Deviation

The standard deviation (SD) is a measurement that tells us how spread out the numbers are in a set of data. It's a crucial concept because it provides insight into the variability within a dataset. A small SD indicates that the data points tend to be close to the mean, whereas a larger SD indicates that the data points are spread out over a wider range of values.

The standard deviation is found by taking the square root of the average squared deviation of each number from its mean. If that sounds complex, consider it as a measure of how much each number differs from the mean and therefore, the reliability of the mean as a representation of the data.

When we multiply all values in a dataset by a constant, as the biologist did by multiplying by 100 after subtracting 7, the standard deviation will also be multiplied by that constant. This leaves the relative dispersion the same, but adjusts the standard deviation to fit the new scale. In our exercise, the original standard deviation was 0.129, and after transformation (multiplying by 100), changed to 12.9 without affecting the distribution of the data.

pH Measurement Analysis

When analyzing pH, which is a measure of acidity or alkalinity, precision is paramount. The pH scale ranges from 0 to 14, with 7 being neutral. Measurements below 7 indicate an acidic substance, while above 7 indicates alkaline.

In biological studies, the pH of samples such as frog tissue can reveal vital information regarding their health and environmental conditions. Through statistical analysis of pH measurements, such as the mean and standard deviation calculations we've seen, scientists can gain insights into the normal pH range for a population and detect outliers or individuals whose pH deviates significantly from the norm.

Our exercise's transformation of pH data by subtracting a fixed value and scaling simplifies comparison and perhaps makes other computations easier. However, the biologist must understand that while this transformation makes the data easier to work with, it does not alter the relationships within the data. Therefore, this type of statistical data transformation is helpful for analysis but does not change the inherent properties of the measurements.