In the context of statistics, how can the $s=\sqrt{\frac{\sum {x-\mu }^2}{n}}$ (standard deviation) be used to specify uncertainty in a set of measurements?

• A. The $s$ (standard deviation) indicates the maximum possible error in any single measurement.
• B. The $s$ (standard deviation) is only used to calculate the $\mu$ (mean) and does not relate to uncertainty.
• C. The $s$ (standard deviation) measures the difference between the largest and smallest values in the data set.
• D. The $s$ (standard deviation) quantifies the typical amount by which individual measurements deviate from the $\mu$ (mean), providing a measure of the uncertainty or spread in the data.

Answer: (D)

Verified step by step guidance

1. Understand that the standard deviation is a measure of spread or dispersion in a data set, indicating how much individual measurements typically differ from the mean.
2. Recall the formula for the standard deviation of a sample: \(\displaystyle s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\), where \(x_i\) are the individual measurements, \(\bar{x}\) is the sample mean, and \(n\) is the number of measurements.
3. Recognize that the standard deviation does not represent the maximum error or the range (difference between largest and smallest values), but rather the average deviation from the mean.
4. Interpret the standard deviation as a quantification of uncertainty by showing how spread out the measurements are around the mean, which helps in understanding the reliability and precision of the data.
5. Use the standard deviation to express uncertainty by stating that most measurements lie within about one standard deviation of the mean, providing a typical error margin rather than an absolute maximum error.