The standard deviation of the values \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(\sigma .\) Calculate the standard deviation of the values \(k x_{1}, k x_{2}, k x_{3}, \ldots, k x_{n}\) where \(k\) is a constant.

The standard deviation formula is a crucial tool in statistics for measuring the amount of variation or dispersion in a set of values. It gives us a quantitative measure of how much the values in the dataset deviate from the mean (average) of the dataset. The formula is expressed as:

\[ \sigma = \sqrt{\frac{\sum_{i=1}^n \left( x_i - \overline{x} \right)^2}{n}} \]
Here, \( n \) is the total number of values in the dataset, \( x_i \) is each individual value, and \( \overline{x} \) is the mean of the dataset. The process involves subtracting the mean from each value to find their deviation from the mean, squaring each deviation to make them positive, summing them up, dividing by the number of values, and finally taking the square root of the result to get the standard deviation.

Dispersion of data refers to how spread out the values in a dataset are. It’s an indication of the variability and can reveal the degree to which the data points differ from each other and from the central value (mean). The standard deviation is one of the key measures of dispersion, and a smaller standard deviation indicates that the data points tend to be closer to the mean, signaling less spread. Conversely, a larger standard deviation shows greater spread from the mean. Understanding dispersion helps in determining the consistency and reliability of the data, which is essential in various fields such as finance, science, and engineering.

The mean of a dataset is the average of all the values contained in that set. It is calculated by summing up all the values and then dividing the total by the number of values. The formula for the mean is:

\[ \overline{x} = \frac{\sum_{i=1}^n x_i}{n} \]
The mean is a type of measure of central tendency and provides a single value representing the center of the data. It's important to note that the mean can be greatly affected by outliers or extremely high or low numbers in the dataset, which may skew the representation of the central location.

When a dataset undergoes a transformation, such as multiplying by a constant, the central tendency and dispersion may also change. Specifically, if each value in the dataset \(x_i\) is multiplied by a constant \(k\), the new mean of the transformed dataset becomes \(k\) multiplied by the original mean:\

\[ \overline{x'} = k\overline{x} \]
However, the standard deviation of the transformed dataset scales with the absolute value of the constant \(k\), due to the squaring process in the standard deviation formula. As shown in the calculations, multiplying all data points by \(k\) results in the standard deviation being multiplied by the absolute value of \(k\) as well:

\[ \sigma' = |k| \sigma \]
It’s important for students to understand how transformations affect statistics so they can analyze the data correctly, even after it's been altered.