In a series of \(2 \mathrm{~m}\) observations half of them equal to \(\mathrm{b}\) and remaining half equal to \(-b\). If the standard deviation of the observations is 3 then \(|\mathrm{b}|=\) (a) 3 (b) \(\sqrt{3}\) (c) \((\sqrt{3} / \mathrm{n})\) (d) \((1 / \mathrm{n})\)

An observational series refers to a set of data points gathered from observations or measurements. In statistics, we often deal with such series to analyze and draw conclusions. In the exercise presented, the observational series consists of a number of values equal to \(b\) and an equal number to \( -b\), making up a total of \(2m\) observations. The importance of this uniform distribution is that it simplifies the process of finding the mean, which, due to the symmetry around zero, ends up being zero. This characteristic of the observational series leads to a simplification when calculating other statistics like variance and standard deviation.

The series reflects real-world phenomena where data can exhibit symmetry. When this occurs, it can significantly streamline statistical calculations and interpretations.

The sum of squares is a key concept in statistics, representing the sum of the squared differences between each observation and the mean. It's a measure of the total variance within a data set. The formula to calculate it is \[\sum (x_i - \bar{x})^2\]. In our exercise, with the mean \(\bar{x}\) being zero, the formula simplifies to \[\sum (x_i)^2\]. Since we have an equal number of \(b\)'s and \( -b\)'s, this will result in \(2mb^2\).

The sum of squares helps us understand how spread out the data is around the mean. It is a foundational stepping stone for calculating variance and standard deviation—a smaller sum of squares indicates that the data points are closer to the mean, and consequently, less spread out.

The arithmetic mean, commonly referred to as the average, is the sum of a collection of numbers divided by the count of numbers in the collection. In simpler terms, \(\bar{x} = \frac{\sum x_i}{n}\) where \(\bar{x}\) is the mean, \(\sum x_i\) is the sum of all observations, and \(n\) is the number of observations.

In the given problem, since half of the numbers are \(b\) and the other half are \( -b\), their sum is zero, leading to a mean of zero. This illustrates an important property: if a data set is symmetrically distributed around a central point, that point is the arithmetic mean of the set. Zero mean implies that the dataset, on average, does not deviate from zero, albeit individual observations might.

Variance is a statistical measure that tells us how much the data in a set are spread out from their mean. The formula to calculate variance is \(\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}\), where \(\sigma^2\) is the variance, \(\sum (x_i - \bar{x})^2\) is the sum of squares, and \(n\) is the total number of observations.

In our textbook exercise, because the arithmetic mean is zero, the variance formula simplifies to \(\sigma^2 = \frac{\sum (x_i)^2}{n}\). Upon solving, we find that the variance is equal to the square of the standard deviation, \(\sigma\), which in this case is given as 3. By squaring this value, we obtain the variance. Variance helps us quantify the dispersion of the dataset, indicating how much the numbers in the set differ from one another and from their mean.