In a distribution, the relationship between the mean and mode indicates the direction of skewness. If the mean is greater than the mode, the distribution has a longer tail on the right side, which is referred to as positively skewed. This indicates that the data contains extreme high values pulling the mean upward.

A perfectly symmetrical distribution, particularly the normal distribution, is characterized by a bell-shaped curve. In this distribution, the mean, median, and mode are identical, and the density of observations decreases symmetrically as one moves away from the center in either direction.

The r-th central moment is defined as the arithmetic mean of the r-th powers of the deviations of the observations from their mean. For a dataset of size n, the formula is the sum of (x - mean) raised to the power of r, divided by n. This measure is fundamental in determining skewness and kurtosis of a distribution.

In statistics, moments are quantitative measures of the shape of a function's graph. The first moment about the origin is the mean, and central moments describe dispersion, skewness, and kurtosis, providing a comprehensive summary of the distribution's properties.

A distribution is considered skewed when it lacks symmetry. If the tail of the distribution extends further to the right, it is positively skewed; if it extends further to the left, it is negatively skewed. This asymmetry indicates that the mean, median, and mode do not coincide.

In a perfectly symmetrical distribution, the central tendency measures (mean, median, and mode) coincide at the center of the distribution. Any deviation from this equality typically indicates that the distribution is skewed. For example, in a positively skewed distribution, the mean is typically greater than the median, which is greater than the mode.

Higher moments, such as skewness (third moment) and kurtosis (fourth moment), provide detailed information about the shape of a distribution. These are often used to describe the characteristics of a population, as they capture nuances beyond the central tendency and dispersion provided by lower moments.

The r-th moment about the origin, often denoted as μ'r, is calculated by taking the sum of the products of frequencies (f) and the r-th power of the values (x), then dividing by the total number of observations (n). This formula provides a way to describe the shape and characteristics of a probability distribution.

In a positively skewed distribution, the tail on the right side (the upper tail) is longer or fatter than the left side. This means that the bulk of the data is concentrated on the left, while the extreme values extend toward the upper tail. The provided answer B suggests the values move toward the upper tail, which is consistent with the direction of the skewness.

The third moment about the mean is the basis for calculating skewness. A zero third moment indicates that the distribution is perfectly balanced around the mean, meaning the left and right sides are mirror images, which defines a symmetrical distribution.