Skewness in this context is defined as the difference between the mean and the mode. Given that Skewness = Mean - Mode, we can substitute the known values: 5 = Mean - 18. Adding 18 to both sides results in a mean of 23.
782
What is the correct formula for calculating the r-th moment about the mean for grouped data?
The r-th moment about the mean for grouped data is defined as the sum of the products of frequencies and the r-th power of the deviations from the mean, divided by the total number of observations: (1/n) * Σf(x - mean)^r.
783
How are the coefficients of skewness developed by Karl Pearson, Professor Kelly, and Professor Bowley classified?
These coefficients are considered relative measures because they are dimensionless. By dividing the absolute measure of skewness by a measure of dispersion (like standard deviation or quartile deviation), these formulas allow for the comparison of skewness across different datasets regardless of the units of measurement used.
784
In a skewed distribution, how do the mean, median, and mode compare to one another?
In a perfectly symmetrical distribution, the mean, median, and mode are identical. However, in a skewed distribution, the asymmetry causes these three measures of central tendency to take on different values, reflecting the pull of the tail on the mean.
785
Which coefficient of skewness is recommended for distributions with open-ended classes or poorly defined modes?
Bowley's coefficient of skewness is based on quartiles rather than the mean, median, or mode. Because it relies on positional values (Q1, Q2, Q3), it is particularly useful for distributions where the mode is ill-defined or where the data is grouped into open-ended intervals, preventing the calculation of the mean.
786
Which type of distribution is characterized by a tail that extends toward the lower values, indicating the presence of outliers on the left side?
A negatively skewed distribution, also known as left-skewed, is characterized by a long tail on the left side of the distribution. This means the mass of the distribution is concentrated on the right, and there are relatively lower values (outliers) pulling the mean below the median and mode. This skewness indicates that the data has a higher frequency of high values.
787
In a negatively skewed distribution, what is the typical relationship between the mean, median, and mode?
In a negatively skewed (left-skewed) distribution, the tail on the left side is longer or fatter than the right side. This pulls the mean towards the left, making it the smallest value, followed by the median, and then the mode, which is the highest peak. Thus, the relationship is mean < median < mode.
788
In a unimodal distribution, what is the skewness classification if the mode is less than the mean?
In a unimodal distribution, the relationship between mean and mode indicates skewness. When the mean is greater than the mode, the distribution is pulled toward the right by extreme high values, resulting in a positive skew. This indicates that the tail on the right side of the distribution is longer or fatter than the left side.
789
What is the value of the first moment about the mean for any dataset?
The first moment about the mean is defined as the sum of (x - mean) divided by the number of observations. Since the sum of deviations of all observations from their arithmetic mean is always zero, the first moment about the mean is mathematically guaranteed to be zero.
790
In a unimodal distribution, what is the skewness if the mode is less than the mean?
In a unimodal distribution, the relationship between mean, median, and mode determines the skewness. If the mean is greater than the mode, the distribution has a long tail extending toward the higher values, which characterizes positive skewness. This indicates that the distribution is skewed to the right.
