Pearson's coefficient of skewness is calculated as (Mean - Mode) / Standard Deviation. With Mean=14 and Mode=12, the numerator is 2. The provided answer assumes the standard deviation is 1, which is not stated. We retain the answer key.
842
Which statistical term is used to describe the lack of symmetry in a distribution?
Skewness is the statistical measure that quantifies the degree of asymmetry in a probability distribution. A perfectly symmetrical distribution has a skewness of zero. If the distribution is skewed, the mean, median, and mode are not equal, and the distribution is stretched toward one of the tails.
843
In the notation system established by R.A. Fisher, what is the square root of beta one (β1) equivalent to?
R.A. Fisher introduced various notations for moments and skewness. In this system, the square root of the first Pearson coefficient of skewness, beta one, is defined as gamma one (γ1), which serves as a measure of the asymmetry of a probability distribution.
844
What term describes the degree of peakedness or flatness of a probability distribution?
Kurtosis is a statistical measure that describes the shape of a distribution's tails in relation to the rest of the distribution. It specifically quantifies the 'peakedness' or the relative flatness of the distribution compared to a normal distribution curve.
845
If the difference between the first quartile and the median is greater than the difference between the median and the third quartile, how is the distribution classified?
In a distribution, skewness is indicated by the relative distances of quartiles from the median. If (Q1 - Median) > (Median - Q3), the left tail is longer, which typically suggests negative skewness. However, the provided answer key indicates 'positively skewed'. This may be due to a specific definition of quartile skewness coefficients. Given the conflict, we note the answer provided by the source.
846
In the context of moments expressed in standard units, what is the fourth alpha (α4) equivalent to?
In descriptive statistics, moments are used to describe the shape of a distribution. The fourth standardized moment, often denoted as α4 or β2 (beta two), measures kurtosis, which describes the 'tailedness' or peakedness of a probability distribution relative to a normal distribution.
847
In the study of kurtosis, how is a frequency distribution curve that is more peaked than a normal distribution classified?
Kurtosis measures the 'tailedness' or peakedness of a distribution. A leptokurtic distribution has a higher peak and fatter tails than a normal distribution (mesokurtic). Conversely, a platykurtic distribution is flatter with thinner tails than a normal distribution.
848
How is a distribution classified if the values of the mean, median, and mode are not identical?
In a perfectly symmetrical distribution, the mean, median, and mode are equal. When these measures differ, the distribution is skewed or asymmetrical, indicating that the data is not balanced around the central value.
849
Who is credited with introducing the relative measure of skewness?
Karl Pearson developed the Pearson's coefficient of skewness, which is a widely used relative measure of asymmetry. It compares the difference between the mean and the mode to the standard deviation, providing a dimensionless value that indicates the direction and magnitude of skewness in a distribution.
850
For a moderately skewed distribution with a median of 8, Q3 of 12, Q1 of 8, and an IQR of 4, what is the relative coefficient of skewness?
Bowley's coefficient of skewness is defined as (Q3 + Q1 - 2*Median) / (Q3 - Q1). Using the provided values: (12 + 8 - 2*8) / (12 - 8) = (20 - 16) / 4 = 4 / 4 = 1. The result is 1, which indicates the direction and magnitude of the skewness.
