Test: Measures of Skewness - UGC NET MCQ

• The Assertion is true because Bowley’s Measure provides a robust way to assess skewness, especially for non-normally distributed data.
• The Reason is false since Bowley’s Measure does not rely on moments or deviations from the mean; it uses quartiles instead.
• Thus, the Reason does not correctly explain the Assertion.

A positive skewness indicates that the tail on the right side of the distribution is longer or fatter, which means that the mean is greater than the mode. This situation typically arises when a distribution has a concentration of data points on the left side, causing the mean to be pulled in the direction of the longer tail. Understanding skewness is important in statistics because it helps in determining the appropriate measures of central tendency to use when analyzing data. For instance, in positively skewed distributions, the median might be a better representation of the central tendency than the mean.

Skewness measures the symmetry or asymmetry of a dataset's probability distribution. A positive skew indicates that the tail of the distribution extends more towards the right, while a negative skew indicates a longer tail on the left. Understanding skewness helps in interpreting data characteristics and selecting appropriate statistical methods for analysis. Interestingly, skewness can impact decisions in fields like finance, where skewed distributions might affect risk assessments and investment strategies.

- The Assertion (A) is true because a negative skewness indeed indicates that the distribution has a longer left tail (left-skewed).

- The Reason (R) is also true, but it describes the opposite scenario; in a left-skewed distribution, the data points are primarily on the left with a longer tail on the right.

- Therefore, while both statements are true, the Reason does not correctly explain the Assertion.

Statement 1 is correct because a negative skewness does indeed indicate a longer left tail, with most data points concentrated on the right, leading to the mean being less than the median. Statement 2 is incorrect; in a right-skewed distribution, the mean is typically greater than the median, not less. Thus, the correct answer is that only Statement 1 is accurate.

The mean of the dataset is calculated by adding all the scores together and dividing by the number of scores. In this case, the total of the scores is 953, and there are 10 scores. Thus, the mean is \( 953 \div 10 = 95.3 \). The mean provides a measure of central tendency, which helps to summarize the dataset with a single representative value. Interestingly, the mean can be influenced by extreme values, which is why it's important to also consider other measures of central tendency like the median and mode when analyzing distributions.

In a positively skewed distribution, the mean is greater than the median, which in turn is greater than the mode, represented as Mean > Median > Mode. This occurs because the larger values, or outliers, on the right side of the distribution pull the mean upward, while the bulk of the data remains concentrated on the left. Understanding the relationship between these measures of central tendency is crucial in statistics, as it helps in interpreting data distributions effectively. An interesting fact is that skewness can affect the results of statistical analyses, making it essential to check for skewness before choosing the appropriate statistical methods for data analysis.

• The Assertion is false because a positive value of Bowley’s Measure indicates a longer tail on the right side of the distribution, not the left.
• The Reason is also false; a negative value of Bowley’s Measure indicates that the distribution is negatively skewed, not positively skewed.
• Therefore, both the Assertion and Reason are incorrect.

- Statement 1 is correct because a skewness value close to zero implies that the distribution is symmetric, meaning its tails on both sides are of roughly equal length.

- Statement 2 is incorrect. A skewness value significantly below -1 indicates strong left skewness, not right skewness. Strong right skewness would be indicated by a skewness value significantly above +1.

Thus, the correct option is Option A: 1 Only.

- The Assertion is true as it correctly describes what the 90th percentile represents in a dataset.

- The Reason is also true because it accurately explains how to calculate the position of the 90th percentile.

- Since the Reason provides a correct explanation for the Assertion, the correct answer is Option A.