Understanding Skewness Calculation Using Student Height Quartiles

This question asks us to find the absolute measure of skewness for a distribution of student heights, given specific values for the quartiles. Skewness is a statistical measure that describes the asymmetry of a probability distribution. A distribution can be symmetric, skewed to the right (positively skewed), or skewed to the left (negatively skewed).

Understanding Quartiles and Skewness

Quartiles are points that divide a dataset into four equal parts. They are denoted as:

• $Q_1$: The first quartile (25th percentile).
• $Q_2$: The second quartile (50th percentile), which is also the median.
• $Q_3$: The third quartile (75th percentile).

The given quartiles for the student height distribution are:

• First Quartile ($Q_1$): 60.125
• Median ($Q_2$): 61.345
• Third Quartile ($Q_3$): 62.688

When only quartiles are available, a common way to measure skewness involves comparing the distances between the median and the quartiles. One such measure, often referred to when dealing with quartiles, is derived from Bowley's coefficient of skewness, or related concepts focusing on the symmetry around the median.

Formula for Absolute Measure of Skewness with Quartiles

While several formulas exist for skewness, the term "absolute measure of skewness" in the context of quartiles, especially when matching one of the provided options precisely, often points towards the calculation of the expression $Q_1 + Q_3 - 2 \times Q_2$. This expression quantifies the asymmetry between the upper and lower halves of the distribution relative to the median.

The calculation is as follows:

Measure = $Q_1 + Q_3 - 2 \times Q_2$

Step-by-Step Calculation

Let's substitute the given quartile values into the formula:

1. Sum of First and Third Quartiles: Calculate the sum of $Q_1$ and $Q_3$.

   Sum = $Q_1 + Q_3 = 60.125 + 62.688 = 122.813$

2. Twice the Median: Calculate two times the median ($Q_2$).

   Twice Median = $2 \times Q_2 = 2 \times 61.345 = 122.690$

3. Calculate the Difference: Find the difference between the sum of outer quartiles and twice the median.

   Measure = $(Q_1 + Q_3) - (2 \times Q_2) = 122.813 - 122.690 = 0.123$

The calculated value is 0.123.

Result Analysis and Option Selection

We compare the calculated value with the given options:

• Option 1: 0.321
• Option 2: 0.312
• Option 3: 0.231
• Option 4: 0.123

Our calculated measure of skewness is 0.123, which exactly matches Option 4.

A positive value like 0.123 suggests that the distribution is slightly skewed to the right, meaning the tail on the right side is longer or fatter than the left side. The magnitude indicates the degree of this asymmetry.