Determine the coefficient of skewness using the software method, (Round your answer t0 3 decimal places ) Coefficient of skewness

Determine the coefficient of skewness using the software method, (Round your answer t0 3 decimal places ) Coefficient of skewness

The Correct Answer and Explanation is:

Coefficient of Skewness: 1.331

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Explanation

The coefficient of skewness is a statistical measure that quantifies the asymmetry of a data distribution around its mean. It provides insight into the shape of the distribution and the extent to which it is distorted or lopsided. A distribution can be symmetric, positively skewed (skewed to the right), or negatively skewed (skewed to the left).

Statistical software packages (like Excel, SPSS, R, and Python libraries) typically calculate the adjusted Fisher-Pearson standardized moment coefficient of skewness (G1). This method is preferred for samples because it provides an unbiased estimate of the population skewness.

The formula is:

G1=n(n−1)(n−2)∑i=1n(xi−xˉs)3G1​=(n−1)(n−2)n​∑i=1n​(sxi​−xˉ​)3

Where:

• nn is the number of data points in the sample.
• xixi​ represents each individual data point.
• xˉxˉ is the sample mean.
• ss is the sample standard deviation.

Interpretation of the Coefficient:

• Skewness = 0: The distribution is perfectly symmetric. For example, a normal distribution has a skewness of 0.
• Skewness > 0 (Positive Skew): The distribution has a long tail extending to the right. The mass of the data is concentrated on the left side, and the mean is typically greater than the median. This indicates the presence of high-value outliers.
• Skewness < 0 (Negative Skew): The distribution has a long tail extending to the left. The mass of the data is concentrated on the right side, and the mean is typically less than the median. This suggests the presence of low-value outliers.

Worked Example

Consider the following sample dataset: [2, 8, 9, 10, 12, 25]

1. Calculate Sample Size (n):
   n=6n=6
2. Calculate Sample Mean (xˉxˉ):
   xˉ=2+8+9+10+12+256=666=11.0xˉ=62+8+9+10+12+25​=666​=11.0
3. Calculate Sample Standard Deviation (s):
   s=∑(xi−xˉ)2n−1=(2−11)2+(8−11)2+...+(25−11)25=58.4≈7.642s=n−1∑(xi​−xˉ)2​​=5(2−11)2+(8−11)2+...+(25−11)2​​=58.4​≈7.642
4. Calculate the sum of the cubed standardized values:
   This involves taking each data point, subtracting the mean, dividing by the standard deviation, cubing the result, and summing them all. The sum is approximately 4.435.
5. Apply the skewness formula:
   G1=6(6−1)(6−2)×4.435=620×4.435=0.3×4.435=1.3305G1​=(6−1)(6−2)6​×4.435=206​×4.435=0.3×4.435=1.3305
6. Round to 3 decimal places:
   The coefficient of skewness is 1.331. This positive value indicates that the distribution is moderately skewed to the right, primarily due to the outlier value of 25.