Understanding Distribution Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It indicates whether the data are concentrated more on one side or the other of the mean.

There are three main types of skewness:

• Symmetric Distribution: In a perfectly symmetric distribution (like the normal distribution), the mean, median, and mode are all equal. The distribution looks the same on both sides of the center.
• Positively Skewed Distribution: Also known as right-skewed. In this type, the tail of the distribution extends towards the right. The mean is typically greater than the median, which is typically greater than the mode (Mean > Median > Mode). This happens when there are extreme values on the right side of the distribution, pulling the mean towards the higher values.
• Negatively Skewed Distribution: Also known as left-skewed. In this type, the tail of the distribution extends towards the left. The mean is typically less than the median, which is typically less than the mode (Mean < Median < Mode). This happens when there are extreme values on the left side of the distribution, pulling the mean towards the lower values.

Analyzing the Given Distribution

We are given the following information for the distribution:

• Mean ($ \bar{x} $): 32
• Mode: 21

Now, let's compare the values of the mean and the mode:

$ \text{Mean} = 32 $

$ \text{Mode} = 21 $

Comparing them, we see that $ 32 > 21 $, which means the Mean is greater than the Mode.

Determining Skewness Based on Mean and Mode

Based on the general relationship between the mean and mode in skewed distributions:

• If Mean > Mode, the distribution is likely positively skewed.
• If Mean < Mode, the distribution is likely negatively skewed.
• If Mean $ \approx $ Mode, the distribution is likely symmetric or close to symmetric.

Since our calculated Mean (32) is greater than the given Mode (21), the distribution is indicated to be positively skewed.

Conclusion on Distribution Skewness

Given that the mean of the distribution is 32 and the mode is 21, the mean is greater than the mode. This relationship between the mean and mode is characteristic of a positively skewed distribution.

Revision Table: Skewness Indicators

Additional Information on Skewness Measurement

While the comparison of mean and mode provides a good indication, skewness can be formally calculated using various coefficients. Common measures include Pearson's coefficient of skewness (Type 1 and Type 2) and the moment coefficient of skewness.

• Pearson's First Coefficient of Skewness: $ Sk_1 = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}} $
• Pearson's Second Coefficient of Skewness: $ Sk_2 = \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}} $

A positive coefficient indicates positive skewness, a negative coefficient indicates negative skewness, and a coefficient near zero indicates symmetry.