For a frequency distribution of a variable x, mean = 32, median = 30. The distribution is:
Positively skewed
Analyzing Frequency Distribution Skewness
The question asks about the shape of a frequency distribution given its mean and median. Specifically, we are told that for a variable x, the mean is 32 and the median is 30. We need to determine if the distribution is positively skewed, negatively skewed, mesokurtic, or platykurtic.
Understanding Skewness vs. Kurtosis
The options include terms related to both skewness (positively skewed, negatively skewed) and kurtosis (mesokurtic, platykurtic). Skewness measures the asymmetry of the distribution, while kurtosis measures the 'tailedness' or peak sharpness of the distribution relative to a normal distribution.
The relationship between the mean, median, and mode is often used to determine the skewness of a distribution:
- If the distribution is symmetric, the Mean, Median, and Mode are approximately equal.
- If the distribution is positively skewed (or right-skewed), the tail is longer on the right side. In this case, the Mean is typically greater than the Median, which is typically greater than the Mode (\(\text{Mean} > \text{Median} > \text{Mode}\)). The mean is pulled towards the higher values in the longer right tail.
- If the distribution is negatively skewed (or left-skewed), the tail is longer on the left side. In this case, the Mean is typically less than the Median, which is typically less than the Mode (\(\text{Mean} < \text{Median} < \text{Mode}\)). The mean is pulled towards the lower values in the longer left tail.
Kurtosis, on the other hand, describes the shape of the distribution's peak and tails. Mesokurtic refers to a distribution with kurtosis similar to a normal distribution. Platykurtic refers to a distribution with a flatter peak and thicker tails than a normal distribution. The given information (mean and median) is directly related to skewness, not kurtosis.
Applying Given Data to Determine Skewness
We are given:
- Mean = 32
- Median = 30
Comparing the mean and median, we see that the Mean (32) is greater than the Median (30).
This relationship, \(\text{Mean} > \text{Median}\), is characteristic of a positively skewed distribution.
| Distribution Shape | Relationship between Mean and Median | Tail Direction |
|---|---|---|
| Symmetric | Mean \(\approx\) Median | Balanced |
| Positively Skewed | Mean > Median | Longer tail on the right |
| Negatively Skewed | Mean < Median | Longer tail on the left |
Conclusion
Since the mean (32) is greater than the median (30) for the given frequency distribution, the distribution is positively skewed.
Revision Table: Skewness Summary
| Metric | Value | Relationship | Skewness Type |
|---|---|---|---|
| Mean | 32 | Mean > Median | Positively Skewed |
| Median | 30 |
Additional Information: Kurtosis
While the question focused on skewness, the options included terms related to kurtosis. Kurtosis measures the shape of the peak and tails of a distribution.
- Mesokurtic: Distributions with kurtosis similar to the normal distribution (e.g., Normal distribution itself).
- Leptokurtic: Distributions with a sharper peak and heavier tails than a normal distribution.
- Platykurtic: Distributions with a flatter peak and lighter tails than a normal distribution.
The values of the mean and median do not provide information about the kurtosis of the distribution.
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