Over many years, rainfall totals for Sacramento, CA in January ranged from a low of about 0.05 inch to a high of about 19.5 inches. The median was about 3.1 inches. Based on this information, explain how you can tell that the distribution of rainfall values in Sacramento in January cannot be bell-shaped.
The Correct Answer and Explanation is :
The distribution of January rainfall in Sacramento, CA, cannot be bell-shaped (normal) because of the significant difference between the highest and lowest values, along with the relatively low median compared to the maximum. A bell-shaped (normal) distribution is symmetric, meaning the data would be evenly spread around the mean, with the majority of values clustering near the center and tapering off equally in both directions. However, the provided rainfall data suggests a strongly right-skewed distribution.
Explanation:
- Skewness in Data:
- The lowest recorded rainfall in January is approximately 0.05 inches, while the highest is 19.5 inches. This extreme range suggests an asymmetry, where most values are likely clustered on the lower end, with a few extreme values pulling the distribution to the right.
- A bell-shaped distribution would have a balanced spread of data on both sides of the mean, which is not the case here.
- Median vs. Maximum Value:
- The median rainfall is 3.1 inches, meaning half the recorded values are below this point and half are above.
- In a normal distribution, the median, mean, and mode are approximately the same. However, in this case, the presence of much higher maximum values (such as 19.5 inches) suggests that the mean is likely greater than the median, further confirming right skewness.
- Presence of Extreme Outliers:
- A bell-shaped distribution follows the empirical rule, where 99.7% of values fall within three standard deviations of the mean. The presence of such a high maximum compared to the typical values suggests that the distribution is not symmetric but rather long-tailed on the right side.
- Rainfall Is Naturally Skewed:
- Precipitation data often exhibits skewed distributions because many months may experience relatively low or moderate rainfall, while occasional storms can result in very high values, creating an asymmetrical spread.
Conclusion:
Since the Sacramento January rainfall data exhibits a low median, an extremely high maximum, and an uneven distribution with more extreme values on one end, it does not follow a bell-shaped normal distribution. Instead, it is right-skewed, with occasional extreme rainfall amounts pulling the data toward higher values.