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 cannot be bell-shaped (normal) because the data is highly skewed rather than symmetrically distributed.
A normal distribution has a symmetric shape, meaning that most values cluster around the mean, and extreme values (low and high) occur with roughly equal frequency. However, in this case, the rainfall totals range from about 0.05 inches (very low) to about 19.5 inches (very high), with a median of 3.1 inches. The key detail here is that the median is much closer to the lower end of the range than the upper end. In a normal distribution, the median and mean are approximately the same, positioned at the center of the data.
The presence of a very high maximum rainfall (19.5 inches) and a very low minimum (0.05 inches) suggests that the data is right-skewed. In right-skewed distributions, extreme high values pull the mean above the median, creating a longer tail on the right side of the distribution. This indicates that while most years have relatively low rainfall, a few years experience very high totals, driving the mean higher.
Furthermore, normal distributions follow the empirical rule (68-95-99.7 rule), where nearly all data points fall within three standard deviations of the mean. Given the extreme variation in rainfall from year to year, a bell-shaped curve would not accurately represent this variability. The presence of such a wide range (from nearly zero to nearly 20 inches) indicates a distribution with a strong positive skew, making a normal distribution an inappropriate model.
Thus, the rainfall data for Sacramento in January does not follow a bell-shaped distribution because of its asymmetry, extreme variation, and right-skewed nature.