The following is a graph of two normal distributions plotted on the same x-axis. The two distributions have equal standard deviations and different means. The two distributions have equal means and different standard deviations. The two distributions have equal means and equal standard deviations. The two distributions have equal means and standard deviations that differ by 9 units.
The correct answer and explanation is:
To accurately answer the question, let’s break down the key characteristics of normal distributions and analyze the situation described in the question.
Key Characteristics of Normal Distributions:
- Mean: The mean determines the center of the distribution. It is the value around which the data points are symmetrically distributed.
- Standard Deviation: The standard deviation measures the spread or dispersion of the data. A larger standard deviation results in a wider distribution (data spread out more), while a smaller standard deviation results in a narrower distribution (data concentrated around the mean).
Different Cases to Consider:
- Equal Standard Deviations and Different Means:
- In this case, the two distributions would be of the same width, but their centers (peaks) would be at different locations on the x-axis. The two curves would not overlap significantly and would be separated by the difference in their means. The spread of each distribution would look the same, but the location of the peak would vary.
- Equal Means and Different Standard Deviations:
- Here, the two distributions would have the same center (the mean) but different spreads. The curve with the larger standard deviation would be wider and flatter, while the one with the smaller standard deviation would be narrower and taller. Despite having the same mean, their shapes would be noticeably different in terms of width.
- Equal Means and Equal Standard Deviations:
- In this case, both distributions would look identical in terms of width and location. The two curves would overlap completely, having the same peak and spread.
- Equal Means and Standard Deviations that Differ by 9 Units:
- If the standard deviations differ by 9 units, the curve with the larger standard deviation would be wider than the one with the smaller standard deviation, assuming both have the same mean. The larger standard deviation would cause the data to be more spread out around the mean.
Correct Answer:
The correct answer depends on which of the described characteristics best matches the graph you see. The key is identifying whether the curves are centered in the same location, how spread out they are, and whether their shapes differ in terms of width.