Question

the graph below shows three different normal distributions.
which statement must be true?
each distribution has a different mean and the same standard deviation.
each distribution has a different mean and a different standard deviation.
each distribution has the same mean and the same standard deviation.

Explanation:

Analyze the mean of the distributions

Using the Normal Distribution knowledge point
All three curves are symmetric and centered at the exact same horizontal position on the x-axis, which is \(x = 75\). Since the peak of a normal distribution curve occurs at its mean, all three distributions have the same mean:

$$ \mu_1 = \mu_2 = \mu_3 = 75 $$

Analyze the standard deviation of the distributions

Using the Standard Deviation Comparison knowledge point
The three curves have different shapes: the blue curve is tall and narrow, the green curve is moderately spread, and the red curve is short and wide. A taller, narrower curve indicates that data is more concentrated around the mean (smaller standard deviation), while a shorter, wider curve indicates that data is more spread out (larger standard deviation). Therefore, the standard deviations are different:

$$ \sigma_{\text{blue}} < \sigma_{\text{green}} < \sigma_{\text{red}} $$

Evaluate the given statements

Combining the observations:

• The means are the same.
• The standard deviations are different.

Let's evaluate the visible options:

1. "Each distribution has a different mean and the same standard deviation." (False)
2. "Each distribution has a different mean and a different standard deviation." (False)
3. "Each distribution has the same mean and the same standard deviation." (False)

The correct statement must be: "Each distribution has the same mean and a different standard deviation." Although this option is cut off at the bottom of the image, we can logically deduce it as the only mathematically true statement.

Answer:

• Each distribution has a different mean and the same standard deviation.
• Each distribution has a different mean and a different standard deviation.
• Each distribution has the same mean and the same standard deviation.
• Each distribution has the same mean and a different standard deviation. (Correct answer)