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.
third option partially obscured

Explanation:

Step1: Recall Normal Distribution Properties

In a normal distribution, the mean ($\mu$) determines the center (peak position) of the curve, and the standard deviation ($\sigma$) determines the spread (width) of the curve. A smaller $\sigma$ means a taller, narrower curve; a larger $\sigma$ means a shorter, wider curve.

Step2: Analyze the Graph

- **Mean Analysis**: All three curves appear to be centered around the same vertical line (e.g., around 75 on the x - axis). So, they have the same mean.
- **Standard Deviation Analysis**: The curves have different widths. The tallest (narrowest) curve has the smallest $\sigma$, the next one has a larger $\sigma$, and the widest (shortest) curve has the largest $\sigma$? Wait, no—wait, the taller the curve, the smaller the standard deviation (since most data is close to the mean). So the three curves have different spreads, meaning different standard deviations, but the same center (mean). Wait, but let's re - check the options. Wait, maybe I misread the x - axis. Wait, the x - axis labels: 40, 50, 60, 75, 80, 90, 100? Wait, no, maybe the peaks are at the same x - value? Wait, the three normal curves: if their peaks are at the same x - coordinate, that means they have the same mean. And their widths (the spread) are different, so different standard deviations. Wait, but let's check the options. Wait, the third option (partially visible) probably says "Each distribution has the same mean and a different standard deviation". Let's analyze the options:

Option 1: "Each distribution has a different mean and the same standard deviation." But the peaks are at the same x (same mean), and spreads are different (different standard deviations), so this is wrong.

Option 2: "Each distribution has a different mean and a different standard deviation." But the peaks are at the same x (same mean), so this is wrong.

The third option (assuming it's "Each distribution has the same mean and a different standard deviation"): Since the peaks are at the same x (so same mean) and the curves have different widths (so different standard deviations), this must be true. But wait, maybe the original problem's third option is that. Wait, maybe I made a mistake. Wait, let's re - examine the graph. The three normal curves: do they have the same center (mean)? Yes, because their peaks are vertically aligned (same x - value for the peak). And their spreads (standard deviations) are different: the tallest curve is the narrowest (smallest $\sigma$), the next one is wider, and the widest is the shortest. So the correct statement is that each has the same mean and different standard deviations. But since the user's options: let's assume the third option is "Each distribution has the same mean and a different standard deviation". But maybe the user's options were cut off. Wait, but based on the graph, the key is: in normal distributions, mean is the center (peak x - value), standard deviation is the spread (width). So same peak x (same mean), different widths (different standard deviations). So the correct statement (the third option, probably) must be true. But since the user's question is about which statement must be true, and given the graph, the correct answer is the one that says same mean and different standard deviations. But let's check the options again. Wait, maybe the user's options were:

1. Each distribution has a different mean and the same standard deviation.

2. Each distribution has a different mean and a different standard deviation.

3. Each distribution has the same mean and a different standard deviation.

So the answer is the third option. But since the user's original problem's third option is partially visible, but based on the graph, the correct statement is that they have the same mean (same peak) and different standard deviations (different spreads). So the correct option is the one that states "Each distribution has the same mean and a different standard deviation" (assuming that's the third option).

Answer:

(Assuming the third option is "Each distribution has the same mean and a different standard deviation") The option stating "Each distribution has the same mean and a different standard deviation" (the third option, likely).