Question

the graph shows a distribution of data. what is the standard deviation of the data?
○ 0.5
○ 1.5
○ 2.0
○ 2.5

Explanation:

Step1: Identify the mean

The peak of the normal distribution (the highest point of the curve) is at \( x = 2 \), so the mean (\( \mu \)) of the data is 2.

Step2: Analyze the spread

In a normal distribution, the standard deviation (\( \sigma \)) represents the spread of the data. Looking at the graph, the distance from the mean (2) to the point where the curve changes concavity (the inflection point) is 0.5? Wait, no, wait. Wait, the inflection points of a normal distribution are at \( \mu \pm \sigma \). Let's check the x - axis. The mean is at 2. Let's look at the points: from 2 to 2.5 is 0.5? Wait, no, wait the options are 0.5, 1.5, 2.0, 2.5. Wait, maybe I made a mistake. Wait, the normal distribution curve: the inflection points are one standard deviation away from the mean. Let's see the x - axis marks: 1, 1.5, 2, 2.5, 3. The mean is at 2. Let's see the distance from the mean to the point where the curve starts to change its curvature. Let's check the difference between 2 and 2.5: \( 2.5 - 2=0.5 \)? No, that can't be. Wait, maybe the scale is different. Wait, no, let's think again. Wait, the normal distribution: the total spread? No, standard deviation is a measure of how spread out the data is. Wait, the mean is 2. Let's look at the options. If the standard deviation is 0.5, then the inflection points are at \( 2\pm0.5 = 1.5 \) and \( 2.5 \). Looking at the graph, the curve has inflection points (where it changes from concave up to concave down) at 1.5 and 2.5, which are 0.5 units away from the mean (2). So the standard deviation is 0.5? Wait, no, wait 2 - 1.5 = 0.5, 2.5 - 2 = 0.5. So the standard deviation is 0.5? Wait, but let's confirm. In a normal distribution, the inflection points are located at \( \mu\pm\sigma \). So if the mean \( \mu = 2 \), and the inflection points are at 1.5 and 2.5, then \( \sigma=2 - 1.5 = 0.5 \).

Answer:

0.5