Question

the graph represents the distribution of the number of questions answered correctly on a 50-question math test. math test scores (graph with x-axis labeled 24, 28, 32, 36, 40, 44, 48) what is the standard deviation of the data? 4, 8, 12, 24

Explanation:

Step1: Analyze the normal distribution graph

The graph is a normal distribution curve. In a normal distribution, the standard deviation can be estimated by looking at the spread of the data. The x - axis has marks at 24, 28, 32, 36, 40, 44, 48. The distance between consecutive marks (e.g., 28 - 24 = 4, 32 - 28 = 4, etc.) seems to represent the standard deviation or a multiple of it. But more importantly, the total range of the data (from around 24 to 48) and the shape of the curve. In a normal distribution, the data is spread around the mean, and the standard deviation is a measure of the spread. Looking at the intervals between the x - axis ticks, each tick is 4 units apart? Wait, no, let's check the distance between the mean (around 36, since the peak is at 36) and the other points. The distance from 36 to 44 is 8, from 36 to 48 is 12, but that's not right. Wait, actually, in a normal distribution, the standard deviation can be estimated by the distance between the mean and the point where the curve changes curvature (the inflection point). The inflection points are usually at mean ± standard deviation. Looking at the graph, the curve starts to change curvature around 36 - 8 = 28 and 36+8 = 44? Wait, no, the x - axis marks: 24, 28, 32, 36, 40, 44, 48. The distance between 36 (the peak, mean) and 44 is 8, and between 36 and 28 is 8. So the standard deviation is 8? Wait, no, let's think again. The normal distribution has inflection points at μ±σ. So if the mean (μ) is at 36, and the inflection points are at 36 - σ and 36+σ. Looking at the graph, the curve starts to bend less (inflection) around 28 and 44. So 36 - 28 = 8, and 44 - 36 = 8. So σ = 8? Wait, but let's check the options. The options are 4, 8, 12, 24. 24 is too big (the minimum is 24, so the range from 24 to 48 is 24, and in a normal distribution, the range is about 6σ (since 99.7% of data is within μ±3σ). So 6σ≈24 (48 - 24), so σ≈4? Wait, that's a contradiction. Wait, 48 - 24 = 24. If 6σ = 24, then σ = 4. But earlier, the inflection points. Wait, maybe I made a mistake. Let's recalculate. The empirical rule says that for a normal distribution, about 99.7% of the data lies within μ - 3σ and μ+3σ. So the total range of the data (from the lowest to the highest) is approximately 6σ. The lowest value on the graph is around 24, and the highest is around 48. So 48 - 24 = 24. Then 6σ = 24, so σ = 4? But that contradicts the inflection point idea. Wait, maybe the mean is not 36. Wait, the peak is at 36, so mean is 36. Then 36 - 3σ = 36 - 12 = 24, and 36+3σ = 36 + 12 = 48. Ah! That makes sense. Because 36 - 3σ = 24 → 3σ = 12 → σ = 4? No, 36 - 3σ = 24 → 3σ = 12 → σ = 4? Wait, 36 - 3*4 = 36 - 12 = 24, and 36+3*4 = 36 + 12 = 48. Yes! So the range from 24 to 48 is μ - 3σ to μ+3σ, so 6σ = 24, so σ = 4? But that's conflicting with the inflection point. Wait, no, the inflection points are at μ - σ and μ+σ. So if σ = 4, then μ - σ = 32, μ+σ = 40. Looking at the graph, the curve at 32 and 40: does the curve change curvature there? The peak is at 36, at 32 (36 - 4) and 40 (36 + 4), the curve does seem to have inflection points there. Let's check the x - axis: 32 is 4 units left of 36, 40 is 4 units right of 36. Then 36 - 3σ = 36 - 12 = 24, 36+3σ = 36 + 12 = 48. Which matches the ends of the curve (around 24 and 48). So the standard deviation is 4? Wait, now I'm confused. Let's use the empirical rule properly. The empirical rule states that for a normal distribution:
- Approximately 68% of the data lies within μ±σ.
- Approximately 95% of the data lies within μ±2σ.
- Approximately 99.7% of the data lies within μ±3σ.

Looking at the graph, the data seems to be from around 24 to 48. So the range of 99.7% of the data is 48 - 24 = 24. Since 99.7% of the data is within μ±3σ, then 6σ = 24 (because μ+3σ-(μ - 3σ)=6σ), so σ = 24/6 = 4. But wait, the peak is at 36, so μ = 36. Then μ - 3σ = 36 - 12 = 24, μ+3σ = 36 + 12 = 48, which matches the x - axis marks (24 and 48 are the ends of the curve). Now, the inflection points (where the curve changes from concave up to concave down) are at μ±σ. So μ±σ = 36±4 = 32 and 40. Looking at the graph, at x = 32 and x = 40, the curve does change its curvature (inflection points). So that makes sense. So the standard deviation is 4? But wait, the options are 4, 8, 12, 24. Wait, maybe I made a mistake in the range. Let's check the x - axis again. The x - axis has marks at 24, 28, 32, 36, 40, 44, 48. The distance between 24 and 48 is 24, but maybe the data is not from 24 to 48, but the curve is just drawn that way. Wait, the peak is at 36, and the distance from 36 to 44 is 8, from 36 to 48 is 12, from 36 to 28 is 8, from 36 to 24 is 12. So if we consider that 95% of the data is within μ±2σ, then μ+2σ = 44 (since 36+8 = 44) and μ - 2σ = 28 (36 - 8 = 28), so 2σ = 8, so σ = 4? No, 2σ = 8 → σ = 4? Wait, 36+2σ = 44 → 2σ = 8 → σ = 4. And 36 - 2σ = 28 → 2σ = 8 → σ = 4. Then 3σ would be 12, so 36+12 = 48, 36 - 12 = 24, which matches the ends of the curve. So the standard deviation is 4? But earlier I thought it was 8. Wait, let's recast:

If σ = 4:
- μ - σ = 32
- μ+σ = 40 (inflection points)
- μ - 2σ = 28
- μ+2σ = 44
- μ - 3σ = 24
- μ+3σ = 48

Looking at the graph, the curve at x = 32 and x = 40 (μ±σ) has inflection points (changes curvature), at x = 28 and x = 44 (μ±2σ) the curve is more spread out, and at x = 24 and x = 48 (μ±3σ) the curve is near the x - axis. This fits the empirical rule. So the standard deviation is 4? But wait, the options include 4, 8, 12, 24. Wait, maybe I messed up the mean. Wait, the peak is at 36, so mean is 36. Then the distance from 36 to 44 is 8, which would be 2σ (since 2σ = 8 → σ = 4) or σ = 8 (if 8 is σ). Wait, no, let's take a step back. The key is that in a normal distribution, the standard deviation is a measure of how spread out the data is. The graph shows that the data is spread from around 24 to 48, with the mean at 36. The range of 99.7% of the data (μ±3σ) is 24 to 48, so 6σ = 24, so σ = 4. But let's check the options. The options are 4, 8, 12, 24. If σ = 4, that's one of the options. But wait, maybe the inflection points are at 36±8, so σ = 8. Let's see: if σ = 8, then μ±σ = 36±8 = 28 and 44, μ±2σ = 36±16 = 20 and 52 (but 52 is more than 50, since it's a 50 - question test, so that can't be). So μ±2σ can't be 20 and 52. So σ = 8 is invalid because 36+2*8 = 52>50. So σ must be 4, because 36+3*4 = 48≤50, and 36 - 3*4 = 24. That makes sense because it's a 50 - question test, so the maximum score is 50, so 48 is close to 50, and 24 is a reasonable minimum. So the standard deviation is 4? Wait, but earlier when I thought about the inflection points at 32 and 40 (36±4), that also makes sense. So the correct answer should be 4? Wait, no, I'm confused again. Wait, let's calculate the range of 6σ. If σ = 4, 6σ = 24, which is the distance from 24 to 48 (48 - 24 = 24). If σ = 8, 6σ = 48, which would mean the range from μ - 3σ to μ+3σ is 48, so if μ = 36, then μ - 3σ = 36 - 24 = 12, and μ+3σ = 36+24 = 60, which is more than 50, so that's impossible. So σ can't be 8, 12, or 24. So σ must be 4.

Step2: Confirm with the empirical rule

Using the empirical rule for normal distributions:
- 99.7% of data is within \( \mu\pm3\sigma \)
The data in the graph ranges approximately from 24 to 48 (since the curve is near the x - axis at these points). Let the mean \( \mu = 36 \) (the peak of the curve). Then:
\( \mu - 3\sigma=24 \) and \( \mu + 3\sigma = 48 \)
Substitute \( \mu = 36 \) into \( \mu-3\sigma = 24 \):
\( 36-3\sigma=24 \)
Subtract 36 from both sides: \( - 3\sigma=24 - 36=-12 \)
Divide both sides by - 3: \( \sigma = 4 \)

Answer:

4