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Visually assessing standard deviation | AP Statistics | Khan Academy

3m read
·Nov 11, 2024

Each dot plot below represents a different set of data. We see that here. Order the dot plots from largest standard deviation (top) to smallest standard deviation (bottom). So pause this video and see if you can do that, or at least if you could rank these from largest standard deviation to smallest standard deviation.

All right, now let's work through this together. I'm doing this on Khan Academy, where I can move these around to order them, but let's just remind ourselves what the standard deviation is or how we can perceive it. You could view the standard deviation as a measure of the typical distance from each of the data points to the mean. So the largest standard deviation, which you want to put on top, would be the one where typically our data points are further from the mean.

Our smallest standard deviation would be the ones where it feels like, on average, our data points are closer to the mean. In all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75. So it's really about how spread apart they are from that.

If you look at this first one, it has these two data points, one on the left and one on the right, that are pretty far. Then you have these two that are a little bit closer, and then these two that are inside this one right over here. To get from this top one to this middle one, you essentially are taking this data point and making it go further and taking this data point and making it go further. So this one is going to have a higher standard deviation than that one.

Let me put it just like that. I just want to make it very clear: keep track of what's the difference between these two things. Here you have this data point and this data point that was closer in, and then if you move it further, that's going to make your typical distance from the middle more, which is exactly what happened there.

Now, what about this one? Well, this one is starting here and then taking this point and taking this point and moving it closer. So that would make our typical distance from the middle, from the mean, shorter. This would have the smallest standard deviation, and this would have the largest.

Let's do another example. Same idea: order the dot plots from largest standard deviation on the top to smallest standard deviation on the bottom. Pause this video and see if you can figure that out.

So this is interesting because these all have different means. Just eyeballing it, the mean for this first one is right around here. The mean for the second one is right around here at around 10, and the mean for the third one looks like the same mean as this top one. So pause this video. How would you order them?

All right, so just eyeballing it, this middle one right over here, your typical data point seems furthest from the mean. You definitely have, if the mean is here, these data points that are quite far from that mean, and even these data points are at least as far as any of the data points that we have in the top or the bottom one. So I would say this has the largest standard deviation.

If I were to compare between these two, if you think about how you would get the difference between these two, it is if you took this data point and moved it and you moved it to the mean. If you took this data point and you moved it to the mean, you would get this third situation. In this third situation, you have the fewest data points that are sitting away from the mean relative to this one.

So I actually like this ordering: that this top one has a larger standard deviation, and this bottom one has the smallest standard deviation.

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