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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 3

Lesson 3: Measuring variability in quantitative data- Interquartile range (IQR)
- Interquartile range (IQR)
- Sample variance
- Sample standard deviation and bias
- Sample standard deviation
- Visually assessing standard deviation
- Visually assess standard deviation
- Mean and standard deviation versus median and IQR

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# Visually assessing standard deviation

Worked examples visually assessing the standard distribution.

## Want to join the conversation?

- You have got to be joking(6 votes)
- so is this like the IQR or nah

please help me(3 votes)- No, standard deviation is not the same as IQR.(4 votes)

- This is weird but I wanted to fully grasp what variance meant like I know it's SD squared and it shows the variability of the graph but I still don't know how it came to be. Please help me with that(1 vote)
- Think about it: remember how the standard deviation squared is the sum of all the points minus the mean squared? Since it is squared, there is no negative numbers, and only the distance from the mean matters on the value of the standard deviation. Therefore, if the distance between points and the mean is large, since it is squared, the standard deviation squared would be larger, and if the distance between points and the mean is small, the standard deviation squared would be smaller. After square rooting it, the standard deviation is still large or small based on the distance between the points.(4 votes)

- If the standard deviation is the typical distance from each of the data points to the mean, then what is the variance?(2 votes)
- The variance is the standard deviation squared.(1 vote)

- I thought that the middle number was called the median and not mean, is that not the case here?(0 votes)
- Middle number of all the data points is called median (the middle number of every number in the range in not median) and the average is called mean, We find the distance from the points and mean. In this example, coincidentally the mean is in the middle of the whole range(5 votes)

- or do you just see wich 1s furthest apart(1 vote)
- how does standard deviation help us measure the variability or spread of data?(1 vote)

## Video transcript

- [Instructor] 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 and 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 and our smallest standard deviation would be the ones where it feels like, on average, our data points
are closer to the mean. Well, 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. And 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, and 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 and so this one is going to
have a higher standard deviation than that one, so let me
put it just like that. And 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
and so that would make our typical distance from the middle, from the mean, shorter, so this would have the
smallest standard deviation and this would have the largest. Let's do another example. So, 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, it looks like the same
mean as this top one. And so, pause this video. How would you order them? All right, so just eyeballing it, these, this middle one right over here, your typical data point seems furthest from the mean, you definitely have, if the mean is here,
you have this data point and this data point that are quite far from that mean, and even this data point and this data point 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 and if I were to compare
between these two, if you think about how you
would get the difference between these two, is if
you took this data point and moved it to the mean and
if you took this data point and you moved it to the mean, you would get this third situation. And so, this third situation
you have the fewest data points that are sitting away from the mean relative to this one. And so, I actually like this ordering that this top one has the
largest standard deviation and this bottom one has the
smallest standard deviation.