Standard Deviation
So far, we have discussed two measure of center, mean and median. There are other measures of center in statistics such as the mode. You do not need to know those for the MCAT. All you need to know for measures of center are mean and median. We've also discussed a few measure of spread so far. Range, quartiles and IQR.
So, notice something interesting. Suppose we change just one data point somewhere the list. The mean would change but none of the others change. What's going on with this? One way to think about this, I'm going to use this term, chunking. This is not an official statistics term.
This is just my own term and so, by just dividing the whole population into a bunch of different chunks. So, for example, what the median does, median is the middle of the list, it simply divides the population into an upper half and a lower half. It chunks the data, that's how we find the median. And, in fact, if we change one value somewhere in the upper list or somewhere in the lower list, change one value slightly, the median is not going to change at all.
Whereas the mean feels every data point, it handles every data point. Change any single data point, and the mean changes. Now we think about measures of spread, all the measures of spread we've talked about so far similarly work on chunking. They divide the whole population into chunks, into different buckets, and they'll tell you where the bucket begins and where the bucket ends.
But if a datapoint within the bucket changes, changes its value, these measures of spread are not gonna change. In other words, they don't feel every single data point. And so we can see here that there is something missing what would be the Measure of Spread that actually feels every data point the way that the mean feels every data point?
And so this Measure of Spread, of course, is the Standard Deviation, what we're gonna be discussing in this video. Before we can discuss the idea of standard deviation, we need to discuss this idea of deviation from the mean. If we take any list, and subtract the mean of the list from every single number on the list, we get a new list, which is the list of deviations from the mean.
So for example, take this nice list of integers, it has a mean of 5. Subtract 5 from every number on that list. We get some positive and negative numbers one of the numbers have to be equal to mean. We got a 0 for that entry. Notice all the numbers below the mean have a negative deviation, the numbers above the mean have a positive deviation and numbers equal to the mean has a zero deviation.
Now we would like to have a sense of the typical size of the deviation. Notice that one thing we cannot do is simply take an average of that list of deviation because that average will always be zero. The positives and negatives will always cancel each other out and it will always be exactly zero. So taking an average of it doesn't work.
The standard deviation is a way to measure the size of a typical deviation. In other words, we are asking the question how far away from the mean are the individual data points. How far away is a typical data point from the mean the standard deviation is the representative answer to that question. In some sense, it is the best single answer we could give to that question.
Now the actual technical calculation for standard deviation is a bit complicated We'll talk about that in the next lesson. Here are some rough and ready facts just to give you some intuition for this very important measure of spread. Fact number one, standard deviation can only be positive or zero, it can't be negative.
Fact number two, the only way that the deviation can equal zero is if all the numbers on the list identical to each other. So, here's a sample list. This is a particularly uninteresting list. Every single number equals seven. Of course, the mean equals 7, so the set of deviations would just be a list of zeroes.
THat means that because every deviation is zero, the standard deviation would be zero. Now of course with real data you never gonna have any list. No interesting scientific list is gonna have every single member is the same. And so we never gonna count this in practice of standard deviation is gonna be greater than zero but, this does give us the sense that if all the numbers are very close to one another, we are going to have a very very small standard deviation.
If all the numbers in the list are exactly the same distance from the mean, that distances the standard deviation. So for example, suppose we have this list {2,2,2,8,8,8}. The mean is 5. Every single entry on the list is three units away from the mean, so the standard deviation would equal 3.
Again, we have a list of integers here. You won't be seeing integers when you're dealing with data, but I'll point out what we have here is what's known as a bimodal distribution. We have a group of numbers on the one side of the mean and an equal group of numbers on the other side of the mean. The mean is exactly between, if you have this situation then the standard deviation is the distance, typical distance from the mean and it would be the size from each hump to the mean.
And so, something to look at if you have a roughly symmetrical bimodal distribution this would be the meaning of the standard deviation in that case. A set with most numbers clustered toward extremes will have a higher standard deviation than a list with most numbers equal to or close to the mean. So let's just compare these two sets, they have the same mean and same median. In fact, they're symmetrical, so mean equals median for both of them.
They both have mean and median of 25. They also have the same range. Notice that in set A, eight of the ten numbers equal the mean. And so all of those would have deviations of zero and there are just these two numbers at the end that are extreme values, far away from everything else. Whereas in B, it's all extreme values.
So you have five low values and five high values. In fact, all of them are ten away from the mean and so we see that the standard deviation from b would be ten. It's a bimodal distribution. We can't just calculate the standard deviation just by inspection for a but we certainly know that the standard deviation of a will be much, much less than ten, it will be much, much less than the standard deviation of b.
Because most of the numbers are equal to the mean. If we add or subtract same thing to every number on a list, then the standard deviation doesn't change. Standard deviation is not affected by addition or subtraction. So some examples with integers these three numbers, we have set A, set B we add 40, set C we add 71.
All three of these have the same standard deviation. In fact any six consecutive integers Would have the same standard deviation as these sets. Here's another set, we have Fibonacci numbers here. If we add to this set, add 30 to every number. Then we get another set with the same standard deviation.
This is interesting, if we subtract each number from then we also get the same standard deviation. Notice that here the spacings are in backwards order, here we have spacings of 1,2, 3, 5, 8. Here we have spacings of one, two, three, five, eight so it's a mirror image going down instead of up but it's the same pattern of spacings and in fact, that's exactly what the standard deviation is measuring.
It's measuring that pattern of spacings, just where they are on the number line, how they're spaced on the number line. It doesn't matter if we slide them up or down. That doesn't matter at all. So imagine the numbers as a set of dots on the number line. We can slide that set of dots up or down, or even reflect it.
As long as the spacings between the dots stay the same, the standard deviation stays the same. And so, these again are the three sets that we talked about on the last slide. And just represented as dots on the number line for comparison. And notice that we have spacings one, two, three, five, eight. One, two, three, five, eight.
One, two, three, five, eight, going the opposite order. That's what the standard deviation is measuring. And so first of all, it's important to have this intuition for what it is that the standard deviation is measuring. Also, very practically, sometimes with real data we add something to a list. or we subtract something from a list.
It's important to know that if we're adding or subtracting, we're not changing the standard deviation at all. That's an important idea. Fact six, if we multiply every number on the list by positive number K, standard deviation also gets multiplied by K. So for example, let's start with this list.
Has a mean of seven. We don't know the standard deviation. Let's just call it Q. Now, let's say we take that list and we multiple it by 3. So here's the new list, everything multiplied by 3. The mean gets multiplied by 3, the standard deviation also gets multiplied by 3.
This is very important with unit changes. Think about most unit changes, we're changing feet to centimeters. And we're changing seconds to hours. Or hours to seconds, what we're doing in effect is we're multiplying by some conversion constant. If we multiply the numbers by that conversion constant, we also multiply the standard deviation by that conversion constant.
So that's very, very important to appreciate. If you know the standard deviation in one set of units You can find it in another set of units as long as you know the conversion factor. Here's a practice problem, pause the video and then we'll talk about About this. Okay, so this is probably not something that the MCAT would ask directly, this is getting a little bit into the details of exactly what the standard deviation does.
But if you understand this, you certainly will understand anything that the MCAT does ask you about standard deviation. So we have the original masses The original samples from the lab, the original spec measures in grams, cube grams, and then R degrees Celsius, R degrees Kelvin. And then lab calculates its own data.
In kilograms and in degree Celsius. And we wanna know how old these standard deviations change. Let's think about this. We know that for example one gram equals a thousand kilograms. The least of the gram values will be a thousand times bigger than the least of the kilogram value.
So therefore, Q will be greater than S. We know that degrees Kelvin = degrees Celsius + a constant. It doesn't even matter, for this question, if we know the value of that constant. As it turns out, that value is 273. But for this question, that doesn't even matter. All that matters is We're adding something.
The size of a degree Celsius, the difference of 1 degree Celsius equals a difference of 1 degree Kelvin. So the size of the units is the same We just slide them up or down the number line, that doesn't change standard deviation at all. And so, r = t. And so we go back to the answer choices, and we choose answer choice D.
In summary, Like the mean, and unlike the other measures of center, the standard deviation, feels each number on the list. And so that's very important, change one number, and we change the standard deviation. If all the numbers on a list are identical, and the standard deviation is 0, obviously, that's not gonna happen with real data.
But if they're very close to each, you're gonna have a very small standard deviation. If all the numbers on a list are the same distance from the mean, the standard deviation equals that distance. This is important to keep in mind for bimodal distributions. We have lots of points close to the mean.
We get a small standard deviation, lots of points far from the mean, we get a larger standard deviation. Outliers increase the standard deviation. Addition does not change the standard deviation, addition or subtraction, very important when we're adding or subtracting standard deviation does not change. But if we multiply by a number, multiply the list by the number, we multiply the standard deviation by a number, that's very important in particular for units in conversion factors If we change all the numbers by multiplying by conversion factor, we also apply that same conversion factor to the standard deviation.
Read full transcriptSo, notice something interesting. Suppose we change just one data point somewhere the list. The mean would change but none of the others change. What's going on with this? One way to think about this, I'm going to use this term, chunking. This is not an official statistics term.
This is just my own term and so, by just dividing the whole population into a bunch of different chunks. So, for example, what the median does, median is the middle of the list, it simply divides the population into an upper half and a lower half. It chunks the data, that's how we find the median. And, in fact, if we change one value somewhere in the upper list or somewhere in the lower list, change one value slightly, the median is not going to change at all.
Whereas the mean feels every data point, it handles every data point. Change any single data point, and the mean changes. Now we think about measures of spread, all the measures of spread we've talked about so far similarly work on chunking. They divide the whole population into chunks, into different buckets, and they'll tell you where the bucket begins and where the bucket ends.
But if a datapoint within the bucket changes, changes its value, these measures of spread are not gonna change. In other words, they don't feel every single data point. And so we can see here that there is something missing what would be the Measure of Spread that actually feels every data point the way that the mean feels every data point?
And so this Measure of Spread, of course, is the Standard Deviation, what we're gonna be discussing in this video. Before we can discuss the idea of standard deviation, we need to discuss this idea of deviation from the mean. If we take any list, and subtract the mean of the list from every single number on the list, we get a new list, which is the list of deviations from the mean.
So for example, take this nice list of integers, it has a mean of 5. Subtract 5 from every number on that list. We get some positive and negative numbers one of the numbers have to be equal to mean. We got a 0 for that entry. Notice all the numbers below the mean have a negative deviation, the numbers above the mean have a positive deviation and numbers equal to the mean has a zero deviation.
Now we would like to have a sense of the typical size of the deviation. Notice that one thing we cannot do is simply take an average of that list of deviation because that average will always be zero. The positives and negatives will always cancel each other out and it will always be exactly zero. So taking an average of it doesn't work.
The standard deviation is a way to measure the size of a typical deviation. In other words, we are asking the question how far away from the mean are the individual data points. How far away is a typical data point from the mean the standard deviation is the representative answer to that question. In some sense, it is the best single answer we could give to that question.
Now the actual technical calculation for standard deviation is a bit complicated We'll talk about that in the next lesson. Here are some rough and ready facts just to give you some intuition for this very important measure of spread. Fact number one, standard deviation can only be positive or zero, it can't be negative.
Fact number two, the only way that the deviation can equal zero is if all the numbers on the list identical to each other. So, here's a sample list. This is a particularly uninteresting list. Every single number equals seven. Of course, the mean equals 7, so the set of deviations would just be a list of zeroes.
THat means that because every deviation is zero, the standard deviation would be zero. Now of course with real data you never gonna have any list. No interesting scientific list is gonna have every single member is the same. And so we never gonna count this in practice of standard deviation is gonna be greater than zero but, this does give us the sense that if all the numbers are very close to one another, we are going to have a very very small standard deviation.
If all the numbers in the list are exactly the same distance from the mean, that distances the standard deviation. So for example, suppose we have this list {2,2,2,8,8,8}. The mean is 5. Every single entry on the list is three units away from the mean, so the standard deviation would equal 3.
Again, we have a list of integers here. You won't be seeing integers when you're dealing with data, but I'll point out what we have here is what's known as a bimodal distribution. We have a group of numbers on the one side of the mean and an equal group of numbers on the other side of the mean. The mean is exactly between, if you have this situation then the standard deviation is the distance, typical distance from the mean and it would be the size from each hump to the mean.
And so, something to look at if you have a roughly symmetrical bimodal distribution this would be the meaning of the standard deviation in that case. A set with most numbers clustered toward extremes will have a higher standard deviation than a list with most numbers equal to or close to the mean. So let's just compare these two sets, they have the same mean and same median. In fact, they're symmetrical, so mean equals median for both of them.
They both have mean and median of 25. They also have the same range. Notice that in set A, eight of the ten numbers equal the mean. And so all of those would have deviations of zero and there are just these two numbers at the end that are extreme values, far away from everything else. Whereas in B, it's all extreme values.
So you have five low values and five high values. In fact, all of them are ten away from the mean and so we see that the standard deviation from b would be ten. It's a bimodal distribution. We can't just calculate the standard deviation just by inspection for a but we certainly know that the standard deviation of a will be much, much less than ten, it will be much, much less than the standard deviation of b.
Because most of the numbers are equal to the mean. If we add or subtract same thing to every number on a list, then the standard deviation doesn't change. Standard deviation is not affected by addition or subtraction. So some examples with integers these three numbers, we have set A, set B we add 40, set C we add 71.
All three of these have the same standard deviation. In fact any six consecutive integers Would have the same standard deviation as these sets. Here's another set, we have Fibonacci numbers here. If we add to this set, add 30 to every number. Then we get another set with the same standard deviation.
This is interesting, if we subtract each number from then we also get the same standard deviation. Notice that here the spacings are in backwards order, here we have spacings of 1,2, 3, 5, 8. Here we have spacings of one, two, three, five, eight so it's a mirror image going down instead of up but it's the same pattern of spacings and in fact, that's exactly what the standard deviation is measuring.
It's measuring that pattern of spacings, just where they are on the number line, how they're spaced on the number line. It doesn't matter if we slide them up or down. That doesn't matter at all. So imagine the numbers as a set of dots on the number line. We can slide that set of dots up or down, or even reflect it.
As long as the spacings between the dots stay the same, the standard deviation stays the same. And so, these again are the three sets that we talked about on the last slide. And just represented as dots on the number line for comparison. And notice that we have spacings one, two, three, five, eight. One, two, three, five, eight.
One, two, three, five, eight, going the opposite order. That's what the standard deviation is measuring. And so first of all, it's important to have this intuition for what it is that the standard deviation is measuring. Also, very practically, sometimes with real data we add something to a list. or we subtract something from a list.
It's important to know that if we're adding or subtracting, we're not changing the standard deviation at all. That's an important idea. Fact six, if we multiply every number on the list by positive number K, standard deviation also gets multiplied by K. So for example, let's start with this list.
Has a mean of seven. We don't know the standard deviation. Let's just call it Q. Now, let's say we take that list and we multiple it by 3. So here's the new list, everything multiplied by 3. The mean gets multiplied by 3, the standard deviation also gets multiplied by 3.
This is very important with unit changes. Think about most unit changes, we're changing feet to centimeters. And we're changing seconds to hours. Or hours to seconds, what we're doing in effect is we're multiplying by some conversion constant. If we multiply the numbers by that conversion constant, we also multiply the standard deviation by that conversion constant.
So that's very, very important to appreciate. If you know the standard deviation in one set of units You can find it in another set of units as long as you know the conversion factor. Here's a practice problem, pause the video and then we'll talk about About this. Okay, so this is probably not something that the MCAT would ask directly, this is getting a little bit into the details of exactly what the standard deviation does.
But if you understand this, you certainly will understand anything that the MCAT does ask you about standard deviation. So we have the original masses The original samples from the lab, the original spec measures in grams, cube grams, and then R degrees Celsius, R degrees Kelvin. And then lab calculates its own data.
In kilograms and in degree Celsius. And we wanna know how old these standard deviations change. Let's think about this. We know that for example one gram equals a thousand kilograms. The least of the gram values will be a thousand times bigger than the least of the kilogram value.
So therefore, Q will be greater than S. We know that degrees Kelvin = degrees Celsius + a constant. It doesn't even matter, for this question, if we know the value of that constant. As it turns out, that value is 273. But for this question, that doesn't even matter. All that matters is We're adding something.
The size of a degree Celsius, the difference of 1 degree Celsius equals a difference of 1 degree Kelvin. So the size of the units is the same We just slide them up or down the number line, that doesn't change standard deviation at all. And so, r = t. And so we go back to the answer choices, and we choose answer choice D.
In summary, Like the mean, and unlike the other measures of center, the standard deviation, feels each number on the list. And so that's very important, change one number, and we change the standard deviation. If all the numbers on a list are identical, and the standard deviation is 0, obviously, that's not gonna happen with real data.
But if they're very close to each, you're gonna have a very small standard deviation. If all the numbers on a list are the same distance from the mean, the standard deviation equals that distance. This is important to keep in mind for bimodal distributions. We have lots of points close to the mean.
We get a small standard deviation, lots of points far from the mean, we get a larger standard deviation. Outliers increase the standard deviation. Addition does not change the standard deviation, addition or subtraction, very important when we're adding or subtracting standard deviation does not change. But if we multiply by a number, multiply the list by the number, we multiply the standard deviation by a number, that's very important in particular for units in conversion factors If we change all the numbers by multiplying by conversion factor, we also apply that same conversion factor to the standard deviation.
