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Name: find a counterexample for the following statement the standard deviation of a set of data is always
Uploaded: 2020-06-03
Duration: 1 min 13 s
Channel: Numerade
Description: Find a counterexample for the following statement. The standard deviation of a set of data is always less than the variance.

Question

Find a counterexample for the following statement. The standard deviation of a set of data is always less than the variance.

Trinity Steen · Accepted Answer

already. So for number 35 we are asked to figure out, um, a way to disprove the standard deviation is always bigger than the variance. Well, if we think back to just the previous problem and number 34 we were able to figure out that it is possible for the standard deviation and the various two equals zero. So if they each equal zero zero, is not, ah, bigger than zero, that is a false statement. So that would be an example there. Now there&#x27;s plenty others. This is one of those answers. May very situations it if you find a decimal. Usually it&#x27;s a number that&#x27;s less than one, um, as your ah center deviation, you take the square root of it. Sometimes your variants will be bigger. So, um, that will also be the case is well

Trinity Steen · Answer

Okay, So for number 34 this is one of those answers May very kind of things in there several different ways. You can go about this and choose this, and it&#x27;s actually pretty straightforward. So you&#x27;re trying to figure out a Senate aviation variants that both equal zero. So if you&#x27;re variants equals zero, the square root of zero is zero. So our main focus is going to be trying to be able to figure out how to work backwards and get a variance that&#x27;s gonna lead to a zero. And it&#x27;s actually pretty straightforward. So I&#x27;m going to do one example. But once you use that you see my table, you can come up with your own relatively straightforward. So basically, you would need to have a list of the same numbers. That is one way you can go about this. So what I mean by that is say, I want a list of lives. Well, here&#x27;s how that would work. If I get a list of five, I need a have five of those in my list because what will then end up happening is if I go and I get my bar, I dio ah five plus five plus five plus five is 25 divided five. You know it&#x27;s five. We&#x27;ll hear a call from previous problems to get your variants. You take all the numbers in your list and you&#x27;re subtracting them. So five minus the mean is going to get me zero. And that&#x27;s what zero squared is still zero. So we&#x27;ve made our goal. So this would also have worked because if I keep going and keep going, I&#x27;m gonna get 0.0 and we all know that still zero, So that is going to be awesome. So if I did a list of four force that could work as well. And so basically, you just do however many numbers of whatever number you pick, so you pick 10 you&#x27;d have 10 tens in your list, and that will still get you your Senate deviation and variation, both being equal to zero

Abigail Carden · Answer

so he were given a mean of 34 in a standard deviation of 3.2. And we want to decide which of the following statements are true or not true. So we&#x27;re gonna start by labeling or curve. We have our mean off 44. At our first inter deviations, we have 40.8 and 47.2. At our second standard deviations, we have 37.6 and 50.4. So now at our percentages, we have 34 percent from the mean to the first inter deviation on either side from the first inter deviation to the second is equal to 13.5% of data and on either side. Beyond the second center deviation is equal to 2.5%. So for our first statement claims that between 37.6 and 5 50.4 it&#x27;s 68% of data. This statement is not true. It actually encompasses 95%. We can find that by adding up 13.5 34 plus 34 post 13.5 again and that gives us 95%. The second statement claims that 13.5% of values are less than 40.8% and that&#x27;s not true again. It actually encompasses 16% and we can find that by adding 13.5 and 2.5%. The last statement that 5% of the values are lower than 37.6 or beyond higher than 50.4 is true and that&#x27;s because that would be 2.5 plus 2.5 and that we give us 5%. So our answer is a that statement&#x27;s one and two are not true.

Aaron Russell · Answer

Hi there. The relationship between variance and standard deviation is that the standard deviation equals the square root of the variants. So using symbols ah, sigma squared, that&#x27;s the variance is equal to the standard deviation.

Shiksha Dutta · Answer

let us now talk about standard deviation on its relationship with variants, so standard deviation is equal to I don&#x27;t under video, since the standard deviation is a measure off. The variation from the I mean the size off. The standard deviation indicates how spread out Dita are from the I mean then.

Bryan Lynn · Answer

there&#x27;s a human to show that the variants okay of two events is not necessarily equal toothy some of their independent variances. Okay, so take, for example, a coin flip s o. The probability of getting heads is going to be equal to the probability of getting a tales, which is 1/2 right. And so we get that the probability of getting a heads plus the probability of getting in tails is going to be the one. It&#x27;s 1/2 plus 1/2 so we consider the expected values. Okay, so the expected value of getting a heads would be equal to the some of the part of each possibility with its probability, right? So it&#x27;s gonna be equal to the some of each possibility multiplied with its probability. Okay, so that&#x27;s going to end up being either you didn&#x27;t get it with a probability of 1/2 or you got your heads with a probability of 1/2 a sequel to 1/2. And by these axiom argument, you will see that this is the expected value of getting a tales as well. Okay. And the expected value of both events would then be equal to DEA. You know they used the exact same definition, but with it by adding them together. Uh, this is gonna be one times one is equal to one. And if we look at the expected value of a square for experience, this is gonna help us set up calculating our variances. Right? Because by definition, the variance axe is equal to the expected value of X squared, my ass. The expected value. Lex squared. Okay, so in company this and this is basically the same appear Experiment Square rx. Right? So this guy&#x27;s being equal to it&#x27;s your A square times 1/2 plus one squared times 1/2 this equal to 1/2 d&#x27;oh and is again by the same argument. See if this is equal to the expected value of getting a tales squared. Now, you want to consider the expected value of getting the heads and tails squared? Okay. Am I the same argument? It&#x27;s gonna be the same as appear, except with a where we are squaring that night as well. And so then we get one square times. One is equal to one. Okay, we have we have all these values now on that we fail. We can put them in, too. The variance property that were given. All right, so we know that Variants X is equal to the expected value of X squared minus the expected value. Lex squared. And from our previous page, you see that the expected value of what the H is here? We had value each 1/2 and they spent the values. A square was also in half. Okay, so when we want to calculate the variance of, hence it&#x27;s going to be equal to 1/2 minus 1/2 squared sequel to 1/4 and buy the exact same argument. You&#x27;ll see it. This is the same value for the variance of tales. Okay, listen, that will look at the variance of heads and tails, and so it&#x27;s gonna be equal to the expected value of our heads and tails squared minus the expected value ever has entails and square the whole thing. Okay. And so from the previous page, we see that this is gonna be equal to one minus one square is equal to zero. Okay, well, we can see you here then, that if we take the variance of heads first, the variance of tales which were both. 1/4 will give us one quarter plus 1/4 is equal to 1/2. But this does not equal zero, which would have been our variants of our heads and tails. Right, So? So these do not match. And so therefore they are this expression idea of the variance of heads Experience of tales be equal to D Variance of heads Close tails while may sometimes be true is not always true.

Problem

Consider the two sets of data. A = {1, 2, 2, 2, …

Problem 35 Medium Difficulty

Find a counterexample for the following statement. The standard deviation of a set of data is always less than the variance.

Answer

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