Problem

$$\begin{array}{|c|c|}\hline Score & {Frequency} …

02:29

Problem 24 Hard Difficulty

$$\begin{array}{|c|c|}\hline Score & {Frequency} \\ \hline 90 & {3} \\ \hline 85 & {2} \\ \hline 80 & {3} \\ \hline 75 & {7} \\ \hline 70 & {6} \\ \hline 65 & {4} \\ \hline\end{array}$$
Find the variance and standard deviation of the scores.

Answer

59.8, 7.7

Related Courses

Intro Stats / AP Statistics

Algebra

Algebra 2

Chapter 12

Probability and Statistics

Section 6

Statistical Measures

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Video Transcript

So for this problem and a few others after this problem, you're going to be needing to look at that frequency table. So frequency means how many times it occurs. So that means there were three different nineties. There were 2 85 3 eighties and so on. And so if we're asked to find the variance Nissinen deviation of these values, we're gonna have to still find the mean first. So if there were 3 to 376 and four that those are the numbers we need to add up to make up our denominator. So we're gonna add up Three plus two plus three plus seven plus six plus four. That's 25. So there were 25 total tests of people or whatever the situation was from the problem. Okay, so now we're ready to add up all its members. Now, the easy way to do this is to kind of think of it like this If there's 3 19 3 times 90 plus two times 85 plus three times 80 and I can help save you some time. And you calculated from adding up all of those numbers so three times 90 plus two times 85 plus three times 80 plus seven times 75 plus six times 70 plus four times 65 gets us 18 baby five. I am ready to divide that by 25 70. Look, 75.4. Okay, So in order to get the variants, we're gonna have to use all of the all three nineties have to be subtracted from the 75.4 and squared. Um, but here's thing. If you go and do that calculation for one of these for each of these, then and then square it, you would basically be ableto then multiply by three multiplied by two and so on. So you're not necessarily repeating at all these steps in your calculator if you can keep track it like that. So we're going to keep going with that same momentum. So Step two is going to be fighting the variance. So what I'm gonna dio is I'm gonna do one of each. I'm gonna dio 90 minus 75.4 and then remember, we're still squaring to get when we do our variants. But then when we're done with that week in times that by three and use that as part of our final answer in the end or final steps of 90 minus 75.4 and then we square it and we get 213 point 16 So that means there's gonna be another to 13 points 16 and yet another two. 13.16 All right, so now the nineties or done Now move on to the 85. So we have a five minus 70 five point for and then square. So 85 minus 75 point or and then squared is 92 0.16. And since there were two of those were going to start another 92.16 down. Now there were three eighties, so we're gonna dio 80 minus 75.4 Army Square A T minus 75.4 squared is 21 0.16. And so there were three of those. So another way to think of that is to just times that by three. But Sri kid do that now or later. There's not really a huge deal or difference, So I'm gonna just do that Now we got 63 point for eight. So later on, we're gonna have to be adding up all the numbers of Catholic it most at surgery 80.48 later and old kind of have to go for 17 many ways to their seven of those That be a lot of writing that we don't stealing you to dio the 75 minus 75.4 squared. So, you know, high by 75.4 is 0.4 square yet 0.16. So since they were setting of those, we're gonna times 0.16 times seven and we get 1.1 to So we'll use that number later when you need it there. Six seventies. So we're gonna dio 70 minus 75.4. I am very square that they're 70 minus 70 car point for squared. It's us 29 0.16 And because there were six seventies, we're gonna have to multiply this by six 1 74 0.96 Last one is 65 therefore of those, So we're gonna dio 65 minus 75.4 squared and that's 108 books 0.16 times four gets us 4 32 0.64 So recall that variance is all adding it all the times that we just calculated and taking the average of it. So we're going to need to do to 13 plus 2 13 plus 2 13 plus. It was to 13.16 to 13.16 plus 2 13.16 plus 2 13.16 plus 92.16 plus 92 0.16 plus. Now we already did this math. So we want the 63.48 getting at it since very did that. So just be careful that you don't over over do that. So then 63.48 plus the 1.12 plus the 1 74.96 plus the 4 32.64 So we end up getting our variance is going to be 14 90 six, divided by the number that was in the list. So we determined earlier that the number the list was 25. So divide by 25 when you get 59 0.84 Exactly, actually, so we don't need the swiftly equals approximation its attacks equal. And then the square root of 59.84 will get us our standard deviation. So senior aviation is gonna be about seven point seven or 7.74

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