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Mean, Median, Mode
Now we can start talking about statistics. In the big picture, statistics is a rather broad subject, but we only have to know a few things for the test. For the purposes of the test, statistics consist of tools, for making sense of data. So we’ll get some data and just be asked about that, this data.
So one of the most fundamental questions we can ask about a data set is, what single number is most representative of the set as a whole? And this turns out to be very important in real world statistics. For example, if we had a list of the household income of everyone in the United States, we’d wanna know, what’s one single number that would be representative, most representative of that entire list.
Well, such numbers are called measures of center, and that’s what we’re going to start talking about in this video. The two most important measures of center are mean and the median. A mean is, is, is simply an ordinary average. To find the mean, say, of seven numbers on a list, you would add up the seven numbers, and then divide this sum by seven.
In general, on a list with N entries, we add up all the entries, and then divide by N that’s the mean. Just an ordinary average. We can write the formula as mean, equals the sum of N entries divided by N. That’s the formula for the mean. Notice that this formula is also quite useful in the following form.
If we just multiply both sides by N, we get N times the mean. In other words, the number of people on the list times the mean has to equal the sum of the entries. Thinking about sums is often the key to many questions about average or mean. So this second form, it’s really underappreciated how powerful this form is.
For example here’s a practice question. Pause the video and then we’ll talk about this. Okay in a class, 18 students took a test and had an average of seven, 70. Alicia and Burt then took the test and the average of all 20 students was 71. If Alicia got a 77, then what was Burt’s grade? All right, so a lot of people would find this a very hard question.
The key to this question is simply thinking about the sums. So first of all, the old sum, those 18 people. I’m just going to take 18 times the mean of 70 multiply that out and that’s 1260. Well now the new sum of all 20 students, that’s going to be 20 times 71 multiply that out that’s 1420. Well think about that, the old sum and the new sum.
What’s the difference between them? The only difference is we added Alicia’s score and Burt’s score to the other 18 scores. So the difference between them should be the sum of the score of Alicia and the score of Burt. So we subtract them that sum.
So Alicia’s score and Burt’s score must add up to 160. Well, if we subtract Alicia’s score of 77. Then we should get, we get 83, and of course that has to be Burt’s grade. So that’s how we answer that question, purely thinking about sums. Now the median. The median is the middle number on a list.
We have to be a little careful here. We have to put the list in ascending order first, that is from smallest to biggest. Technically, the median is the middle number on an ordered list. So, we can’t just write them in any jumbled order and then say. Okay, the median is the one that happened to be sitting in the middle of that jumble.
No, we have to put the list in order from smallest to biggest. And often, incidentally, the test will not do that. They’ll give you the numbers in a jumbled order, and then you yourself have to put them in order, and then find the median. So you have to be careful with that. So if we have this list, for example, the number right smack dab in the middle is 4.
So, that’s the median. There are three numbers below it, three numbers above it. Clearly in the middle. Now this list is a little bit interesting, because there’s an even number of terms on the list, not an odd number. So there’s no one number at the middle.
At the middle, we have these two numbers, four and five, and the can’t both be the median. So what we do, if there’s an even number on a list, we average the two middle numbers. And so the median is between four and five, we take the average of four and five which is 4.5.
That is the median. But if we’re given this list, well, the very first thing we have to do, is put that list in order. So now it’s in order. So now the median, there are 1, 2, 3, 4, 5, 6, 7, 8 numbers on this list.
And so, the median has to be between eight and 13. So we’re going to average eight and 13. 8 plus 13 divided by 2, 21 divided by 2, 10.5. 10.5 is the median of list K. Notice that the median only takes into account the number or numbers at the very center.
We could change the numbers at either end of the list. And this change wouldn’t affect the median at all. So for example, in that particular list, suppose we changed the 55 to 55,000. The median wouldn’t change at all. The mean would change, but not the median. We’ll talk about that in the next video.
Here’s a practice problem. Pause the video and then we’ll talk about this. Okay. The median of list B, is exactly four higher than the median of list A. All right, so the very first thing we need to do is find the median of list A, we put the terms in order.
The median is the average of 7 and 10 which is 8.5, that’s the median of list A. The median of list B has to be four higher than this, so it has to be 12.5, that’s 8.5 plus 4. All right, so now let’s think about this. There are three numbers that are less than that median. Four, seven and ten are all less than the median.
18, 25 and x must be greater than the median. And in fact, that 12.5 must be the average of 10 and x. So, 12.5 is the average of 10 and x. When multiplied by 2, what we get is 12.5 times 2 is 25, subtract 10, we get x equals 15.
And so, that’s the value of x. If x has a value of 15, then set B would have a median of 12.5, which is exactly 4 higher than 8.5. One final measure of center is the mode. You often her, hear these three said together. Mean, median, and mode.
What’s the mode? The mode is the most frequently appearing number on the list. The number that makes the greatest number of appearances. This is far less important than either the mean or the median, for a variety of reasons. Some lists have a single mode, so for example, there are a lot of 3s on this list.
There are three 3s and every number only appears, every other number only appears once. So the mode is clearly three. Some lists have two modes. For example, in this list, we have a pair of 2s and also a pair of 5s. So the modes are two and five.
If all the numbers on the list are different from one another, as is usually the case. Then there simply is no mode. So sometimes there is a mode, sometimes there’s more than one, and often there’s just no mode at all. So think about this.
Every single list on the planet has a mean. Every single list on the planet has a median. But, only some lists have modes. Some lists have more than one mode, and many have no mode at all. So this is one of the big reasons why the mode is not nearly as important as the mean or median.
In summary, the mean is the simple average often in questions about mean, it’s helpful to think in terms of the sums. So the sum of the entry equals the number of entries times the mean. We just rewrite that formula on the top to get this. The median is the middle number of an ordered list. If two numbers are in the middle, then we simply average those numbers.
And the mode, the most frequently appearing number, is just less important. Some lists have, have a mode. Some have more than one, and many have no mode at all. Theoretically the tests could ask you about a mode, but it’s much, much less common than, than questions about mean or median.
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