بیشتر در میانگین و میانه

More on Mean and Median. In the last video, we noted that changing the highest or lowest number on a list would not change the median, but it would change the mean. Numbers far away from the center of the list are called outliers. The mean is heavily influenced by outliers, and the median is entirely unaffected by outliers.

And this is important. First of all, when are the mean and the median the same? Well, if the list consists of evenly spaced numbers, then the mean equals the median. Consecutive integers and consecutive multiples of the same number are examples of evenly spaced lists.

So for example, we take this list, the mean and the median is 11. Also, mean and median are equal whenever the list is entirely symmetrical. What do I mean by that? Well consider this list. Of course the median is the middle number, 25. Now think about the other numbers.

23 and 27 are both exactly 2 away from 25. 13 and 37 are 12 away from 25. 8 and 42 are 17 away from 25. And 4 and 46 are 21 away. Another way to see that incidentally is that 13 plus 37 is 50, 8 plus 42 is 50, and 4 plus 46 is 50.

All of which are double of 25. And so that means that everything is symmetrical, perfectly symmetrical around 25.

If we put these as dots on a number line, it would be a completely symmetrical pattern of dots. And so, because the entire list is symmetrical around 25, it means that the mean equals the median.

Both are 25. When the list is asymmetrical, then the mean and the median differ. In particular, when there’s a distinct outlier or set of outliers in one direction, those outliers pull the mean away from the median.

So it’s as if the median sits still and the mean gets pulled away from it. And so, here we have a symmetrical list, mean = median = 4.

Now, if we take that last number and make it much, much higher, well the median is still exactly the same, but now the mean is gonna be much, much higher than 4.

If a test question asks you to compare the mean to the median, you don’t necessarily have to calculate the mean. This is very important. Often it’s enough to notice in which direction the most pronounced outliers lie.

The mean follows the outliers. High-value outliers cause the mean to be higher than the median. So here’s a practice question. Pause the video and then we’ll talk about this. Okay, on a test in a class of more than 40 students, the scores had mean = median = mode = 81.

Two absent students then took the test, so 42 took it altogether. And these two students got grades of 83 and 47. What are the new mean and median? Well first of all, notice that those two grades are both symmetrical around the median. So we have lots of numbers at 81, the median is 81.

If we add something higher than 81 and something lower than 81, the median is still gonna be 81. So the median’s not gonna change, so we can eliminate C and D right away.

So the median is still 81, what about the mean? Well, notice that before the students took the test, the mean and the median were both 81.

Well, now we added one grade that was just two above the mean, but then we added a grade that was significantly lower than the mean. So that 47 is an outlier.

And that’s gonna pull the mean down, pull it below the median. And so the mean is not gonna stay the same, and in fact, we’re gonna have a lower mean while the median stays the same, answer choice B.

In summary, if all the numbers on a list are evenly spaced, or if the list is symmetrically distributed, then the mean = median.

Outliers pull the mean away from the median. And we can often compare mean and median, or infer which one got bigger or smaller without doing a calculation, purely by observing the direction of the outliers