Explain mean greater than the median

GarIT

I've been looking at the 2010 Maths SEC sample paper Q2(a). It shows a histogram labelled A that is skewed right. In the answers it shows that the mean is greater than the median but I just don't get it. I thought if the mean was greater than the median it should be to the right of the median.

Actually is the mean in a histogram in relation to the x or y axis?

ehshup

think about it like this: the median is the middle value ie half are above and half are below, so say if you have a perfectly symmetrical distribution for example of wages, with mean and median of 40k, then you add in another data point very far to the right making it positively skewed, like maybe two people got over 1 million. Well this would suddenly make the mean bigger, however as it's only two people the midway point would still be roughly in the same place... does that make any sense lol? however if you remember that median is always in between mode and mean, that might help (apart from when they're all equal)

emmamurphy233

When graphs are positively skewed (i.e the one in that question) the mean>median>mode ALWAYS. For negatively skewed ones the opposite is true (i.e mean<median<mode). Hope that helps.

GarIT

I got full marks in the mocks in stats and I can do all the exam questions except this one. Its so frustrating. It seems to be backwards in my head. Say if i have LC points on the x axis and number of students on the y axis, is the median 300 points or is it the number of people that got 300 points. And what is the average i would have thought it would be every person x their points added together and then devided by the number of people.

GarIT

I'm getting even more confused by the minute now. I hate histograms. Just tell me what the median is to me and I can work from there. I think the mode is the value on the x axis of the largest y value. So say if more people get 350 points than anything else then 350 is the mode. So then is the median the amount of points that the middle amount of people got or just the middle x value.

emmamurphy233

Woah. Now you've confused me too. :P If you have the texts and texts strand 1 book it's all explained in that?

mcpaddington

GarIT wrote: »

I'm getting even more confused by the minute now. I hate histograms. Just tell me what the median is to me and I can work from there. I think the mode is the value on the x axis of the largest y value. So say if more people get 350 points than anything else then 350 is the mode. So then is the median the amount of points that the middle amount of people got or just the middle x value.

The most frequent of a single piece of data basically.

i.e in this: 1,2,3,4,5,5,5,8,9
5 is the median


For any graf the mode will always be the highest point, the mean will be the lowest and the median will always be in the middle.

cocopopsxx

I dunno if this will help you or not but it does help me cos I find this confusing too. My teacher told us this-

Mean always follows the tail, mode is the most popular ( I.e. highest value on y axis) and median is always between mean and mode.

Basically,For a positive skew (skewed to the right) mean>median>mode and for negative skew (skewed to left) mean<median<mode always.

Hope I didn't confuse you even more.

GarIT

This is freaking me out cause I just did out a negatively skewed one and it worked out that mode>mean>median. Just say in the LC 100 people got 100 points 200 people got 300 points and 50,000 got 600 points. The mode would be 600, median would be 300 because its in the middle and the average(mean) would be 599 points. So in that case mode>mean>median.

Wesc.

GarIT wrote: »

This is freaking me out cause I just did out a negatively skewed one and it worked out that mode>mean>median. Just say in the LC 100 people got 100 points 200 people got 300 points and 50,000 got 600 points. The mode would be 600, median would be 300 because its in the middle and the average(mean) would be 599 points. So in that case mode>mean>median.

Would the median not be 600?

What I do is just imagine spreading out all the data. So you've them all side by side.. there'll be a massive amount 600's so in the middle it will be 600 which is the median.

FaoiSin

Means the lowest. Modes the highest. Medians in between.

I haven't the brain capacity to know why but the examiner won't care

GarIT

I understand it now, my. Notes are wrong. I thought if I had 1,2,3,3,3 2 was the mean because it would be in the middle of the histogram, but when you write out each number on its own 3 is in the middle.

Wesc.

GarIT wrote: »

I understand it now, my. Notes are wrong. I thought if I had 1,2,3,3,3 2 was the mean because it would be in the middle of the histogram, but when you write out each number on its own 3 is in the middle.

If it helps, this is the formula we were given for the median:

Median = (n+1)/2 where n is the sample size.

FaoiSin

Wesc. wrote: »

If it helps, this is the formula we were given for the median:

Median = (n+1)/2 where n is the sample size.

Is the median not the middle term when the list is in order?

GarIT

Wesc. wrote: »

Would the median not be 600?

What I do is just imagine spreading out all the data. So you've them all side by side.. there'll be a massive amount 600's so in the middle it will be 600 which is the median.

Yeah your right, stress, tiredness and bad note taking got to me. I just wrote down the number in the middle is the median, so when I read it again I took it as the middle number along the bottom. I understand now, thanks.

Wesc.

YeaaahFaoiSin wrote: »

Is the median not the middle term when the list is in order?

I think that formula still applies.. If there was an odd set of numbers then you add the middle and the number after the middle and divide by 2.
But yeah my teacher says in all data sets that applies so I'll take his word for it, did some research online to make sure and it's right

GarIT

Wesc. wrote: »

If it helps, this is the formula we were given for the median:

Median = (n+1)/2 where n is the sample size.

That isn't the median. That is how many numbers in the median is. So say if you used that and got 5, 5 isn't the median, 5 is the number you count in to find the median.

Say if I have 1 2 3 4 6 6 7 8 9 10, that formula will give 5.5 as the median, but 6 is the median, you count in 5.5. Its just as easy to count the whole thing than go in halfway.

MathsManiac

If you're a visual thinker, this might help relate the mode, mean and median to the picture of the distribution:

• The mode is the (x-coordinate of the) highest point. If there are two or more (locally) highest points, these are all modes.
  
• If you draw a vertical line through the median, half of the area is to the left of it and half is to the right.
  
• If you imagine trying to balance the graph on top of a needle placed somewhere on the x-axis, the mean is the place where it will balance.

Now, similar to what ehshup said: imagine a symmetric distribution; the mean and the median are both in the middle. Now imagine adding two extra things - one on each side. One of the things you put in is close the middle, but the one you put on the other side is very far away . The median stays where it is, because you still have the same area on each side. But you can see that the new thing far away will tilt the scales more than the one close by, so you'd have to move the needle across a bit in that direction to keep the balance.

This is why in a skewed distribution, the mean tends to get pulled towards the long tail more than the median does.

(By the way, the median isn't ALWAYS between the mean and the mode - it just USUALLY is. You have to pick a fairly peculiar distribution for it not to be true.)

emmamurphy233

GarIT wrote: »

This is freaking me out cause I just did out a negatively skewed one and it worked out that mode>mean>median. Just say in the LC 100 people got 100 points 200 people got 300 points and 50,000 got 600 points. The mode would be 600, median would be 300 because its in the middle and the average(mean) would be 599 points. So in that case mode>mean>median.

That would be a negatively skewed graph. So yes of course it would be Mean<median<mode.

finality

YeaaahFaoiSin wrote: »

Is the median not the middle term when the list is in order?

They're the same. Just think, if you have 4 numbers

3569

what's the median? You can't get it by looking at it, instead it will be (4+1)/2 =2.5, so you add the second and third numbers and divide by 2, meaning the median is 5.5

But if you have 5 numbers,

23579

5 is in the middle, and (5+1)/2 = 3, so the third number is the median.

basically if you have an even number of numbers, the median may not be one of the values so you need (n+1)/2

MathsManiac

The formula (n+1)/2 does NOT give the median. It gives the position of the median in an ordered list.

If the numbers are, say: 23, 25, 27, 28, 28, 30, 31. Then n is 7. The formula tells you that the median is in position (7+1)/2 = 4. i.e., the median is the fourth number: 28.

(n+1)/2 is just a formula (as if you needed one!) that tells you how far you have to count into a list of n numbers to get to the middle one.

Wesc.

Sorry guys I should have made that more clear.