-
- "Ana played five rounds of golf
-
"and her lowest score was an 80.
-
" The scores of the first four rounds and the lowest round
-
"are shown in the following dot plot."
-
And we see it right over here.
-
The lowest round she scores an 80,
-
she also scores a 90 once, a 92 once,
-
a 94 once, and a 96 once.
-
"It was discovered that Ana broke some rules when she scored
-
"80, so that score", so I guess cheating didn't help her,
-
"so that score will be removed from the data set."
-
So they removed that 80 right over there.
-
We're just left with the scores from the other four rounds.
-
"How will the removal of the lowest round
-
"affect the mean and the median?"
-
So let's actually think about the median first.
-
So the median is the middle number.
-
So over here when you had five data points
-
the middle data point is gonna be the one
-
that has two to the left and two to the right.
-
So the median up here is going to be 92.
-
The median up there is 92.
-
And what's the median once you remove this?
-
Now you only have four data points.
-
When you're trying to find the median
-
of an even number of numbers
-
you look at the middle two numbers.
-
So that's a 92 and a 94.
-
And then you take the average of them.
-
You go halfway between them to figure out the median.
-
So the median here is going to be,
-
let me do that a little bit clearer.
-
The median over here is going to be
-
halfway between 92 and 94 which is 93.
-
So the median, the median is 93.
-
Median is 93.
-
So removing the lowest data point
-
in this case increased the median.
-
So the median, let me write it down here.
-
So the median increased by a little bit.
-
The median increases.
-
Now what's going to happen to the mean?
-
What's going to happen to the mean?
-
Well one way to think about it without having to do
-
any calculations is if you remove a number
-
that is lower than the mean, lower than the existing mean,
-
and I haven't calculated what the existing mean is,
-
but if you remove that the mean is going to go up.
-
The mean is going to go up.
-
So hopefully that gives you some intuition.
-
If you removed a number that's larger than the mean
-
your mean is, your mean is going to go down
-
cause you don't have that large number anymore.
-
If you remove a number that's lower than the mean,
-
well you take that out, you don't have that small number
-
bringing the average down and so the mean will go up.
-
But let's verify it mathematically.
-
So let's calculate the mean over here.
-
So we're gonna add 80, plus 90,
-
plus 92, plus 94, plus 96.
-
Those are our data points.
-
And that gets us: two plus four is six, plus six is 12.
-
And then we have one plus eight is nine, and this is,
-
so these are nine and then you have another nine,
-
another nine, another nine, another nine.
-
You essentially have, this is five nines right over here.
-
So this is going to be 452.
-
So that's the sum of the scores of these five rounds,
-
and then you divide it by the number of rounds you have.
-
So it would be 452 divided by five.
-
So 452 divided by five
-
is going to give us, five goes into,
-
it doesn't go into four, it goes into 45 nine times.
-
Nine times five is 45, you subtract, get zero,
-
bring down the two.
-
Five goes into two zero times, zero times five is,
-
zero times five is zero, subtract.
-
You have two left over, so you can say that the mean here,
-
the mean here is 90 and 2/5.
-
Not nine and 2/5, 90 and 2/5.
-
So the mean is right around here.
-
So that's the mean of these data points right over there.
-
And if you remove it what is the mean going to be?
-
So here we're just going to take our 90, plus our 92,
-
plus our 94, plus our 96, add 'em together.
-
So let's see, two plus four plus six is 12.
-
And then you add these together you're gonna get 37.
-
372 divided by four, cause I have
-
four data points now, not five.
-
Four goes into, let me do this
-
in a place where you can see it.
-
So four goes into 372,
-
goes into 37 nine times.
-
Nine times four is 36, subtract, you get a one.
-
Bring down the two, it goes exactly three times.
-
Three times four is 12.
-
You have no remainder.
-
So the median and the mean here
-
are both, so this is also the mean.
-
The mean here is also 93.
-
So you see that the median,
-
the median went from 92 to 93, it increased.
-
The mean went from 90 and 2/5 to 93.
-
So the mean increased by more than the median.
-
They both increased but the mean increased by more.
-
And it makes sense cause this number was way,
-
way below all of these over here.
-
So you could imagine if you take this out
-
the mean should increase by a good amount.
-
But let's see which of these choices
-
are what we just described.
-
"Both the mean and the median will decrease", nope.
-
"Both the mean and the median will decrease", nope.
-
"Both the mean and the median will increase,
-
"but the mean will increase by more than the median."
-
That's exactly, that's exactly, what happened.
-
The mean went from 90 and 2/5 or 90.4,
-
went from 90.4 or 90 and 2/5 to 93.
-
And then the median only increased by one.
-
So this is the right answer.