We polled 300 Art of Problem Solving students
about what facial skills they have. 111 can wiggle both of their ears, while 57
can wiggle just one ear. 89 of them can raise one eyebrow without raising
the other, while 123 can flare their nostrils. 137 can curl their tongue. 274 – almost all of them – can make fish lips,
while only 5 of them can do that. Very special skill. Now, imagine I ask you to order these from
the most common skill to the least common skill. Well, you could do it with this, but you’re
not going to be happy about it. If you see a paragraph like this in a book
you’re reading, I know what you’re going to do. You’re going to skip it, because that’s painful
to look at. Hard to pick out the numbers, hard to pick
out the data. You want to see something that maybe looks
a little bit more like this. This is a table where we have one row for
each skill. You can pick out the skills really easily. You can pick out the numbers really easily. So I can look through here. 274, that’s the highest. Fish lips is the most common. 5, that’s the lowest. Weird lips, that’s the least common. But, you know, if all we really care about
is the order, don’t care about the exact numbers here, there’s an even faster way to look at
these data. We can do this. This is a bar chart, or you call it a bar
graph. Here’s one bar for each skill. And we can just glance at this and see immediately:
fish lips, most common, most students can do that. Weird lips, that’s the hardest, almost no
one can do that. You have to be a real freak to be able to
do that. Now, you can also compare all . . .any two
of these. You can quickly see, curl tongue, that’s easier
than wiggle one ear. Flare nostrils, that’s easier than wiggle
both ears. We can put them in order, just at a glance. Tallest bar, fish lips. Next is curl tongue. Next is flare nostrils, next is wiggle both
ears, then raise one eyebrow, then wiggle one ear, and then last, the hardest, weird
lips. So the bar chart here allows us to compare
pairs very easily. We can see the relative comparison, too. See? Fish lips is about, you know, on the order
of twice as common as these. And it’s just way, way easier than the weird
lips. So the bar chart gives us a nice graphical
quick-glance way to compare different skills to each other. Let’s take a look at another type of chart. Here. We had another poll of all our students and
we asked them, what’s your favorite subject? Remember, this is Art of Problem Solving students,
so there is some selection bias here. 56 percent said math. And 24 percent said science, and math is the
language of science, so that’s okay by us. 7 percent said anything but history (I’m sure
math was second there, too, so that’s okay as well). 11 percent is English. I can forgive those students; I had some really
good English teachers, too. 2 percent said speech because, well, everybody
hates speech, well, except for me. You know I love to run my mouth. So this was the result of our poll. And we draw this nice little pie chart here
where each slice of the pie is the size relative to how many people gave that answer. So 56 percent said math, slightly more than
half the pie is the math slice. 24 percent said science, slightly less than
a quarter of the pie is science. Then same for these. You know, speech, almost no one said speech. It’s a little tiny sliver. So I can tell just by looking at this graph
what portion of the whole – without even looking closely at the numbers – I can tell at a glance
a little more than half the people like math, around a quarter of them like science, almost
nobody likes speech. And that’s what a pie chart is good for, is
revealing parts of a whole, and that’s why we’re not going to use a pie chart back here. These different skills don’t, all together,
make up a whole. Back here, each student is going to fall into
one — exactly one — of these slots. And all together, when we add all these up,
we’ll get all of the results, all of our answers, to our poll. Whereas over here, some students, some people
are going to be in one or two of these, some of them are going to be none. They’re going to be no skills at all, can’t
do any of them. And then there are going to be total freaks
that can do all seven of them. So, we can’t put these all together and say
that these are all parts of some whole because some people are in a bunch of these slots, whereas here, each person is in exactly one
slot. Only one slot. So we can use a pie chart when we want to
display parts of a whole. And that’s what we’re doing here. Let’s go ahead and tackle a quick problem
using the data here. Imagine 21 [students] said, “Anything but
history.” How many chose science? Now, we can tackle this problem in a bunch
of different ways. First, well, 21 is 7 percent of everybody. So we could just say, x is the total and 7
percent of x has to be 21. So we get an equation. 0.07x equals 21. Divide both sides by 0.07, 21 over 0.07 and
. . .you divide 7 into 21, you get 3, you’ve got the decimal place, you deal with that,
it’ll come out to be 300. So once you know that there are 300 total,
well, 24 percent of them said science, so 24 percent of 300 is 0.24 times 300. 3 times 24 is 72. So that’s one way we could have tackled the
problem. Now, another way we could tackle that is say,
7 percent is 21. Well, if I divide by 7, that means 1 percent
is 3. So if 1 percent is 3, then 24 percent is 24
times 3, that gives us 72. And yet another way we could do this is say,
well, y is the number in science, and we can set up a ratio here. We know that the ratio of science to “anything
but history” is 24 to 7, and that’s going to be equal to y to 21. Because y is the total number of students
in science and 21 said anything but history. So this proportion here comes from our – this
ratio here comes from our percents. 24 to 7. And this is the actual number of students:
y in science and 21 said anything but history. And now we can, clearly, multiply both sides
of this ratio by 3, we’ll get 72 again. And you might be wondering, why didn’t we
just show this as a bar chart. And we definitely could, could easily have
taken these, the same responses here, and stuck them in a bar chart here. Now, what we can’t tell from the bar chart
is, is math more or less than half the total? Well, we’d have to do some thinking here. We can clearly order them using the bar chart,
just like we did those wicked skills. We can see math was the best, science was
the next best, then English, then not history, and then speech. But we can’t see very easily what the proportion
of the whole is. We can’t see that science actually is about
a quarter of the total. The pie chart tells us that right at a glance. We see the quarter-pie slice, we know that
science is around a quarter of the total. So the bar chart is great for us to compare
these to each other, the pie chart is great for comparing them to the entire thing. Now that you understand the difference between
a pie chart and a bar chart, I know exactly what you’re thinking. You’re thinking, how did he do that weird
lip thing?