Normal distribution
-
Introduction to the Normal Distribution
-
Normal Distribution Excel Exercise
-
ck12.org Normal Distribution Problems: Qualitative sense of normal distributions
-
ck12.org Normal Distribution Problems: Empirical Rule
-
ck12.org Normal Distribution Problems: z-score
-
ck12.org Exercise: Standard Normal Distribution and the Empirical Rule
-
Empirical rule
-
ck12.org: More Empirical Rule and Z-score practice
-
Z scores 1
-
Z scores 2
-
Z scores 3
ck12.org Normal Distribution Problems: Qualitative sense of normal distributions Discussion of how "normal" a distribution might be
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- You could never have too much practice dealing with the
- normal distribution, because it's really one of those super
- important building blocks for the rest of statistics.
- And really, a lot of your life.
- So what I've done here is I've taken some sample problems--
- this is from ck12.org's open source flex book.
- Their AP statistics flex book.
- And I've taken the problems from their normal
- distribution chapter.
- So you could go to their site and actually look
- up these same problems.
- So this first problem: which of the following data sets is most
- likely to be normally distributed?
- For the other choices, explain why you believe they would not
- follow a normal distribution.
- So let's see, choice a.
- So this is a really-- this is, you know, my
- beliefs come into play.
- So this is unusual in the math context.
- It's more of a, what do I think?
- It's kind of an essay question.
- So let's see what they have here.
- A, the hand span.
- Measured from the tip of the thumb to the tip of the
- extended fifth finger.
- So I think they're talking about-- let me see if
- I can draw a hand.
- So that's the index finger, and then you've got the middle
- finger, and then you've got your ring finger, and then
- you've got your pinky.
- And a hand will look something like that.
- I think they're talking about this distance.
- From the tip of the thumb to the tip of the
- extended fifth finger.
- Which is a fancy way of saying the pinky, I think.
- They're talking about that distance, right there.
- They're saying, if I were to measure it on a random sample
- of high school seniors, what would it look like?
- Well, you know, how far this is.
- This is a combination of genetics and
- environmental factors.
- Maybe how much milk you drank or how much you hung from your
- pinky from a bar when you were growing up.
- So I would think that it is a sum of a huge number
- of random processes.
- So I would guess that it is roughly normally distributed.
- You know, if I look at my own hand, and my hand I don't
- think has grown much since I was a high school senior.
- It looks like roughly 9 inches, or so.
- I play guitar.
- Maybe that helped me a stretch my hand.
- But if that, you know, it's really an essay question.
- So I just have to say what I feel.
- So I would guess that the distribution would look
- something like this.
- I don't know.
- I've never done this.
- But, you know, maybe it has a mean of 8 inches or 9 inches,
- and it's distributed something like this.
- Maybe it probably does look like a normal distribution.
- But it probably won't be a perfect-- in fact, I can
- guarantee it won't be a perfect normal distribution.
- Because one, no one can have negative length of that span.
- This distance can never be negative.
- So they're going to have a-- I guess, you could have no hand.
- So that would maybe be counted as 0.
- But the distribution wouldn't go into the negative domain.
- So it wouldn't be a perfect normal distribution on
- the left hand side.
- It would really just end here at 0.
- And even on the right hand side, there are some physically
- impossible hand lengths.
- No one can have a hand that's larger than the height of
- the earth's atmosphere.
- Or, you know, an astronomical unit.
- You would start touching the sun.
- There is some point which is physically
- impossible to get to.
- And you know, in a true normal distribution, if I were to flip
- a bunch of points, there's some very, very small probability
- that I could get 1,000,000 heads in the row, in a row.
- It's almost 0, but there's some probability.
- But in the case of hand span, there's no way-- out here-- you
- know, the probability of a human being who happens to be a
- high school senior having a, I don't know, 1 mile length
- hand span, that's 0.
- So it's not going to be a perfect normal distribution of
- the outliers, or as we get further and further
- away from the mean.
- But I think it'll be a pretty good-- in our everyday world,
- as good as we're going to get-- approximation.
- The normal distribution's going to be a pretty good
- approximation for the distribution that we see.
- I guess one thing that, you know-- it's funny.
- This is high school seniors.
- When I did this, it was kind of from my point of view, a guy.
- And I would argue that, you know, high school senior guys
- probably have larger hands than women, so that it's possible
- that you actually have a bimodal distribution.
- So instead of having it like this, it's possible that the
- distribution looks like this.
- That you have 1 peak for guys, maybe at 8 inches.
- And then maybe a slight-- another peak for women
- that, you know, I don't know, at 7 inches.
- And then the distribution falls off like that.
- So it's also possible it could be bimodal.
- But in general, a normal distribution is going to be
- a pretty good approximation for part a of this problem.
- Let's see what part b-- what they're asking us to describe.
- The annual salaries of all employees of a
- large shipping company.
- So if we're talking about annual salaries, we have
- minimum wage laws, whatnot.
- So I would guess at any corporation, if we're talking
- about full-time workers at least, there's going to be
- some minimum salary that people have.
- So I would say-- and probably a lot of people will have
- that minimum salary.
- Because it'll be probably the most labor intensive jobs.
- You probably have-- most people are down there at the low
- end of the pay scale.
- And then you have your different middle level
- managers and whatnot.
- And then you probably have this big gap.
- And then you probably have your true executives.
- Maybe your CEO or whatnot.
- If this mean right here is maybe $40,000 a year, and this
- is probably $80,000, where some of the mid-level managers lie.
- But this out here, this'll probably be-- you know,
- actually if you were to draw it real, the way I've scaled it
- right now, this would be $80,000, this would
- be about $200,000.
- Which is actually a reasonable salary for a CEO.
- But the reality is that this actually might get pushed
- way out from there.
- It might look something like that.
- It might be way off the charts.
- You know, let's say the CEO made $5,000,000 in a year,
- because he cashed in a bunch of options or something.
- So it could be way over here.
- And maybe it's a CEO and a couple of other people,
- the CFO or the founders.
- So my guess is it definitely wouldn't be a normal
- distribution.
- And it definitely would have a second-- it would be bimodal.
- You would have another peak over here for senior management
- up at the-- unless we're-- well, they're not saying, you
- know, if we're in maybe Europe, this would probably be
- closer to the left.
- But it won't be a perfect normal distribution, and you're
- not going to have any values below a certain threshold.
- Below that kind of minimum wage level.
- So I would call this-- when you have a tail that goes more to
- the right than to the left-- call this a right
- skewed distribution.
- Since it has 2 humps, right here-- 1 there and 1 there-- we
- could also say it's bimodal.
- I mean, it depends on what kind of company this is.
- But that would be my guess of a lot of large shipping
- companies' salaries.
- Let's look at choice c.
- Or problem, part c.
- The annual salaries of a random sample of 50 CEOs
- of major companies.
- 25 women and 25 men.
- The fact that they wrote this here, I think they maybe are
- implying that maybe men and women-- you know, the gender
- gap has not been closed fully, and there is some discrepancy.
- So if it was just purely 50 CEOs of major companies, I
- would say it's probably close to a normal distribution.
- It's probably something like-- well, you know, once again,
- there's going to be some level below which no CEO is
- willing to work for.
- Although I've heard of some cases where they work for free,
- but they're really getting paid in other ways.
- If you include all of those things, there's probably some
- base salary that all CEOs make at least that much.
- And then it goes up to some value, you know, the
- highest probability value.
- And then it probably has a long tail to the right.
- And this is if there were no gender gap.
- So this would just be a purely right skewed distribution,
- where you have a long tail to the right.
- Now if you assume that there's some gender gap, then you might
- have 2 humps here, which would be a bimodal distribution.
- So if you assume there's some gender gap-- this is part
- c right here-- then maybe there's 1 hump for women.
- And if you assume that women are less than men, then
- another hump for men.
- And there are 25 of each, so there wouldn't be necessarily
- more men than women.
- And then it would skew all the way off to the right.
- And, in fact, I think there would probably be a chance that
- you have this other notion here, where you have these
- super CEOs, or mega CEOs, who make millions, while most
- probably just make, you know, just-- I'll put it in quotation
- marks-- a few hundred thousand dollars, while there's a small
- subset that are way off, many standard deviations
- to the right.
- So it could even be a trimodal distribution here.
- So that's choice c.
- And then, so far, choice a looks like the best candidate
- for pure-- or the closest to being a normal distribution.
- Let's see what choice d is.
- The dates of 100 pennies taken from a cash drawer
- in a convenience store.
- 100 pennies.
- So that's actually an interesting experiment.
- But I would guess, once again, this is really a question where
- I get to express my feelings about these things.
- You know, as long as your answer is reasonable, I
- would say that it is right.
- Most pennies are newer pennies, because they
- go out of commission.
- They get traded out.
- They get worn out as they age.
- They get lost.
- Or, you know, they get pushed-- you know, pressed-- at the
- little tourist place into the little souvenir things.
- I'm not even sure if that's legal, if you can do
- that to money legally.
- So my guess is that if you were to plot it, you would have
- a ton of pennies that are within the last few years.
- So if we were-- so the dates of 100 pennies, not their age.
- So the dates.
- So if this is 2010, I would guess that right now, you're
- not going to find any 2010 pennies.
- But you're probably going to find a ton of 2009 pennies.
- And it probably just goes down from there.
- And, of course, you're not going to find pennies that are
- older than, say, the United States, or before they even
- started printing pennies.
- So obviously this tail isn't going to go to
- the left forever.
- But my guess is you're going to have a left
- skewed distribution.
- Where you have the bulk of the distribution on the right, but
- the tail goes off to the left.
- That's why it's called a left skewed distribution.
- Sometimes this is called a negatively skewed distribution.
- And similarly, this right skewed distribution
- is sometimes called positively skewed.
- And if you have only 1 hump-- you don't have a multimodal
- distribution like this-- in a left skewed distribution,
- your mean is going to be to the left of your median.
- So in this case, maybe your median might be
- someplace over here.
- But since you have this long tail to the left, your mean
- might be some place over here.
- And likewise in this distribution, your median--
- your middle value-- might be some place like this.
- But because it's right skewed, and for the most part only has
- 1 big hump-- this hump won't change things too much, because
- it's small-- your mean is going to be to the right of it.
- So that's another reason why it's called a right skewed,
- or positively skewed, distribution.
- So to answer the question, these are my feelings
- about all of them.
- But I would say, you know, for the other choices explain why
- you believe they would not follow-- or they said, which of
- the following data sets is most likely to be normally
- distributed?
- Well, I would say choice a.
- But it's really, you know, a matter of opinion, at
- least in this question.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.