# Density Curves, Empirical Rule & Normality, Z-score Intro

Hello again! So, Mr. Tarrou. I got 15 minutes

to cover density curves, what it actually means to be normal, standard normal distribution

calculations, and give you some examples. So let’s see if I can get through this.

Hit pause any time you like and copy these notes down.

Let’s talk about density curves. When you’ve seen me do a lot of discussions lately about

the shape of distributions: left-skewed, right-skewed, unimodal, symmetric. And I keep drawing these

nice sort of smooth lines over these histograms so that I don’t have to spend so much time,

you know, drawing these histograms with all the vertical lines and equal spacing of numbers

and stuff. Well, these smoothed-over lines are not just out of convenience. They’re

also how we’re going to do a lot of calculations this year. So let’s talk about really what

these density curves are. Well it’s a smooth line curve that’s placed over a histogram,

stem plot, or dot plot and normally it’s going to be histograms. You’ll see examples

similar to this in your textbook. Now, the area of a density curve has to be an area

of one. Well, you know, this histogram goes from five to twenty-one and we don’t even

know what the height is, so I’m not going to say that the area of this histogram is

equal to one. I’m sure it’s not. But once we smooth over the shape and if we’re

happy that that distribution is normal, well you know what- there is only one true distribution

that’s Normal. Now we’re going to talk about that in a minute. So normal is normal

is normal. So think of it like a figure in geometry. When you have a triangle- and you

might be talking about similar figures in geometry- well you know I could take a triangle

that’s like this big and make it, you know, bigger and yet it’d still be similar, the

same shape. Well that’s kind of what we’re doing with these density curves. I’m drawing

this curve over this histogram, but then I’m going to, for definition of density curve,

shrink it or enlarge it (if that happens to be the case) until the area is exactly one.

And density curves do not have to be a bell shape. You can even have a density curve that’s

in the area of a rectangle to display completely random data. Like maybe this goes from zero

to two, but since the area has to be one it would only have a height of .5 to make sure

the area equals one. It does not describe or display outliers and our most common use

of density curves is for calculating z-scores and finding probabilities so outliers are

not part of that normal distribution. You can have right-skewed distribution as well,

but again, you know, we’re trying to smooth over the fuzzy kind of outline of the data

and get an overall picture. So outliers will not be discussed or displayed in those density

curves. Since the area is equal to one, the area is easily converted into a p-value. What

the heck is that? A p-value is a probability. We’re going to be doing tons of these calculations

this year once we get through all the building block chapters. And the p-value is a probability

of observing a certain outcome or one that’s more extreme. We’ll talk more about that

later, but it is a probability. And the areas in that bell curve are going to be converted

into a probability. We’ll say some conclusions- well I’ll do those conclusions there at

the end of this introduction. So density curves, smoothed over lines over the histogram. We

need those areas to be one or they may be even some weird geometric shape. Let’s finally

define what normal really is. Woo! Gotta run out of the way- in a hurry! Duh-nah-nah-nah…

Alright, there we go. We’re going to learn about a graph that

helps us to verify normality. This is not it, but this is the definition of normality.

So, it’s just not a very easy definition to use for very fine normality. So a normal

distribution is, for the, y’know, 50th time now, a bell-shaped curve. You will have with

bell-shaped curves the three measures of center being equal mean, median, and mode. When you

have a normal distribution, this is short-hand notation for describing a normal distribution

where it’s capital N for normal, two numbers in parentheses: the first number will be the

mean and the second number will be the standard deviation. And if I’m using mu or sigma

I’m talking about mean and standard deviation of a population.

If I use x-bar and s of x, I’m talking about the mean of a sample and the standard deviation

of the sample, but yet that sampling distribution is still normal. So this is what normal data

falls into. Now if you are doing a sample question in your textbook that only has 15

or 20 numbers you’re not going to follow into this distribution exactly, but if you

have a lot of numbers- a few hundred or a thousand numbers- then you will fall close

into this distribution of data if it truly is normal. So, here you go: the center of

a bell curve is always the mean- well actually mean, median, and mode, but we’re concerned

about mean because standard deviation is average distance from mean and we use standard deviation

for these calculations. So the mean is in the center. If you go from the center of the

bell curve (the mean) left and right one standard deviation, you are going to contain within

that interval (plus or minus one standard deviation) 68% of the data. If you go two

standard deviations (one-two to the left and one-two to the right) that entire interval

is going to hold 95% of the data. Your actual, real-life data will not be exactly 95%, but

as a mathematical model it will be exactly 95%- well actually not exactly, these are

all a little bit rounded off, but basically 95%. And if you go to the left three standard

deviations and to the right three standard deviations- if the data is normal- you will

contain approximately 99.7% of the data. You might have exercises in your textbook having

you check this, but there is a graph that the calculator does for us called the normal

probability plot that will automatically check normality for us. We’re not going to explore

that in today’s video. Alright, now we’re going to move on to standard normal distributions

and standardizing values. Alright, there we go. Standard normal distribution. Okay: standard

normal distribution. This is the chart in your textbook. You’ll see a z-score chart

in the front of your textbook. It will have a normal distribution- it is the normal bell

curve. It will have a mean of zero and a standard deviation of one. Well, we’re going to have

questions about you know, everything: medical studies, weight of chickens, speed of light,

you pick it there it’s going to be in that book. Those all have very wide-ranging units

of measure yet there is one perfect unit of normal distribution and we need to make our

data fit that table in the front of the book so we can find these probabilities that I’m

talking about, these p-values which you may not know exactly what I mean by that, but

you will in a minute. We’re going to run through this thing called a z-score. I’ve

written it the way it will look like on your formula sheet. A z-score is found by taking

the estimate. That’s going to be the numerical value that you’re given in the question

like, “something has an average of 50, what’s the probability of scoring under a 40?”

Well the 40 would be your estimate. Minus the parameter, the population parameter, something

we know about the entire population. Again, that’s going to be mu. And the book will

tell us what that mean of that population is and the standard deviation of the estimate.

These are in vague terms because we’re going to work with all different kinds of variables

this year: proportions and means and counts and y’know, so we gotta fill in. This is

sort of like the outline… hmm, yeah that’s good- the outline of this formula. Z-scores,

okay. Z-score’s unit of measure is the number of standard deviations away from the mean

you are. If you have a piece of data that’s less than the mean, then the z-score is going

to be negative. Maybe your negative 1 standard deviation is to the left of the mean or the

negative 1.5 standard deviation is to the left of the mean. And if you’re to the right

of the mean- if your data is above the mean- you’re going to have a positive z-score.

Negative z-scores: left of the mean, positive z-scores: right of the mean. And the unit

of measure, you know, whether you’re talking about the weight of chicks or the miles you’ve

traveled, every question in the textbook, through this process of z-score will turn

the units of measure into, again, the number of standard deviations you are away from the

mean. And again, negatives are left, positives are right. This formula- y’know I’m seeing

minus- has a subtraction in it and a division in it. So this is a form of a linear transformation,

that’s what we talked about earlier- in an earlier video. So linear transformations

that change the original units of measure into, again, standard deviations away from

the mean. And this linear transformation, the z-score formula, does allow us to use

one standard normal distribution (that’s the z-score chart that’s in the front of

your book) for every single problem in the book until we get into t-score calculations,

but that’s many, many chapters from now. And maybe high square if you want to argue

another test, but the vast majority of the work in the textbook for quite a while is

going to be z-score calculations- at least my textbook. So we’re going to use the same

chart, the same standard normal distribution, for a large number of questions this year.

Now, I’m going to move into some examples of how this z-score works and how we take

areas out of a bell curve and then call that a probability, or a p-value, but I can’t

stand up here at a chalkboard and show you how to read a page in a textbook, so ask your

teacher. And I do plan next week to do a little video on how to read that z-score chart in

your book, but I’m just now going to run through this formula come up with the z-scores,

but I’m just going to give you the p-value that I’ve gotten out of the chart. If you

have trouble reading that, ask your teacher or check out a later video that I work on.

Okay, I have 5 minutes left I think. Oh, 4 minutes. Let’s run through as many examples over here as we can. Okay. We have a limited amount of time. I’ll get through as much

of this as I can. We have a bell curve whose center is at 10 and whose standard deviation is 3. That’s some data. I want to find the probability

of getting a piece of data that’s less than 7. That’s going to be represented by a bell

curve centered around 10 and we’re going to find the probability of getting a score

of 7 or less, so I’m going to shade the left-hand tail which is how the shading is

set up in your textbook. How do you do this? Well, you find the z-score,

which is going to be 7. The estimate minus the population parameter of 10, over the standard

deviation which I’ve made up to be 3. Well 7 minus 10 is negative 3. Over 3. Which is

a z-score of negative 1. So in this population if you’ve scored 7 or collected the data

point of 7, that’s one standard deviation to the left of the mean. If you look the z-score

up of negative 1 in your textbook, you will find that the probability, or the area to

the left of that value (the p-value) is equal to .1587. So, that means that the probability

of observing a data less than 7 in this continuous, random, quantitative variable, this normal

distribution with the mean of 10 and the standard deviation of 3, the probability of seeing

a number less than 7 is 15.9%. The probability of observing a value of less than 7 is 15.9%.

This p-value becomes the probability. Remember this whole area had an area of 1? So .1587-

if you just move the decimal over 2 places- that’s the probability of observing that

event. Probably will just have on another video talking about sampling distributions

and where these bell curves really come from, but I wanted to run through a couple of quick

calculations for you as an example for the end of this video. Probability that x is less

than or equal to 7? Well, I don’t need to do any more work for that because this is

a continuous random variable. So the probability of seeing exactly 7, that’s exactly one

outcome out of (if it’s continuous and the variable can truly take on any value) an infinite

number of possibilities. Well what’s one over infinity? That’s 0. So the probability

of seeing a particular value in a continuous random variable is 0. So what does that mean?

It means the probability of getting a value less than 7, which was .1587, is exactly the

same as finding the probability of getting x is less than or equal to 7 because the probability

that x equals 7 is 0. So the answer to this question, in a bell curve calculation- and

that’s important because it’s not always that, is the same. So the probability of this

is .1587. Or, as a summary statement, the probability

of observing a value of 7 or less is 15.9%.

Thank you:) I hope you do well this year and that these videos help when needed. Good luck on your AP test at the end of the year.

How about a "Like" vote as a return:):)

@wazooman123456

You can count on me continuing to upload these videos. My students have given me great feedback as well. Thank You Very Much for letting know that these are helping you as well. Keep up the great work.

I love your hop on screen intro!! So much energy. You are doing a great job!!

@drusilanormie Thank you. I try not to be the droning monotone teacher. Sometimes the math can be boring enough without the teacher helping:)

@sabrinalwilliams

Well…I will certainly keep my fingers crossed that your grades improve dramatically:):)

What I would give to have you as my stat teacher. Thanks so much for the help!

great energy kept me awake and i definitely learned something

@fred11895 Thank you very much. Thanks for watching. I hope you do very well in your class:)

@merdbor :):) Thank you very much! I'm thrilled I could help.

Really like ur zest..i too have developed a liking to stats watching ur vids….i am much more confident now

Definately will help me in my endeavour to become a knowledgable n successful human being…god bless u

These stats concepts will help me in my roadmap to a successful six sigma career..ur an inspiration

@abhi124s Thank you very much. I am very happy you are finding these videos helpful:)

@abhi124s 🙂

this video underlines the fact that knowledge knows no boundaries…its universal…any person who has an appetite and thirst for knowledge can try quench it watching quality videos like yours..!! sound bit philosophical but its true..

@abhi124s Very Deep:) Thanks

@Angeliniwini I like you already:)

I have a Q. I need to find if a set of 150 values follows normal distrib. im using the kolgomorov-smirnov test for normality. the null hypothesis is "Normal distrib is shown". after doing the test the KS value is 0.075 and the P-value is 0.043. this is LESS than the critical value at 149 degrees of freedom, therefore the null hypothesis, with 95% confidence can be rejected… but the histogram shows NORMAL DISTRIBUTION at intervals of 10. im baffled, should I reduce the intervals for more bars?

@StikManJones I am sorry to reply that I don't have any knowledge of the specific test you are doing. I will say though, that a histogram that is unimodal and symmetric is not automatically Normal. T-distributions have those same characteristics but are not Normal…they have greater area in the tails compared to a Normal Distribution. I am sorry I can't be of more help.

@profrobbob so your advice is basically ignore the graph, regardless if it is bell shape 'n' trust the test? thanks for that, i dont know why I got it into my head that histograms automatically show normal distribution. P.S I finished the stuff you teach in stats about 3 years ago. Even though stats was the maths module i scored highest in, it was the most boring topic. In medschool we are getting stats stuffed down our throats, but luckily for me you make it pretty damn fun to revise thank you.

@StikManJones THANK YOU:):):) I really appreciate the positive feedback. I just started doing these videos 6/7 months ago. I am sure next year may want to redo a few videos and definitely add more. I wish you great success in Med school! Thank you again for watching my vids and supporting my efforts.

@StikManJones Is StikMan related to you going into medicine? Great name:)

@profrobbob No problem and thanks. And no, lol, I got into medicine because I've seen the positive affect it has on the patient AND families life when it gets it right. But I did not realise how much medicine loves using P-Values, normality, confidence levels and stats for pharmacological studies and drug testing. I was one of those idiots who thought after my exams I'd never ever encounter that subject in my life again… how life is cruel. Now I'm seeing stats EVERYWHERE

O, how I've missed you. Ha. Back for more ;).

Statistics in Psychology this semester at Rowan.

nnnaa nnnaa na na.

I missed you too:):):)

I hope you do great on your finals:) I don't do private tutoring…sorry. Between my full time job and building my channel I just don't have any more time available:(

🙂

Subscribed today, have my statistics final on Friday and you have helped me tremendously already; wish my stats teacher taught as you do!

First may I say…Thank You for subscribing and your comment. I will send some positive energy your way for an "A" on that final! Congrats for seeking out the extra help, I hope all that extra effort pays off….so BAM!…go study some more and make me proud!!!!

Sir, you are a GENIUS. Our Teachers at my university can learn A LOT from you. I got my test in two weeks and i am so so so so happy that i bumped into your videos!

Thanks a lot, this my man is really appreciated!!!

Thank you for the enthusiasm…please feel free to share my channel with your class. I also have a printable postcard available on my facebook page that you could print and share at your university if you are so inclined. I truly appreciate all the support I get from my viewers, cause if it were not for viewers like you there would be no me!

These videos are so helpful!! I have my final next week and you explain things so much better than my teacher!!!

Thank you for subscribing and I hope my videos have helped you enough to ace your final!…just when you think you've studied enough, study just a little bit more, you'll be surprised at the outcome…so BAM!…go do your homework and pass that final:))

Final in 2 days and I forgot all of this. Thanks so much man, you rock!

You're welcome and Thanks for watching…hope you ace that final now!!!!

THANK YOU!

WOW…and THANK YOU for "liking" so many of my videos! I really appreciate the support. Please help me grow by sharing my channel with others:D

Please don't stop making videos like this. You are saving me and I really mean it. God bless you. "many thanks from Kuwait"

There's no stopping me know..teaching on YouTube has become very gratifying:)

Also, many thanks from Florida for watching and subscribing…I appreciate the support from Kuwait!

You're a blessing for all AP statistic students. My exam is this Friday. I've maintained a 95 average throughout the year but your videos are giving me added confidence for the AP exam! Thank you for taking your time to thoroughly explain each lesson 🙂 If every teacher was as good as you our nation would be filled with geniuses lol

Thank you for sending such a gratifying comment. As a teacher, I love to hear from such self motivated students! To have such a high GPA and still keep striving for the best grade you can make on your exam…well you are every teachers model student:) If every "student" was like you the world WOULD be full of geniuses!

Therefore all thats left to say is: BAM!!!…go with confidence and get that A you are so deserving of…and tell when you do!

Thank you very much! You have been a blessing to me! I used to be soo scared of maths , i watch your videos right after class and before my exams and quizes and I am improving. I even surprise myself! God richly bless you. Please dont stop what you are doing.

You are so very welcome…and THANK YOU for being such a loyal viewer! I'm glad I was able to lift some of your math fears and show you that math can be fun while learning and once you learn and that confidence kicks in, there's no stopping the possibilities:) As long I I keep growing and viewers like you keep watching and learning and commenting, I promise to keep making videos:D

THANK YOU for such clarifying vidoes! You really helped me out since now I make A's on my quizzes! Wish my teacher is just as great…

You're welcome. CONGRATULATIONS on the A's…now you see that all it takes is a little extra effort on your part and BAM!!! next thing you know you have all A's:) Thanks for choosing my channel to be that extra effort and I hope you continue to watch, learn, share, and get A's 😀

Thank you so much! As someone who is taking online classes, this is the holy grail! Thanks =D

THANK YOU for choosing my channel to learn from and I hope you'll your online teacher about my channel and maybe they could use it as a free tutoring resource for other students:)

the fact that you make videos completely for strangers in need is awesome, thanks man you are a true teacher.

And THANK YOU for taking the time to notice:) I do love teaching, I'm in my 18th year of classroom teaching. When I started this channel it was to help one of my students, I never dreamed that I would still be making videos 2 years later. I've been asked many times to join the masses and sell my videos…but I just want to follow Kahn's path and keep them FREE for everyone who's willing to seek some outside help:)…now, if I could just figure out how to get his subscriber #'s!!!

Hey just wanted to say your absolutely amazing to put this up! this has helped me a lot to efficiently and quickly learn and/or recap for my exams. I'm all the way up in iceland so I hope you know that your helping people literally all over the world! with much gratitude from myself! Best wishes!

Brrrrr!!!! I'm cold just thinking about teaching in Iceland:)…this thin florida blood wouldn't know what to do in the cold. Thanks for sharing where you are from…it really hits home when I hear the places I'm reaching. Be sure to share my channel with your friends and tell them to support and subscribe and continue to share and help to make "Tarrou's Chalk Talk" the #1 place to go for FREE Math Help in Iceland…BAM!!!

Shouldn't the z-score formula be y – yhat/theta not Sx because when theta is not given you have to use the t-test?

#statistics

why are you so amazing?

I rely on youtube so much because some of these teachers in schools need to really take the time and re-evaluate their teaching style.. and their life while they're at it.

:'(( why did the video cut off?

Poop… I was hoping that you'd talk about Chebychev's Theorem, because that's what I'm really, really struggling with. But thanks for a little bit of clarification!

Very nice to include the times of the video that each section is covered!! Thank you! 🙂

Honestly, thank you so much for these videos… really helps for my final. I wish I had you for my teacher

Omg thanks so much for showing this!!! I'm seriously taking all your notes!! I will ace my stat exams!! 🙂

But why are your videos time limited? 🙁 I was hoping to see more examples ESP when they ask for proportion between two values which is an odd percentage, and that confuses me a lot 🙁

Thank you very much Summer Rae Newhouse for providing the Closed Captions for this lesson. Your help is greatly appreciated!!!

When I finished calculus I was like awwwe man no more ProfRobBob but here I am again studying statistics haha. Some of the best and clear math tutorials Ive ever come across. Thumbs up!

Thanks ProfRobBobPlease continue making math videos And you have Great #math Energy

More Statistics Videos Please My Professor lectures a lot about Bell Shape and The Chebyshev's Theorum .. will tell my relatives friends and my classmates in this Stats class .. I'm currently taking .

thnx for uploading these lectures, i'm cramming them in for my upcoming test in biostatistics nxt week -_-

my textbook defines the mean of a density curve as the point at which it would balance if it was made of solid material but I have NO idea what this means. I have a vague idea, but it's just not clicking. Can anyone help me understand what my book means by this and put it into more mathematical terms? I'm having trouble distinguishing it from the median of a density curve which is the point at which the areas to the left and right of the median are equal. How are these two different?

I'd love to hear the rationale behind why the 4 'dislikers' disliked this video .. man!

YESSSSSS!!!! I'm so happy you have statistic videos.

Good job and love the shirt!