- 00:01
SPEAKER 1: Hello.Welcome to this lesson on Mastering Statistics.We'll continue solving problems here,so you get practice with the normal distribution.So, what we have here is a question.IQ scores have a mean of 100 and a standard deviation of 15.By the way, I don't know this to be actually completely true,but it's probably ballpark.

- 00:22
SPEAKER 1 [continued]: What percentage of people have an IQ lower than 85?So, there's two parts to this.First, identify what you're given.You you're told that it has a mean of 100and a standard deviation of 15.You also are told-- I didn't write it on the board--but you're told that it's normally distributed.Usually things that are a very large characteristic

- 00:43
SPEAKER 1 [continued]: of a large group of people, usually,are normally distributed.So, go ahead and say that this is normally distributed.I'm giving you that fact.The mean is 100 .The standard deviation is 15.Now, that's the characteristic of the population of people.Now, what percentage of people have an IQ lower than 85?When you're given these kinds of problems,sometimes you're asked: what's the probability

- 01:04
SPEAKER 1 [continued]: of choosing a person at random and theyhave an IQ of less than 85?Sometimes, you'll be asked: what'sthe percentage of people that have an IQ of less than 85?It's really the same question.You attack it the same way.So, the first thing we want to dois just say and write down what we know.The mean is 100.And the standard deviation is 15.

- 01:26
SPEAKER 1 [continued]: And, if we really wanted to draw this, then, the waywe do that is draw our little axis here.And our normal distribution would be something like this.Right?Now, the mean, we already know is 100.We could put the standard deviations in here.

- 01:46
SPEAKER 1 [continued]: But ultimately, all we want to knowis what percentage of people have an IQ lower than 85.So, let's say 85 somewhere over here.And so, usually, when we're trying to find something lower,we're trying to find the area all in this tail here.So, ultimately, we're going to read it right off of the chart.But we have to convert it to a z-score first.

- 02:07
SPEAKER 1 [continued]: So, ultimately, what we want to findis the probability that our random variable xis less than 85.So, I'll put that is equal to a question mark.That's what we're trying to do.Remember, random variable is justsort of the outcome of your experiment.You can think of it in terms of you'repulling something, what's the value you're going to get.You do that experiment again and again.

- 02:28
SPEAKER 1 [continued]: It represents your unknown value.So, what is the probability that if I pull somebodyfrom the population that their score isgoing to be less than 85.Well, we can't do it as is.We have to convert these z-scores because wehave to use the table.And so, what we're going to do is convert 85.So, it's the value minus the mean

- 02:50
SPEAKER 1 [continued]: over the standard deviation.We have a value of 85, minus 100.The standard deviation is given as 15.So, when you think about it, thisis negative 15 divided by 15.You're going to get a z-score of negative 1.All right.So, ultimately, we want to know the areato the left of the z-score, which is exactly what is

- 03:10
SPEAKER 1 [continued]: given to us in the table.So, the way you kind of write it all out--there's many ways youcan do it.I'm just trying to get you familiar with things.The probability that, when we draw somebody at randomand their score is less than 85, our random variableis less than 85, is the same answeras the probability of getting a z-score less than negative 1.

- 03:32
SPEAKER 1 [continued]: Less than negative 1.Because we convert that and get negative 1.So, we put negative 1 in the chartand we get an answer of 0.1587.This is a decimal.It's a probability.That's what we get out of the chart.So, if the problem were phrase differently and it said,what is the probability that a randomly drawn personhas an IQ of less than 85?

- 03:53
SPEAKER 1 [continued]: This would be the probability.Because that's what we did.We looked here.We're getting this area.We convert to a z-score.This is the probability.But, that's not what the problem said.It said what percentage of peoplehave an IQ lower than 85.Think about weather forecasts.Weather forecasts are just percentages, too.I mean, if it's 50 percent chance of rain,that's a 0.5 probability of happening, basically.

- 04:14
SPEAKER 1 [continued]: So, all you need to do, when you want to go to a percentage,is multiply by 100.So, when you take this and multiply by 100,you can write the answer directly.15.87 percent of people have IQ less than 85.

- 04:37
SPEAKER 1 [continued]: And that's the answer.If we were asking for the probability of a person drawn,you would circle that.If we're asking for the percentage of peoplein our population, then you can easily find that, as well.Because, when you think about it,if this represents your entire population of everybody,if you add everybody up, then that's100 percent of the people covered.

- 04:57
SPEAKER 1 [continued]: So, the area represents the probabilityof pulling a single person and getting a score in that range.It also can represent sort of like a percentage,like a weather forecast.So, the second part here says, in a similar way,what percentage of people have an IQ greater than 130.So, if I wanted to draw that, I have a normal distribution

- 05:22
SPEAKER 1 [continued]: here.Right?And I have, what is the average score?It's 100.And so, IQ greater than 130.This is 130.So, I'm trying to find, essentially,what is that surface area right there.What is the surface area.So, let's convert to a z-score.Z is equal to x minus the mean over the standard deviation.

- 05:45
SPEAKER 1 [continued]: 130 is the number I care about, minus 100over a standard deviation of 15.So, when I do this math, this becomes 30on the top divided by 15.You get a z-score of 2.So, we try to write our answer down.See where we get.So, what we're trying to find outis the probability that our random variable x is

- 06:06
SPEAKER 1 [continued]: going to be greater than 130.Let's find that probability.But we use the z-score idea to figure outthat that's the same thing as the probability of havinga z-score less than 2 or greater than 2.So, we're basically converting this probleminto a problem with z-scores, which is whatwe've been doing all along.But we cannot look this up in the chart.We can't, because the chart gives this shaded

- 06:29
SPEAKER 1 [continued]: areas to the left.So, because of that, we need to further change this and say zis less than negative 2.We've discussed that many, many times.So, whenever you do this, what youget, when you look negative 2 up in that table,and you look at the answer you get, 0.0228.So, this is what we would circle if I asked you.

- 06:50
SPEAKER 1 [continued]: Pull a person at random.What is the probability that their IQ is greater than 130.This is a probability.But that's not what we're asked.We're asked for a percentage.So, just multiply by 100.Move the decimal spot two times.So, we'll say 2.28 percent of people

- 07:12
SPEAKER 1 [continued]: have IQ greater than 130.2.28 percent.That's a fairly high IQ.Anything greater than that, you'restarting to get into serious high IQs.OK.So, that's the basic idea.A lot of these problems are goingto look similar after a while.That's exactly what I'm driving for.I want you get comfortable with that,

- 07:32
SPEAKER 1 [continued]: so when you're on an exam or test,you're trying to calculate something,you're not worried about it.You need to be able to look at data, look at the mean,look at the standard deviation, know that it is normallydistributed.And then, draw a picture of that, so that,whenever you calculate the z-score,you know exactly what shaded area you're trying to get.And you use those charts and it gets totally, totally

- 07:52
SPEAKER 1 [continued]: like the back of your hand.Because, as we go on in statistics,we're going to be getting into other thingsthat can trip a lot of students up.But, if you have the foundational material,rock solid, then that's going to build a huge skillset for you to achieve and do well in statistics.

### Video Info

**Series Name:** Mastering Statistics, Vol 2

**Episode:** 14

**Publisher:** Math Tutor DVD

**Publication Year:** 2013

**Video Type:**Tutorial

**Methods:** Normal distribution, Probability

**Keywords:** IQ; mathematical concepts; mathematics

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Instructor Jason Gibson demonstrates how to calculate the probability of IQ scores within a population. This includes calculating z-values and converting them into percentages.