Breakout Session #6Numerical Statistics

Presented byDr. Del Ferster

What’s in store for today?

›We’re going to wrap up our monthly meetings by considering numerical statistics.–Next time, we’ll be working on a review.

›We’ll look at mean, mode, median, and range. –We’ll also look at standard deviation ( a measure of “spread” or dispersion)

What’s in store for today?

›We’ll review how to read a Z CHART (a standard normal chart), to compare scores relative a population’s mean and standard deviation.

›We’ll wrap it up by looking at some practice problems.

First, a picture of Conor!!!Come on now, you knew that I’d do that!

Distributions of DataAn introduction to the Normal Curve(sometimes called the Bell curve)

Different Distributions of Data› Data can be distributed or “spread out” in different ways.

› It can be spread out so that more data falls on the left.

We would say that this distribution is SKEWED RIGHT

(interestingly, the SKEW is NOT where the data clusters, it’s where the tail falls.

Different Distributions of Data› It can be spread out so that more data falls on the right

We would say that this distribution is SKEWED LEFT

Different Distributions of Data› Of course, the data might fall all over the place.

In this case, there doesn’t seem to be any discernible pattern.

Different Distributions of Data› Today, let’s look at the case where data tends to

cluster around a central value, with no tendency to the left or right. It gets closer to a NORMAL DISTRIBUTION, like this:

The Normal DistributionA closer look at key ideas.

The Bell Curve› The Bell curve is a Normal Distribution

› The yellow histogram shows some data that follows the curve closely, but not perfectly. (This is USUAL, as it’s seldom that a data set is perfectly normal.)

What’s normal, Del?› Many things in life closely follow a normal distribution.–The heights of a group of people.–The size of widgets that are produced by a machine.

–Errors in measurements.–Blood pressure.–Grades on a test

A bit more on the normal distribution

› We say that the data is Normally Distributed

› In a normal distriubtion– The mean, mode, and median are the SAME NUMBER

– The graph is symmetric with respect to the center

– 50% of the values fall above the mean, and 50% of the values fall below the mean

A cool online app› This Quincunx or Galton Board app will simulate a

normal curve.

› Let’s look at it.

› Here’s the link

› http://www.mathsisfun.com/data/quincunx.html

Standard Deviation› Standard Deviation is a measure of how SPREAD OUT the data are. Some books refer to Standard Deviation as a measure of dispersion.

› You’re familiar with another measure of dispersion—RANGE.

› The formula for calculating standard deviation is pretty ugly, and luckily, many software packages along with graphing calculators provide a fast, easy way to find it.

Calculating Standard Deviation Rocking Math Old School!!!› Calculate the average (mean) of your set of numbers

› Calculate the difference between each number and the mean

› Square the differences

› Add up the square of all the differences

› Divide this by one less than the number of numbers in your set - this is called the variance

› Take the square root of the variance and you've got the standard deviation

More on Standard Deviation› When you calculate the standard deviation of your

data, you will find, in general:

68% of the data falls within 1 standard deviation of the mean95% of the data falls within 2 standard deviations of the mean99.7% of the data falls within 3 standard deviations of the mean

Statements regarding Data and Standard Deviation› In general, we can say that any value in a data set is:

› “likely” to be within 1 standard deviation of the mean

› 68 out of 100 are

› “very likely” to be within 2 standard deviations of the mean.

› 95 out of 100 are

› “almost certain” to be within 3 standard deviations of the mean.

› 997 our of 1000 are

Z scores› A z score tells us the number of STANDARD DEVIATIONS that a given data point falls above or below the mean.

› Some books refer to these as STANDARD SCORES

› Our formula for finding z scores

› This process is sometimes called STANDARDIZING

xz

Example: Determining a z score› Beth scored a 92 percent on her last statistics exam. The class average was 80 percent and the standard deviation was 5.

› Calculate her z score.

xz

92 80

5z

2.4z

Beth’s score exceeded the mean by 2.4 standard deviations.

Why standardize?› Most data sets have very different means and standard deviations.

› The process of standardizing (or calculating z scores) allows us to convert all normal distributions to one (the STANDARD NORMAL DISTRIBUTION)

Let’s do a problem together› Dr. Ferster has just given a quiz to his Math 204

class. Out of 60 points, his students recorded the following grades:

20, 15, 26, 32, 18, 28, 35, 14, 26, 22, and 17

›Dr. F. is bummed , these students must not have studied very much.

›Perhaps they were watching the Packers.

Continuing our problem

20, 15, 26, 32, 18, 28, 35, 14, 26, 22, and 17

› If Dr. F grades normally, it will be pretty ugly.

›Suppose he decides to grade on a normal curve, so that the only people who fail will be those who score more than 1 standard deviation below the mean.

Continuing our problem

20, 15, 26, 32, 18, 28, 35, 14, 26, 22, and 17

›Calculate the mean› I’ll tell you that the standard deviation is 6.6

›Calcualate the z score for –SUZY who scored 20.–SPARKY who scored 15.–STELLA who scored 26.

xz

SUZY—scored 20

› Suzy scores .45 standard deviations below the mean, however she still passes (she isn’t more than 1 standard deviation below the mean—that would be z = -1)

xz

20 23

6.6z

0.45z

23

. . 6.6

mean

st dev

SPARKY—scored 15

›Sparky scores 1.21 standard deviations below the mean, So he fails!

xz

15 23

6.6z

1.21z

23

. . 6.6

mean

st dev

STELLA—scored 26

› Suzy scores .45 standard deviations above the mean, so she passes. Her score falls above the mean.

xz

26 23

6.6z

0.45z

23

. . 6.6

mean

st dev

A picture for that last problem

Sparky

Suzy Stella

A look at the standard normal curve and areas› We can determine the percentage of our data that

falls – Below a given z score– Above a given z score– Between a given z score and the mean– Between 2 different z scores.

› In the old day, we used tables or charts.

› I’m going to show you how to read one later on…I know that you’re psyched for that!

› We can also look at a visual, which we’ll do on the next slide.

A look at the standard normal curve and areas› The picture below illustrates how the percentages of area

under a normal curve are related to different z scores.

A look at the standard normal curve and areas› Of course, in this age of technology, you shouldn’t

be too surprised that we can find an online app that determines areas, too.

› Let’s look at one. Here’s the link

› http://www.mathsisfun.com/data/standard-normal-distribution-table.html

Old School Stats with Del—Reading a z chart› I’m partial to the charts that read areas ALL THE WAY TO THE LEFT...or below the z score that we’re reading.

› I’m going to give you 2 charts—1 is useful for positive z scores, the other is used for negative z scores.–We’ll read both charts the same way.–Suppose we want to find the area to the LEFT of2.16z

Reading a z chart› We seek the area below

› First, grab chart that reads positive z scores.–Now, locate the 2.1 on the left-most column.

–Read across the top of the page till you find the column that is headed 0.06.

–Read the intersection and you find our area

–Did you get 0.9846?

2.16z

Let’s look at a problem.› Skippy’s Widget factory manufactures widgets that

have a length of life that is normally distributed with a mean of 12.3 years, and a standard deviation of 2.5 years.

› What percentage of the widgets that Skippy’s Widget factory produces will last:

› 1. Less than 9.425 years

› 2. More than 16.925 years

› 3. Between 7.75 years and 17.9 years.

Solution to our problem› mean of 12.3 years, standard deviation of 2.5

years.

› What percentage of the widgets that Skippy’s Widget factory produces will last:

› 1. Less than 9.425 years

12.3 2.5

xz

9.425 12.3

2.5z

1.15z

Solution to our problem (continued)› Now, we pick up our z chart and locate

› We’re looking to find the area below this z (remember we want to find the percentage of widgets that last less than the given number of years)

› so, we read our z chart and find

›.1251›So, 12.51% of the widgets last

less than 9.425 years

1.15z

Solution to our problem› mean of 12.3 years, standard deviation of 2.5

years.

› What percentage of the widgets that Skippy’s Widget factory produces will last:

› 2. More than 16.925 years

12.3 2.5

xz

16.925 12.3

2.5z

1.85z

Solution to our problem (continued)› Now, we pick up our z chart and locate

› We’re looking to find the area above this z (remember we want to find the percentage of widgets that last more than the given number of years)

› so, we read our z chart and find

› .9678

› So, 96.78% of the widgets last less than 16.925 years

› This means that 100%-96.78% or

› 3.22% of the widgets will last more than 16.925 years

1.85z

Solution to our problem› mean of 12.3 years, standard deviation of 2.5

years.

› What percentage of the widgets that Skippy’s Widget factory produces will last:

› 3. Between 7.75 years and 17.9 years.

12.3 2.5

xz

7.75 12.3

2.5z

1.82z

Solution to our problem› mean of 12.3 years, standard deviation of 2.5

years.

› What percentage of the widgets that Skippy’s Widget factory produces will last:

› 3. Between 7.75 years and 17.9 years.

12.3 2.5

xz

17.9 12.3

2.5z

2.24z

Solution to our problem (continued)› Now, we pick up our z chart and locate

› We want to find the AREA between these 2 z-scores

› Let’s read our z charts—Fun, eh??

1.82z 2.24z

2.24z 1.82z

Now, let’s work on that solution

We seek this shaded areaSo, we’ll read our 2 z-scores, and subtract the areas that we get from the z chart.

Now, let’s work on that solution

Reading z= 2.24, we get 0.9875Reading z= -1.82, we get 0.0344So, the area shaded in red can be found by subtracting these values (I get 0.9531)

So, 95.31% of the widgets will last between 7.75 years and 17.9 years

“Test-Like Problems”Dealing with Numerical Statistics

Practice Problems› Let’s take some time and work on some problems

that deal with numerical statistics.

› The last page of the handout contains a “partial” z chart.– Of course, you can use the full z chart that we

used earlier.

› On a happy note!– I spoke with Dr. Morgan about a formula sheet for

the exam that you’ll take in May– Let’s say that I’m cautiously optomistic!

Solutions to Practice Problems

› 1. Mean time spent with dad =

› 2. Mean time spent with mom=

› 3. Range of time spent with mom=

› 4. Range of time spent with dad=

› 5. Median time spent with mom=

› 6. I actually calculated the standard deviations, the greater spread is with the time spent with MOM

38.3 min .

242.5 min.

25 min.

20min.

242.5 min.

6.96

8.54Dad

Mom

Solutions to practice problems (continued)

› 7.

› 8. raw score (x) =49

› 9A. English: Science:

History:9B. English (biggest z score)9C. .6915 or 69.15%9D. 1-.9452 = .0548 0r 5.48%

2.50z

1.60z 0.50z 1.25z

Wrapping Up›Thanks for your attention and participation.

› It’s been a tough winter—not just the snow, but all of the adjusted schedules.

› I have the utmost respect for what you do as a professional!

›Here’s hoping that Spring brings us sun and smiles.

› If I can help in any way, don’t hesitate to shoot me an email or give me a call.