Z scoresAnd whatnot

This is a NORMAL DISTRIBUTION

Most people are near the hump

So the Mean, Median, and Mode fall in the middle too

Note the symmetry off the curve

Normal Distributions standardize information from different data sets so

the Mid-point is 0

0

Each point is just like standard counting numbers, so to speak

1 2 3-1-2-3

The segments on the left are negative

0 1 2 3-1-2-3

Its essentially a number line

You typically won’t see higher than 4

Next we’ll start getting into numbers

Here are the formulas…

Z = Z =X - μ

σX - μ

S

X = μ + (Z)(σ)

Learn them, Love them

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

A problem may be given and from it you can pull out X (some score), Mu (the mean), and Sigma

(standard deviation)

Here are some numbers for example

X = 300 μ = 200 σ = 50

You pretty much just plug in the numbers from there

Remember these numbers, I’ll refer to them a lot

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 200 μ = 300 σ = 50

Z =X - μ

σChose a formula that has the matching

components

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 300 μ = 200 σ = 50

Z =X - μ

σ

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 300 μ = 200 σ = 50

Z =300 - 200

50

Solve

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 300 μ = 200 σ = 50

Z =300 - 200

501002

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 300 μ = 200 σ = 50 Z = 2

0 1 2 3-1-2-3

Lets bring back that normal distribution now

You put the Z score right on the graph

And that’s it

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

You can practice with these numbers

X= 50

μ =30

σ =5

X= 8

μ =6

σ =2

X =80

μ =100

σ =5

X =18

μ =36

σ =6

4 1 -4 -3

(click for answers)

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)You might be

wondering about this formula

In that case you’d be given Z and solve for X

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

Z = 1 μ = 10 σ = 2

You would be given this

X = μ + (Z)(σ)

X = 10 + (1)(2)

X = 10 + 2

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X = 12

Easy enough

A few more things about the normal distribution

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There can be more than one Z score on the graph

It can be negative

And it can have decimals, in that case you just estimate where it sits

Z =

Z =

X - μ

σ

X - μ

S

X = μ + (Z)(σ)

X= 50

μ =30

σ =5

X= 8

μ =6

σ =2

X =80

μ =100

σ =5

X =18

μ =36

σ =6

4 1 -4 -3

Remember these answers from before?

4 1 -4 -3

0 1 2 3-1-2-3

Its uncommon to go beyond 3, but it is possible as is the

case here

-4 4

These numbers were from different data sets but can share the same graph

Ok, so what if you’re asked to find a percent of data under or above a

certain score?

X = 300 μ = 200 σ = 50 Z = 2

0 1 2 3-1-2-3

Lets bring back the data we worked with earlier to save some time

What if you need to know what percent is below the Z-score?

Well, at this point you go into that nifty page at the back of your Stat-psych book and look up under

column (A) the Z score “2”

You can look it up yourself if you want, its on page 529

You go to your z score, in our case its 2, then you take the number from the column of the

area you want to know.

(B) Is body

(c) Is tail

(D) Is proportion between the Mean and Z

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This is the body

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This is a tail

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We want the body

That gives us the number .9772

What does that mean!?!

Nothing when you want the percent

So you just multiply that number (.9772) by 100 and you have your percent of

area.

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.9772 x 100 = 97.72So about 98%

Ok, that’s awesome when you want all the data under or above a

Z-score, but what if you want to see… lets say, 60% of the data

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Hmmmm…

No picture for this yet

Whatever, I’ll skip it

So you bust out that stylish book of yours and go to page 528

We want 60% of the data so we’re going to look at column D, the proportion between the mean and Z-score and you’re going to

want the number closest to .3000

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Ok, now we can use the picture

So, You want to look up the closest number to 30%

or .3000

If you look up the closest to .6000 (60%) in column D it

will be looking for 60% on both sides

And that just doesn’t work

That will put 30% on each side adding up to a total 60%

0 1 2 3-1-2-3

The number happens to be .2995

Z-score of 0.84

So, lets draw that on

That’s what it looks like, but now what if you need the score to be within the

60% of the data?

X = μ + (Z)(σ)

Remember this formula?

Just plug in your numbers and you’ll have your answer

X = μ + (Z)(σ)μ = 200 σ = 50 Z = + .84Data from before

X = 200 + (.84)(50)

X = 200 + 42

X = 242

X = 200 + (-.84)(50)

X = 200 - 42

X = 158

There's the low and the high scores to be within 60% of the data

I’ll wrap this up with a quick review of the Central limit theorem as It was taught on

Tuesday and I’m not really sure whether I’m suppose to include it or not in my lesson

Its for small samples, but you generally want a sample in the 20s or so, a “n” of 3 doesn’t really work so well

And you know to use this when you see “n” which we haven’t seen at all in Z-scores

So here is the formula

Z =M - μ

σm

You’ll be given μ, M, σ, and n so it really makes things simple.

It only looks complicated but its really quite easy

Just take it one step at a time.

μ = 95

M = 100

σ = 20

N = 25

Take your numbers and plug them in Z =

M - μ

σm

M - μ 100 – 95 = 5

σm = σ

N√=

20

√25205

= = 4 4

Z = 1.25

= ZM - μ

σm

1. (Here is the formula)

2. (start with the top)

3. (next do the bottom)

4. (Put it together)

= 5

5. (that’s all)

And there you have it

Z scoresAnd whatnot