Week 2 Chapter 4: The Normal Distribution and Standard Scores

Distribution meaning

Describes how scores/data are spread out

Distribution tells us what?

Where are the most scores?

Are the scores concentrated around the middle?

Are there unusually high/low scores?

Is the distribution balanced?

Does is have a shape?

Normal Distribution shape

Bell shaped curve

Normal Distribution Description of curve

Symmetric around the mean

Unimodal

Bell-shaped

Mean=median=mode

For which of the following distributions will the skew news value be zero?

N(0,1)

N(0,2)

N(10,50)

The z table tells what

The area below a particular z score

Z score example

z = 1.00

P(z = < 1.00) = .8413

=84.13%

Negative z score

z = -1.00

P(z < -1.00) = P(z > 1.00)

P(z < 100) = .8413

1 - .8413 = .1587

Area between two z scores

z = -2.50

z = 1.00

P(-2.50 < z < 1.00)

P(z<1.00)=.8413

P(z < -2.50)=.0062

.8413 - .0062 =0.835

=83.51%

What a z score tells

How many standard deviations a score is above or below the mean

Z score formula

z = (X - $\mu$)/$\sigma$

Z score formula z meaning

z = (X - $\mu$)/$\sigma$

Standardized score

Z score formula X meaning

z = (X - $\mu$)/$\sigma$

Individual raw score

Z score formula $\mu$ meaning

z = (X - $\mu$)/$\sigma$

Population mean

Z score formula $\sigma$ meaning

z = (X - $\mu$)/$\sigma$

Population standard deviation

Z score example

X = 75

$\mu$ = 60

$\sigma$ = 15

How far X is from mean: individual raw score - population mean (75-60=15)

Score is 15 points above mean

Divide by 1 SD: 15/15 =1

Z = +1 The person is 1 standard deviation above the mean

Positive z

X > $\mu$

z = +2

2 SD above the mean

Negative z

X < $\mu$

z = -1.5

1.5 SD below the mean

Z = 0

X = $\mu$

Z score → raw score

z = (X - $\mu$)/$\sigma$

z = (X - $\mu$)/$\sigma$ → X = $\mu$ + z$\sigma$

Z score → raw score example

$\mu$ = 100, $\sigma$ = 15, z = 2

X = 100 + (2)(15)

z = 2 → X = 130

Positive skew

Mode < Median < Mean

Mean gets pulled towards the tail on the right

Negative skew

Mode > Median > Mean

Mean gets pulled towards the low tail on the left

Kurtosis

Peakedness/flatness

Leptokurtic, Mesokurtic, Platykurtic

Leptokurtic

Positive kurtosis, tall/peaked

g2 > 0

Mesokurtic

Normal-like, neither particularly flat not particularly peaked, middle

g2 = 0

Platykurtic

Negative kurtosis, flat

g2 < 0