Sampling Distributions

What is a sampling distribution?

Grab a sample of size N

Compute a statistic (mean, variance, etc.)

Record it

Do it again (until all possible outcomes are recorded or infinitely)

The resulting distribution is a sampling distribution

Articulation of the Sample Space

Concept of the Effect Size

Related to Study Outcome

Indicates relations between X and Y (relations between IV and DV)

Indicates magnitude of effectSize of effect, Effect Size

Two Common Effect Sizes

Correlation, r Standardized Mean Difference, d

N

zzr YX

pooledSD

XXd 21

value

populationvalue

population

ES Sampling Distributions

If delta = 0, distribution approx normal

If rho = 0, distribution approx normal

If not zero, distributions are not normal. Customary to apply fixes for this (discussed later).

Boxplot1

17N =

D1

9

8

7

6

5

4

3

2

1

Median

25 %tile

75 %tile

Middle50 Percent

Largest Case not an Outlier

Smallest Case not an Outlier

Whiskerortail

Whiskerortail

21N =

20

10

0

-10

19

21

20

22

Outlier

Extreme Outlier

Outlier

Extreme Outlier

Boxplot 2

227N =volcano heights

30000

20000

10000

0

10000

222227223226225224

Boxplot 3

Empirical (Monte Carlo) Sampling Distributions

Examine R programsIn running R, you will want to save your outputs in separate files that let you keep records. The graph is replaced (overwritten) each time you run a graphical command

Form groups and complete exercise

Some results show here; others in a separate window shown on additional slides.

You don’t need to understand the computations unless you want to write your own programs.

You need to input parameters.

Results of running the sim (histogram)

N = 120; rho = .8

Results of running the sim (boxplot)

N = 120; rho = .8

You input the parameters. If you start with a standardized mean difference (e.g., d = 1), you can just set one mean to zero, the other to the value of d, and the standard deviations within each group to 1.0. The program is written to give you more flexibility (e.g., you can see what happens if the variances and sample sizes are unequal across groups).

d = 1N1=N2=15

(M1=14 M2=15)(SD1 = SD2 =1)

d = 1N1=N2=15

(M1=14 M2=15)(SD1 = SD2 =1)

Exercise 3aWhat happens to the standard error of the mean (square root of the sampling variance) of r as rho increases from near zero to near 1

(use rho = 0, .3, .6, .9,

samplesize=100,

Nsamples=10000

What happens to the shape of the sampling distribution of r (particularly skew) as rho increases from near zero to near 1

(use the same values for the simulation)?

Create a table and an illustrative graph or series of graphs to tell your story. Prepare to present to the class.

Exercise 3b

What happens to the standard error of the mean of d as delta increases?

Use delta= 0, .5, 1, 2 (use SD=1 and choose means)

N1=N2=25

Nsamples = 100000

What happens to the shape of the sampling distribution of d as delta increases (use the same values)?

Create a table and an illustrative graph or series of graphs to tell your story. Prepare to present to the group.