Sampling Distributions
Jan 04, 2016
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.