Download Epidemiology 9509 - Principle of Biostatistics Chapter 7: Sampling

Epidemiology 9509
sampling distributions (again)
Epidemiology 9509
Principle of Biostatistics
Chapter 7: Sampling Distributions (continued again)
John Koval
Department of Epidemiology and Biostatistics
University of Western Ontario
Epidemiology 9509
sampling distributions (again)
Distribution of median
for normal distribution
2
variance of X̄ is σn
that of median, XM , is
for n = 2m + 1
πσ2
4m
for large samples, the ratio of the variance of the median
to the variance of the sample mean is about
π
2
≈ 1.57
Epidemiology 9509
sampling distributions (again)
Plot of distributions of Sample mean and Median
Epidemiology 9509
sampling distributions (again)
sampling from distribution of Sample Median
1. variance decreases as sample size increases
2. distribution is symmetric
3. distribution becomes ”Normal” as sample size increases
4. variance is larger than that of Sample Mean
Question: Because it is easier to compute than the sample mean,
should we use the sample median to estimate
the centrality of the distribution?
Epidemiology 9509
distribution of sample variance
instead of calculating the sample mean,
calculate the sample variance
and look at its distribution
sampling distributions (again)
Epidemiology 9509
sampling distributions (again)
distribution of sample variance (continued)
following descriptive statistics
Var Nsam
Mean
Std Dev
Minimum
Maximum
---------------------------------------------------xbar
10 2.0040000 0.3874715 0.7000000 3.3000000
xbar
30 1.9993000 0.2281213 1.3333333 2.8333333
xbar 100 2.0067300 0.1250946 1.6500000 2.3600000
xbar 1000 1.9997290 0.0399812 1.8890000 2.1500000
---------------------------------------------------var
10 1.6202222 0.7428275 0.2222222 5.1111111
var
30 1.6055644 0.4191585 0.6160920 3.6965517
var
100 1.6098704 0.2260165 1.0079798 2.4544444
var 1000 1.6003043 0.0703617 1.4183373 1.8162162
---------------------------------------------------still skewed for nsam = 100
requires several hundred for symmetry (normality)
Epidemiology 9509
plots of distribution of sample variance
sampling distributions (again)
Epidemiology 9509
distribution of sample variance (nsam=30)
sampling distributions (again)
Epidemiology 9509
distribution of sample variance (nsam=100)
sampling distributions (again)
Epidemiology 9509
distribution of sample variance (nsam=1000)
sampling distributions (again)
Epidemiology 9509
sampling distributions (again)
sampling from distribution of the sample variance
1. variance decreases as sample size increases
2. distribution is asymmetric
3. becomes more symmetric (Normal) as sample size increases
4. even as nsam = 100, still quit asymmetric
Epidemiology 9509
sampling distributions (again)
distribution of sample standard deviation
as for variance but more symmetric at smaller sample sizes
Epidemiology 9509
sampling distributions (again)
Conclusion
1. distribution of most statistics become ”Normal”
2. some require very large sample size (100’s)
3. some (estimates of centrality) require smaller sample (30?)
size to attain normality