Sampling Distributions of Sample Mean
Philadelphia Rainfall Data
Population: 540 months of rainfall in Philadelphia (100^{ths} of inches) from January, 1825December, 1869.
Histogram with superimposed Normal curve:
Descriptive Statistics:
Descriptive Statistics

N 
Minimum 
Maximum 
Mean 
Std. Deviation 
RAIN 
540 
19.00 
1582.00 
367.6796 
191.77305 
Valid N (listwise) 
540 




Samples: Took 1000 random samples of size 30. Theoretical Mean and Standard Deviation (Standard Error) of Distribution of Sample Means:
Histogram of 1000 sample means (n=30) with superimposed normal curve
Descriptive Statistics:
Descriptive Statistics

N 
Minimum 
Maximum 
Mean 
Std. Deviation 
YBAR 
1000 
259.10 
469.30 
366.4056 
33.58160 
Valid N (listwise) 
1000 




Note: The mean of the sample means (366.41) is very close to the population mean (367.68). However, the standard deviation of the sample means (33.58) is further away, specifically below the theoretical standard deviation of the sampling distribution (35.0). This is (partly) due to the fact that we are sampling from a finite population, and that our sample size is relatively large as compared to the population size. When these sizes are known, we can incorporate the finite population correction factor for the standard deviation (standard error) of the sampling distribution of the sample mean:
Note that in practice, either the population size is unknown or or very large (even infinite), relative to the sample size, and this correction is ignored.