# Sampling Distributions of Sample Mean

```Sampling Distributions of Sample Mean
Population: 540 months of rainfall in Philadelphia (100ths of
inches) from January, 1825-December, 1869.
Histogram with superimposed Normal curve:
70
60
50
40
30
20
Std. Dev = 191.77
10
Mean = 367.7
0
0.
0
N = 540.00
10 20 30 40 50 60 70 80 90 10 11 12 13 14 15 16
0
0
0
0
0
0
0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 00.
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
RA IN
Descriptive Statistics:
Descriptive Statistics
N
RAIN
540
Valid N (listwise)
540
Minimum
19.00
Maximum
1582.00
Mean
367.6796
Std. Deviation
191.77305
Samples: Took 1000 random samples of size 30. Theoretical Mean
and Standard Deviation (Standard Error) of Distribution of Sample
Means:
Mean :  Y    367.68 Std. Dev. :  Y 

n

191.77
30
 35.0 Shape :Approximat ely Normal
Histogram of 1000 sample means (n=30) with superimposed normal
curve
140
120
100
80
60
40
Std. Dev = 33.58
20
Mean = 366.4
N = 1000.00
0
0
0.
46
0
0.
44
0
0.
42
0
0.
40
0
0.
38
0
0.
36
0
0.
34
0
0.
32
0
0.
30
0
0.
28
0
0.
26
Y BAR
Descriptive Statistics:
Descriptive Statistics
N
1000
YBAR
Valid N (listwise)
Minimum
259.10
Maximum
469.30
Mean
366.4056
Std. Deviation
33.58160
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:
Y 

n
N  n 191.77 540  30

 35.0(.9727)  34.1
N 1
540  1
30
Note that in practice, either the population size is unknown or very
large (even infinite), relative to the sample size, and this
correction is ignored.
```