1. 3.3 Experimental Standard Deviation
assume a normal pdf for the data, xi
what is the pdf of deviations, di?
what is the pdf of the deviations squared?
what is the pdf of the sum of di2?
what is the pdf after dividing by N-1?
what is the pdf after taking the square root?
the moments of f(s)
miscellaneous observations about s
3.3 : 1/6

2. Random Variable and Deviations
The experimental standard deviation is computed using the equation at right.
Assume that each xi comes from a normal pdf, n[m,s]. The deviations, , have a "shifted" normal pdf, n[0,s].
In the example below, x has the pdf, n[5,0.1] while the deviations, d, have the pdf, n[0,0.1]. Theoretical pdfs are shown in blue.
3.3 : 2/6

3. Squaring and Summing
On slide we derived the pdf for the square of n[0,s] and saw that it was a gamma density G[a,l] = G[1/2,1/(2s2)].
In the standard deviation equation, the summation of squares is composed of only N-1 statistically independent values. The sum of N-1 random variables with gamma density G[a,l] is G[(N-1)a,l].
For the example with which we have been working, d2 has the pdf G[1/2,1/0.02], while the sum of three squared deviations (N = 3) has the pdf G[1,1/0.02]. Theoretical pdfs are shown in blue.
3.3 : 3/6

4. Dividing by N-1 and Taking the Root
A gamma density, G[a,l], divided by the constant, N-1, has the gamma density, G[a,(N-1)l]. For the example with which we have been working, the result for N = 3 is G[(N-1)/2,(N-1)/(2s2)] = G[1,1/s2].
On slide we showed that the square root of a gamma variable, G[a,l], had the form of a normal moment. Note that the density is a factor of two larger because there are no negative s values.
The mean of s with
N = 3 is This value is statistically different (lower) than 0.1.
3.3 : 4/6

5. f(s) and N
The probability density function of s is given by the following, where N is the number of terms in the summation. N 3 5 10 20 ms
0.088
0.094
0.097
0.099 ss
0.046
0.034
0.023
0.016
a3(s)
0.063
0.041
0.025
0.017
red, N = 3; blue, N = 5; green, N = 10; magenta, N = 20; all for s = 0.1
3.3 : 5/6

6. Miscellaneous Facts About s
As N  ∞, ms  s, ss  0 and a3(s)  ss  0.
Thus, as N gets large the mean of f(s) approaches s, the pdf becomes symmetric and its width (uncertainty) approaches zero. The remaining question is, "How large does N need to be in practice?"
Section 5 of the Statistical Tables shows that for N = 100, 95% of the computed standard deviations fall within the range of 0.876σ to 1.16σ. This demonstrates how slowly s approaches σ.
The true standard deviation, s, is seldom known in a research environment. However, analytical laboratories making routine measurements will most often have an excellent estimate of s. Examples would be clinical and quality control labs.
3.3 : 6/6