Campbell Hydrosense, Instruction Manual

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Appendix B. How Many Soil Water Content Measurements Are Enough
B-2
The relationship between confidence interval and probability can be described
using the following expressions.
µ
σ
µ
σ
− < < +
k
n
x k
n
[1]
with
µ
the actual population mean,
σ
the actual population standard deviation
and n the number of values used to calculate the mean,
x
. The confidence
coefficient, k , is used to specify a probability value. A probability of 90% or
0.9 could be specified and [1] would then define the range below and above the
mean which is the confidence interval. The probability is 90% that a measured
mean will fall within the defined interval.
x k
n
x k
n
− < < +
σ
µ
σ
[2]
Equation 2 is algebraically identical to Equation 1, but the rearranged form now
describes the likelihood that an interval will bracket the actual population
mean. The difference is subtle but important.
Equation 2 is based on theory which assumes the actual population mean and
standard deviation are known. Actual values are seldom known. If the values
were known there would be no reason to measure them. Assuming a normal
distribution, correcting for the fact that the actual population standard
deviation,
σ
, must be estimated from measurements, and applying the theory of
the Student's t-distribution, equation 2 becomes
x t
s
n
x t
s
n
− < < + µ
[3]
with
s
the estimate of
σ
based on the measurements and t the value obtained
from a table of Student's t -values. Using the Student's t-distribution corrects
the normal distribution for number of measurements taken or degrees of
freedom.
Consider the data presented in Section B.2. If a probability value of 0.90 is
chosen as acceptable, equation 3 can be used to define the confidence interval.
For this data, the mean is 11.9 and the standard deviation is 1.74. With 11
degrees of freedom, the Student's t value is 2.201. The calculated confidence
interval is 10.76,12.97. In summary, there is a 90% probability that the mean
water content lies in the water content range from 10.76% to 12.97%
B.4 Determining the Minimum Number of Samples
Knowing the confidence interval and the sample mean and standard deviation,
equation 4 can be used to determine the minimum number of randomly taken
measurements.
N
t s
L
=
4
2 2
2
[4]