How many soil water content measurements are enough? (app. note 2s-g)
How many soil water content TION NOTE measurements are enough?
Copyright (C) 2001Campbell Scientific, Inc. How many soil water content measurements are enough? This application note provides basic statistical methods used to determine the number ofmeasurements needed to provide a specific level of confidence that the soil water con-tent of a given area has been adequately characterized.Spatial Variability of Soil Water Content
Soil water content can vary significantly among several locationswhich are near each other and apparently similar. Water contentmeasurements using the most accurate methods available provideevidence of significant differences in soil structure and textureeven when the measurements are limited to an area of only onesquare meter. The degree of variability is dependent on many fac-tors including presence of coarse fragments (rocks), micro andmacro-faunal activity, erosion, management practice, and plantroot activity. The difference in soil physical properties from loca-tion to location and the subsequent difference in soil water charac-teristics is commonly referred to as spatial variability of soilhydraulic properties. Any characterization of an area used tomanage or inventory water resources in that area must account forthis variability. Example Data Set
As an example, Table 1 lists the results of 12 volumetric watercontent measurements taken within a two meter radius on a well-established and level lawn. Below the table are the mean, stan-dard deviation, and range of the table’s data. Table 1. Sample Water Content Data Copyright 2001 Campbell Scientific, Inc.815 W. 1800 N., Logan, UT 84321-1784 (435) 753-2342How many soil water content measurements are enough?Confidence Intervals
Statistical approaches are based on probability theory. Usingconfidence intervals is a method of expressing probability. Consider measuring the water content at several different loca-tions in a volume of soil. The measurements will differ in valuebut will be distributed about a mean value. If another measure-ment is then taken, you can calculate the probability that this mea-surement will fall within a specific amount below and above themean value. Conversely, for a specific probability value, you cancalculate the water content range that the measured value willlikely fall within. The sum of the values that lie below and abovethe mean value is known as the confidence interval. The confi-dence interval becomes smaller as the probability increases.
The relationship between confidence interval and probability canbe described using the following expressions.
Where µ is the actual population mean, σ is the actual populationstandard deviation and n is the number of values used to calculatethe mean, x
ෆ. The confidence coefficient, k, is used to specify a
probability value. A probability of 95% or 0.95 could be speci-fied and [1] would then define the range below and above themean that is the confidence interval. The probability is 95% thata measured mean will fall within the defined interval.
Equation 2 is algebraically identical to Equation 1, but therearranged form now describes the likelihood that the interval willbracket the actual population mean. The difference is subtle butimportant.
Equation 2 is based on the assumption the actual population meanand standard deviation are known. Actual values are seldomknown. If the values were known, you wouldn’t have to measurethem. Assuming a normal distribution, correcting for the fact thatthe actual population standard deviation, σ, must be estimatedfrom measurements, and applying the theory of the Student's t-distribution, Equation 2 becomes:
Copyright 2001 Campbell Scientific, Inc.815 W. 1800 N., Logan, UT 84321 (435) 753-2342How many soil water content measurements are enough?
Where s is the estimate of σ that’s based on the measurementsand t is the value obtained from a table of Student's t -values. Using the Student's t-distribution corrects the normal distributionfor the number of measurements taken or degrees of freedom.
Consider the data presented in Table 1. If a probability value of0.95 is acceptable, Equation 3 can be used to define the confi-dence interval. For this data, the mean is 11.9 and the standarddeviation is 1.74. With 11 degrees of freedom, the Student's tvalue is 2.201. The Student's t-distribution is symmetrical, andthe probability of 0.95 implies 0.025 is on both sides of the mean. For most tables, to choose a t value for 0.95, you use the valuelisted for 0.975 (i.e., 1.0-0.025). However, some tables do it dif-ferently and you’d use the value listed for 0.025. Entering thosevalues into Equation 3, you get confidence interval values of10.79 and 13.00. In summary, there is a 95% probability that themean water content lies between the water content range of10.79% to 13.00%. For this example, the depth of water associat-ed with the range is 0.44 cm in a 20 cm profile. Minimum Number of Samples
When the the sample mean and standard deviation are known anda confidence interval is chosen, Equation 4 can be used to deter-mine the minimum number of randomly taken measurements.
Where N is the number of water content measurements, and L isthe acceptable water content range as defined by the user. Forexample, the acceptable water content range might be ±2.5% so Lwould be 5.0.
Equation 4 relies on a good estimate of the population standarddeviation. A useful approach of estimating this is to take severalwater content measurements then calculate the standard deviationfor these measurements. This standard deviation estimate and theassociated t value are used in Equation 4. The range is specifiedby the user and the calculated N value defines the required num-ber of measurements. Using the same s and t values, the number
Copyright 2001 Campbell Scientific, Inc.815 W. 1800 N., Logan, UT 84321-1784 (435) 753-2342How many soil water content measurements are enough?
of measurements can be determined for several water contentranges.
The calculation of N using the sample standard deviation showsthat three measurements are required to give a 95% probabilitythat the sample mean gives the desired ±2.5%.
Copyright 2001 Campbell Scientific, Inc.815 W. 1800 N., Logan, UT 84321 (435) 753-2342
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