Changes

The purpose of statistics is to project or infer, from limited samples, the character of a population. In most cases, particularly in oil and gas investigations, geological information is not derived from carefully designed sample schemes but, by design, represents anomalies. What successful company would drill on a regional trend as opposed to the top of a structure, on a bright spot, or at the crest of a reef? Statistical procedures presume that sufficient data are randomly sampled from a population and that the average sample value approximates the population average. This is only possible if both high and low values are sampled without bias and enough samples are taken to stabilize the calculations. While proper sampling techniques are essential to formal statistical inference, geological samples are much too difficult or costly to obtain and cannot be discarded. Therefore, the robust testing of hypotheses and calculation of confidence intervals for statistical projections must be viewed in the restrictive light of geological data. Nonetheless, quantitative description and relationship inferences can be made with the underlying awareness of the constraint of data quality.

The purpose of statistics is to project or infer, from limited samples, the character of a population. In most cases, particularly in oil and gas investigations, geological information is not derived from carefully designed sample schemes but, by design, represents anomalies. What successful company would drill on a regional trend as opposed to the top of a structure, on a bright spot, or at the crest of a reef? Statistical procedures presume that sufficient data are randomly sampled from a population and that the average sample value approximates the population average. This is only possible if both high and low values are sampled without bias and enough samples are taken to stabilize the calculations. While proper sampling techniques are essential to formal statistical inference, geological samples are much too difficult or costly to obtain and cannot be discarded. Therefore, the robust testing of hypotheses and calculation of confidence intervals for statistical projections must be viewed in the restrictive light of geological data. Nonetheless, quantitative description and relationship inferences can be made with the underlying awareness of the constraint of data quality.

It is also important to remember the effect of resolution and precision in analyzing quantitative geological data. J. C. Davis put it eloquently in his introduction to his classic text (Davis, 1986<ref name = pt06r24>Davis, J. C., 1986, Statistics and data analysis in geology: New York, John Wiley, 646 p.</ref>):

It is also important to remember the effect of resolution and precision in analyzing quantitative geological data. J. C. Davis put it eloquently in his introduction to his classic text<ref name = pt06r24>Davis, J. C., 1986, Statistics and data analysis in geology: New York, John Wiley, 646 p.</ref>:

<blockquote>If you pursue the following topics, you will become involved with mathematical methods that have a certain aura of exactitude, that express relationships with apparent precision, and that are implemented on devices which have a popular reputation of infallibility.

<blockquote>If you pursue the following topics, you will become involved with mathematical methods that have a certain aura of exactitude, that express relationships with apparent precision, and that are implemented on devices which have a popular reputation of infallibility.

Even given this extreme difficulty, geological and statistical procedures share the common principle of parsimony: the simplest explanation is superior to a complex solution to a problem. Recognition of this relationship can form a basis for proper selection and application of the multitude of statistics available to the scientist.

Even given this extreme difficulty, geological and statistical procedures share the common principle of parsimony: the simplest explanation is superior to a complex solution to a problem. Recognition of this relationship can form a basis for proper selection and application of the multitude of statistics available to the scientist.

Prior to beginning any statistical investigation, be sure to review any one of a number of overview texts in geological applications in the oil and gas industry, including Davis (1986<ref name = pt06r24 />), Harbaugh et al. (1977<ref name = pt06r47>Harbaugh, J. W., Doveton, J. H., Davis, J. C., 1977, Probability methods in oil exploration: New York, John Wiley, 269 p.</ref>), and Krumbein and Graybill (1965<ref name = pt06r68>Krumbein, W. C., Graybill, F. A., 1965, An introduction to statistical models in geology: New York, McGraw-Hill, 475 p.</ref>).

Prior to beginning any statistical investigation, be sure to review any one of a number of overview texts in geological applications in the oil and gas industry, including <ref name = pt06r24 />), <ref name = pt06r47>Harbaugh, J. W., Doveton, J. H., Davis, J. C., 1977, Probability methods in oil exploration: New York, John Wiley, 269 p.</ref>), and <ref name = pt06r68>Krumbein, W. C., Graybill, F. A., 1965, An introduction to statistical models in geology: New York, McGraw-Hill, 475 p.</ref>).

==Central tendency==

==Central tendency==

The simplest and most commonly overlooked statistical procedure is to plot the data. (Atkinson, 1985<ref name = pt06r7>Atkinson, A. C., 1985, Plots, transformations, and regression: Oxford, U., K., Oxford Press, 282 p.</ref>). Often a simple crosspiot reveals the essential characteristics of a data set and allows for interpretation as well as proper selection of additional methods. In most cases, plotting of data reveals the nature of the data set and outliers or anomalous data points to review for accuracy or measurement error and can indicate the spread or variability of the data. Eliminating measurement error is not uncommon even in commercial data sets. For example, in a data set composed of well information, if the kelly bushing is not known or uniformly subtracted from all wells, the resulting map will develop a severe case of volcanoes!

The simplest and most commonly overlooked statistical procedure is to plot the data.<ref name = pt06r7>Atkinson, A. C., 1985, Plots, transformations, and regression: Oxford, U., K., Oxford Press, 282 p.</ref>. Often a simple crosspiot reveals the essential characteristics of a data set and allows for interpretation as well as proper selection of additional methods. In most cases, plotting of data reveals the nature of the data set and outliers or anomalous data points to review for accuracy or measurement error and can indicate the spread or variability of the data. Eliminating measurement error is not uncommon even in commercial data sets. For example, in a data set composed of well information, if the kelly bushing is not known or uniformly subtracted from all wells, the resulting map will develop a severe case of volcanoes!

There are three measures of characterizing a population by describing the average value, or its central tendency. The most familiar measure is the ''arithmetic mean'', which is simply the sum of the values divided by their number. The ''mode'' is the value that occurs with the greatest frequency, and the ''median'' is the value that has as many values above it as below it (Figure 1). As an example of comparing some of the statistics discussed in previous chapters, consider the following values of [[porosity]] (in percent) that have been measured on ten different sandstone samples: 15.1, 16.5, 18.8, 19.0, 22.0, 23.0, 25.0, 24.9, 31.9, and 43.0. Of the measures of central tendency, the arithmetic mean is the sum of all these numbers divided in this case by 10, or 239.2 ö 10 = 23.93. The median is 22.5 (halfway between 22.0 and 23.0), the value below which half the [[porosity]] values fall. The mid-range value is 29.05. The mode is the most frequently occurring value. Of the measures of dispersion, the range is computed to be 27.9, the variance is 61.79, and the standard deviation (the square root of the variance) is 7.86.

There are three measures of characterizing a population by describing the average value, or its central tendency. The most familiar measure is the ''arithmetic mean'', which is simply the sum of the values divided by their number. The ''mode'' is the value that occurs with the greatest frequency, and the ''median'' is the value that has as many values above it as below it (Figure 1). As an example of comparing some of the statistics discussed in previous chapters, consider the following values of [[porosity]] (in percent) that have been measured on ten different sandstone samples: 15.1, 16.5, 18.8, 19.0, 22.0, 23.0, 25.0, 24.9, 31.9, and 43.0. Of the measures of central tendency, the arithmetic mean is the sum of all these numbers divided in this case by 10, or 239.2 ö 10 = 23.93. The median is 22.5 (halfway between 22.0 and 23.0), the value below which half the porosity values fall. The mid-range value is 29.05. The mode is the most frequently occurring value. Of the measures of dispersion, the range is computed to be 27.9, the variance is 61.79, and the standard deviation (the square root of the variance) is 7.86.

[[file:statistics-overview_fig1.png|thumb|{{figure_number|1}}An asymmetrical data set. The three measures of central tendency are different.]]

[[file:statistics-overview_fig1.png|thumb|{{figure_number|1}}An asymmetrical data set. The three measures of central tendency are different.]]

Although the mean, median, and mode convey the same general notion of centrality, their values are often different, as just demonstrated, because they represent different functions of the same data. Statistically, each has its strengths and weaknesses. Although it is sensitive to extreme values, the arithmetic mean is most generally used, partially because of convention and partially because of its computational versatility in other statistical calculations.

Although the mean, median, and mode convey the same general notion of centrality, their values are often different, as just demonstrated, because they represent different functions of the same data. Statistically, each has its strengths and weaknesses. Although it is sensitive to extreme values, the arithmetic mean is most generally used, partially because of convention and partially because of its computational versatility in other statistical calculations.

The differences among these measures are a function of the frequency distribution of the samples. The frequency distribution is nothing more than a plot of the values versus the number of times the value occurs, and it is often depicted as a histogram. Most values cluster around some central value, and the frequency of occurrence declines toward extreme values. There are several shapes of frequency distributions that commonly occur in nature. Data sets that are symmetrical about a central value develop the familiar “bell-shaped” ''normal'' distribution (Figure 2). Data sets that have numerous small values and a few large values develop an asymmetrical curve shape. Comparison of histograms plays a vital role in the study of various geological properties. For example, construction of a histogram might be used to determine if a particular oil field exhibits a multimodal [[porosity]] distribution, indicating the presence of multiple lithologies. Another situation might involve a comparison of the distributions of petroleum field sizes discovered worldwide in foreland and rift basins.

The differences among these measures are a function of the frequency distribution of the samples. The frequency distribution is nothing more than a plot of the values versus the number of times the value occurs, and it is often depicted as a histogram. Most values cluster around some central value, and the frequency of occurrence declines toward extreme values. There are several shapes of frequency distributions that commonly occur in nature. Data sets that are symmetrical about a central value develop the familiar “bell-shaped” ''normal'' distribution (Figure 2). Data sets that have numerous small values and a few large values develop an asymmetrical curve shape. Comparison of histograms plays a vital role in the study of various geological properties. For example, construction of a histogram might be used to determine if a particular oil field exhibits a multimodal porosity distribution, indicating the presence of multiple lithologies. Another situation might involve a comparison of the distributions of petroleum field sizes discovered worldwide in foreland and rift basins.

[[file:statistics-overview_fig2.png|thumb|{{figure_number|2}}A symmetrical data set. The three measures of central tendency are identical.]]

[[file:statistics-overview_fig2.png|thumb|{{figure_number|2}}A symmetrical data set. The three measures of central tendency are identical.]]

The three measures of central tendency are identical in symmetrical data sets (Figure 2) and are very different in asymmetrical data sets (Figure 1). This difference is crucial in arriving at essential estimates. For example, what is the ''most likely'' value for reserves for the next well we drill? If, as in most producing basins, there are a few huge fields and many subcommercial small fields, the most likely discovery is not the mean but the mode. Determining the shape of the frequency distribution is critical to understanding which statistic to use. (For an excellent discussion of the characteristics of petroleum data population distributions, see Harbaugh et al., 1977<ref name = pt06r47 />.)

The three measures of central tendency are identical in symmetrical data sets (Figure 2) and are very different in asymmetrical data sets (Figure 1). This difference is crucial in arriving at essential estimates. For example, what is the ''most likely'' value for reserves for the next well we drill? If, as in most producing basins, there are a few huge fields and many subcommercial small fields, the most likely discovery is not the mean but the mode. Determining the shape of the frequency distribution is critical to understanding which statistic to use. (For an excellent discussion of the characteristics of petroleum data population distributions, see <ref name = pt06r47 />.)

Different geological properties and phenomena exhibit rather diverse distributions. For example, [[porosity]] is generally believed to be normally distributed, while [[permeability]] often tends to be lognormally distributed (that is, the logarithm of [[permeability]] tends to be normally distributed). Knowledge of the general form of the distribution is important to the selection of [[summary]] statistics because it helps prevent incorrect interpretations of the data. As a case in point, use of the arithmetic mean to represent average [[permeability]] is generally inappropriate because of the lognormality and high skewness of that property. Thus, the geometric mean, which identifies the median of a lognormal distribution, is better suited to this situation. In geology, not all quantities of interest approximate a normal distribution, and for that reason, uniform use of a particular statistic simply as a matter of convenience should be avoided. Table 1 lists formulas that are commonly used to derive effective [[permeability]].

Different geological properties and phenomena exhibit rather diverse distributions. For example, porosity is generally believed to be normally distributed, while [[permeability]] often tends to be lognormally distributed (that is, the logarithm of permeability tends to be normally distributed). Knowledge of the general form of the distribution is important to the selection of summary statistics because it helps prevent incorrect interpretations of the data. As a case in point, use of the arithmetic mean to represent average permeability is generally inappropriate because of the lognormality and high skewness of that property. Thus, the geometric mean, which identifies the median of a lognormal distribution, is better suited to this situation. In geology, not all quantities of interest approximate a normal distribution, and for that reason, uniform use of a particular statistic simply as a matter of convenience should be avoided. Table 1 lists formulas that are commonly used to derive effective permeability.

{| class = "wikitable"

{| class = "wikitable"

|-

|-

|+ {{table_number|1}}Commonly used formulas to derive effective [[permeability]]

|+ {{table_number|1}}Commonly used formulas to derive effective permeability

|-

|-

! Name

! Name

|-

|-

| Arithmetic mean

| Arithmetic mean

| <inline-formula> <tex-math notation="TeX"> $\begin{align*}\bar{k} = \displaystyle\frac{1}{H_{t}} \displaystyle\sum\limits_{i = 1}^{N}k_{i}h_{i}\end{align*}$ </tex-math> </inline-formula>

| <math>\bar{k} = \displaystyle\frac{1}{H_{t}} \displaystyle\sum\limits_{i = 1}^{N}k_{i}h_{i}</math>

| Average of uniform, horizontal, parallel layers in linear flow. ''k''_''j'' and ''h''_''j'' are the [[permeability]] and thickness of layer ''j'' . ''H''_''t'' is the total thickness.

| Average of uniform, horizontal, parallel layers in linear flow. ''k''_''j'' and ''h''_''j'' are the permeability and thickness of layer ''j'' . ''H''_''t'' is the total thickness.

|-

|-

| Harmonic mean

| Harmonic mean

| <inline-formula> <tex-math notation="TeX"> $\begin{align*}\bar{k} = H_{t} \left(\displaystyle\sum\limits_{i = 1}^{N}\displaystyle\frac{h_{i}}{k_{i}}\right)^{-1}\end{align*}$ </tex-math> </inline-formula>

| <math>\bar{k} = H_{t} \left(\displaystyle\sum\limits_{i = 1}^{N}\displaystyle\frac{h_{i}}{k_{i}}\right)^{-1}</math>

| Average of uniform, horizontal, serial layers in linear flow. Used for vertical [[permeability]] estimates in shale-free sands.

| Average of uniform, horizontal, serial layers in linear flow. Used for vertical permeability estimates in shale-free sands.

|-

|-

| Geometric mean

| Geometric mean

| <inline-formula> <tex-math notation="TeX"> $\begin{align*}\bar{k} = \left(\displaystyle\prod\limits_{i = 1}^{N}k_{i}\right)^{1/N}\end{align*}$ </tex-math> </inline-formula>

| <math>\bar{k} = \left(\displaystyle\prod\limits_{i = 1}^{N}k_{i}\right)^{1/N}</math>

| Approximate average of an ensemble of uncorrelated random permeabilities in globally linear flow. ''k''_''j'' is the [[permeability]] of each element in the ensemble

| Approximate average of an ensemble of uncorrelated random permeabilities in globally linear flow. ''k''_''j'' is the permeability of each element in the ensemble

|-

|-

| Radial flow

| Radial flow

| <inline-formula> <tex-math notation="TeX"> $\begin{align*}\bar{k} = (k_{\rm max} \cdot k_{\rm min})^{1/2}\end{align*}$ </tex-math> </inline-formula>

| <math>\bar{k} = (k_{\rm max} \cdot k_{\rm min})^{1/2}</math>

| Radial inflow (well) [[permeability]] in homogeneous, anisotropic media. ''k''_max and ''k''_min are the major and minor axes permeabilities.

| Radial inflow (well) permeability in homogeneous, anisotropic media. ''k''_max and ''k''_min are the major and minor axes permeabilities.

|-

|-

| Cross bedding

| Cross bedding

| <inline-formula> <tex-math notation="TeX"> $\begin{align*}\bar{k} = \frac{\cos^{2}\alpha}{k_{0}} + \frac{\sin^{2}\alpha}{k_{90}}\end{align*}$ </tex-math> </inline-formula>

| <math>\bar{k} = \frac{\cos^{2}\alpha}{k_{0}} + \frac{\sin^{2}\alpha}{k_{90}}</math>

| [[Permeability]] in a direction at an angle a to cross bedding. ''k''₀ and ''k''₉₀ are the permeabilities parallel and perpendicular to cross bedding.

| [[Permeability]] in a direction at an angle a to cross bedding. ''k''₀ and ''k''₉₀ are the permeabilities parallel and perpendicular to cross bedding.

|}

|}

There are two basic types of measured data: discrete and continuous variables. ''Discrete variables'' are measurements that can only be represented by counted values. For example, the number of limestone beds in a formation or the number of producing wells in a field can only be whole numbers. ''Continuous variables'' can have any value within the scale of measurement. Gamma ray log values, the [[porosity]] or [[permeability]] of a rock, or the subsea elevation of a formation are examples of continuous variables. They can have fractional values and can even have values the same as a previous sample.

There are two basic types of measured data: discrete and continuous variables. ''Discrete variables'' are measurements that can only be represented by counted values. For example, the number of limestone beds in a formation or the number of producing wells in a field can only be whole numbers. ''Continuous variables'' can have any value within the scale of measurement. Gamma ray log values, the porosity or permeability of a rock, or the subsea elevation of a formation are examples of continuous variables. They can have fractional values and can even have values the same as a previous sample.

==Variability==

==Variability==

* <math>\bar{x}</math> = sample mean

* <math>\bar{x}</math> = sample mean

* ''n'' = number of samples

* ''n'' = number of samples

The ''standard deviation'' is also used to describe the dispersion about the mean, and it is simply the square root of the variance. This statistic gives a measure of the variation in units of the variable instead of in squared units. For example, the variance of data measured in feet would be square feet or area. The standard deviation is the square root of this number, expressed in feet, which makes more sense for data measured in length.

The ''standard deviation'' is also used to describe the dispersion about the mean, and it is simply the square root of the variance. This statistic gives a measure of the variation in units of the variable instead of in squared units. For example, the variance of data measured in feet would be square feet or area. The standard deviation is the square root of this number, expressed in feet, which makes more sense for data measured in length.

* ''Z''_''i'' = ''i''th transformed variable

* ''Z''_''i'' = ''i''th transformed variable

* ''s'' = sample standard deviation

* ''s'' = sample standard deviation

Confidence intervals around the population estimate also reflect the significance of a calculated statistic. A ''confidence interval'' is the range of possible values that contains the true value of the population estimate with some specified level of confidence or probability. For example, the confidence interval about the mean of a normal distribution can be represented by

Confidence intervals around the population estimate also reflect the significance of a calculated statistic. A ''confidence interval'' is the range of possible values that contains the true value of the population estimate with some specified level of confidence or probability. For example, the confidence interval about the mean of a normal distribution can be represented by

* ''s'' = sample standard deviation

* ''s'' = sample standard deviation

* μ = true population mean

* μ = true population mean

The probability used for defining the ''t''-distribution statistic also defines the range containing the true population value of the mean (μ).

The probability used for defining the ''t''-distribution statistic also defines the range containing the true population value of the mean (μ).