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Analytical Chemistry Easy Definition Essay

For the journal, see Analytical Chemistry (journal).

Analytical chemistry studies and uses instruments and methods used to separate, identify, and quantify matter.[1] In practice separation, identification or quantification may constitute the entire analysis or be combined with another method. Separation isolates analytes. Qualitative analysis identifies analytes, while quantitative analysis determines the numerical amount or concentration.

Analytical chemistry consists of classical, wet chemical methods and modern, instrumental methods.[2] Classical qualitative methods use separations such as precipitation, extraction, and distillation. Identification may be based on differences in color, odor, melting point, boiling point, radioactivity or reactivity. Classical quantitative analysis uses mass or volume changes to quantify amount. Instrumental methods may be used to separate samples using chromatography, electrophoresis or field flow fractionation. Then qualitative and quantitative analysis can be performed, often with the same instrument and may use light interaction, heat interaction, electric fields or magnetic fields . Often the same instrument can separate, identify and quantify an analyte.

Analytical chemistry is also focused on improvements in experimental design, chemometrics, and the creation of new measurement tools. Analytical chemistry has broad applications to forensics, medicine, science and engineering.

History[edit]

Analytical chemistry has been important since the early days of chemistry, providing methods for determining which elements and chemicals are present in the object in question. During this period significant contributions to analytical chemistry include the development of systematic elemental analysis by Justus von Liebig and systematized organic analysis based on the specific reactions of functional groups.

The first instrumental analysis was flame emissive spectrometry developed by Robert Bunsen and Gustav Kirchhoff who discovered rubidium (Rb) and caesium (Cs) in 1860.[3]

Most of the major developments in analytical chemistry take place after 1900. During this period instrumental analysis becomes progressively dominant in the field. In particular many of the basic spectroscopic and spectrometric techniques were discovered in the early 20th century and refined in the late 20th century.[4]

The separation sciences follow a similar time line of development and also become increasingly transformed into high performance instruments.[5] In the 1970s many of these techniques began to be used together as hybrid techniques to achieve a complete characterization of samples.

Starting in approximately the 1970s into the present day analytical chemistry has progressively become more inclusive of biological questions (bioanalytical chemistry), whereas it had previously been largely focused on inorganic or small organic molecules. Lasers have been increasingly used in chemistry as probes and even to initiate and influence a wide variety of reactions. The late 20th century also saw an expansion of the application of analytical chemistry from somewhat academic chemical questions to forensic, environmental, industrial and medical questions, such as in histology.[6]

Modern analytical chemistry is dominated by instrumental analysis. Many analytical chemists focus on a single type of instrument. Academics tend to either focus on new applications and discoveries or on new methods of analysis. The discovery of a chemical present in blood that increases the risk of cancer would be a discovery that an analytical chemist might be involved in. An effort to develop a new method might involve the use of a tunable laser to increase the specificity and sensitivity of a spectrometric method. Many methods, once developed, are kept purposely static so that data can be compared over long periods of time. This is particularly true in industrial quality assurance (QA), forensic and environmental applications. Analytical chemistry plays an increasingly important role in the pharmaceutical industry where, aside from QA, it is used in discovery of new drug candidates and in clinical applications where understanding the interactions between the drug and the patient are critical.

Classical methods[edit]

Although modern analytical chemistry is dominated by sophisticated instrumentation, the roots of analytical chemistry and some of the principles used in modern instruments are from traditional techniques many of which are still used today. These techniques also tend to form the backbone of most undergraduate analytical chemistry educational labs.

Qualitative analysis[edit]

A qualitative analysis determines the presence or absence of a particular compound, but not the mass or concentration. By definition, qualitative analyses do not measure quantity.

Chemical tests[edit]

Further information: Chemical test

There are numerous qualitative chemical tests, for example, the acid test for gold and the Kastle-Meyer test for the presence of blood.

Flame test[edit]

Further information: Flame test

Inorganic qualitative analysis generally refers to a systematic scheme to confirm the presence of certain, usually aqueous, ions or elements by performing a series of reactions that eliminate ranges of possibilities and then confirms suspected ions with a confirming test. Sometimes small carbon containing ions are included in such schemes. With modern instrumentation these tests are rarely used but can be useful for educational purposes and in field work or other situations where access to state-of-the-art instruments are not available or expedient.

Quantitative analysis[edit]

Further information: Quantitative analysis (chemistry)

Quantitative analysis is the measurement of the quantities of particular chemical constituents present in a substance.

Gravimetric analysis[edit]

Further information: Gravimetric analysis

Gravimetric analysis involves determining the amount of material present by weighing the sample before and/or after some transformation. A common example used in undergraduate education is the determination of the amount of water in a hydrate by heating the sample to remove the water such that the difference in weight is due to the loss of water.

Volumetric analysis[edit]

Further information: Titration

Titration involves the addition of a reactant to a solution being analyzed until some equivalence point is reached. Often the amount of material in the solution being analyzed may be determined. Most familiar to those who have taken chemistry during secondary education is the acid-base titration involving a color changing indicator. There are many other types of titrations, for example potentiometric titrations. These titrations may use different types of indicators to reach some equivalence point.

Instrumental methods[edit]

Main article: Instrumental analysis

Spectroscopy[edit]

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Further information: Spectroscopy

Spectroscopy measures the interaction of the molecules with electromagnetic radiation. Spectroscopy consists of many different applications such as atomic absorption spectroscopy, atomic emission spectroscopy, ultraviolet-visible spectroscopy, x-ray fluorescence spectroscopy, infrared spectroscopy, Raman spectroscopy, dual polarization interferometry, nuclear magnetic resonance spectroscopy, photoemission spectroscopy, Mössbauer spectroscopy and so on.

Mass spectrometry[edit]

Further information: Mass spectrometry

Mass spectrometry measures mass-to-charge ratio of molecules using electric and magnetic fields. There are several ionization methods: electron impact, chemical ionization, electrospray, fast atom bombardment, matrix assisted laser desorption ionization, and others. Also, mass spectrometry is categorized by approaches of mass analyzers: magnetic-sector, quadrupole mass analyzer, quadrupole ion trap, time-of-flight, Fourier transform ion cyclotron resonance, and so on.

Electrochemical analysis[edit]

Further information: Electroanalytical method

Electroanalytical methods measure the potential (volts) and/or current (amps) in an electrochemical cell containing the analyte.[7][8] These methods can be categorized according to which aspects of the cell are controlled and which are measured. The four main categories are potentiometry (the difference in electrode potentials is measured), coulometry (the transferred charge is measured over time), amperometry (the cell's current is measured over time), and voltammetry (the cell's current is measured while actively altering the cell's potential).

Thermal analysis[edit]

Further information: Calorimetry and Thermal analysis

Calorimetry and thermogravimetric analysis measure the interaction of a material and heat.

Separation[edit]

Further information: Separation process, Chromatography, and Electrophoresis

Separation processes are used to decrease the complexity of material mixtures. Chromatography, electrophoresis and Field Flow Fractionation are representative of this field.

Hybrid techniques[edit]

Combinations of the above techniques produce a "hybrid" or "hyphenated" technique.[9][10][11][12][13] Several examples are in popular use today and new hybrid techniques are under development. For example, gas chromatography-mass spectrometry, gas chromatography-infrared spectroscopy, liquid chromatography-mass spectrometry, liquid chromatography-NMR spectroscopy. liquid chromagraphy-infrared spectroscopy and capillary electrophoresis-mass spectrometry.

Hyphenated separation techniques refers to a combination of two (or more) techniques to detect and separate chemicals from solutions. Most often the other technique is some form of chromatography. Hyphenated techniques are widely used in chemistry and biochemistry. A slash is sometimes used instead of hyphen, especially if the name of one of the methods contains a hyphen itself.

Microscopy[edit]

Further information: Microscopy

The visualization of single molecules, single cells, biological tissues and nanomaterials is an important and attractive approach in analytical science. Also, hybridization with other traditional analytical tools is revolutionizing analytical science. Microscopy can be categorized into three different fields: optical microscopy, electron microscopy, and scanning probe microscopy. Recently, this field is rapidly progressing because of the rapid development of the computer and camera industries.

Lab-on-a-chip[edit]

Further information: Microfluidics and Lab-on-a-chip

Devices that integrate (multiple) laboratory functions on a single chip of only millimeters to a few square centimeters in size and that are capable of handling extremely small fluid volumes down to less than picoliters.

Errors[edit]

Main article: Approximation error

Error can be defined as numerical difference between observed value and true value.[15]

In error the true value and observed value in chemical analysis can be related with each other by the equation

where

E = absolute error,
O = observed value,
T = true value.

Error of a measurement is an inverse measure of accurate measurement i.e. smaller the error greater the accuracy of the measurement. Errors are expressed relatively as:

× 100 = % error,
× 1000 = per thousand error

Standards[edit]

See also: Analytical quality control

Standard curve[edit]

A general method for analysis of concentration involves the creation of a calibration curve. This allows for determination of the amount of a chemical in a material by comparing the results of unknown sample to those of a series of known standards. If the concentration of element or compound in a sample is too high for the detection range of the technique, it can simply be diluted in a pure solvent. If the amount in the sample is below an instrument's range of measurement, the method of addition can be used. In this method a known quantity of the element or compound under study is added, and the difference between the concentration added, and the concentration observed is the amount actually in the sample.

Internal standards[edit]

Sometimes an internal standard is added at a known concentration directly to an analytical sample to aid in quantitation. The amount of analyte present is then determined relative to the internal standard as a calibrant. An ideal internal standard is isotopically-enriched analyte which gives rise to the method of isotope dilution.

Standard addition[edit]

The method of standard addition is used in instrumental analysis to determine concentration of a substance (analyte) in an unknown sample by comparison to a set of samples of known concentration, similar to using a calibration curve. Standard addition can be applied to most analytical techniques and is used instead of a calibration curve to solve the matrix effect problem.

Signals and noise[edit]

One of the most important components of analytical chemistry is maximizing the desired signal while minimizing the associated noise.[16] The analytical figure of merit is known as the signal-to-noise ratio (S/N or SNR).

Noise can arise from environmental factors as well as from fundamental physical processes.

Thermal noise[edit]

Main article: Johnson–Nyquist noise

Thermal noise results from the motion of charge carriers (usually electrons) in an electrical circuit generated by their thermal motion. Thermal noise is white noise meaning that the power spectral density is constant throughout the frequency spectrum.

The root mean square value of the thermal noise in a resistor is given by[16]

where kB is Boltzmann's constant, T is the temperature, R is the resistance, and is the bandwidth of the frequency .

Shot noise[edit]

Main article: Shot noise

Shot noise is a type of electronic noise that occurs when the finite number of particles (such as electrons in an electronic circuit or photons in an optical device) is small enough to give rise to statistical fluctuations in a signal.

Shot noise is a Poisson process and the charge carriers that make up the current follow a Poisson distribution. The root mean square current fluctuation is given by[16]

where e is the elementary charge and I is the average current. Shot noise is white noise.

Flicker noise[edit]

Main article: flicker noise

Flicker noise is electronic noise with a 1/ƒ frequency spectrum; as f increases, the noise decreases. Flicker noise arises from a variety of sources, such as impurities in a conductive channel, generation and recombination noise in a transistor due to base current, and so on. This noise can be avoided by modulation of the signal at a higher frequency, for example through the use of a lock-in amplifier.

Environmental noise[edit]

Environmental noise arises from the surroundings of the analytical instrument. Sources of electromagnetic noise are power lines, radio and television stations, wireless devices, Compact fluorescent lamps[17] and electric motors. Many of these noise sources are narrow bandwidth and therefore can be avoided. Temperature and vibration isolation may be required for some instruments.

Noise reduction[edit]

Noise reduction can be accomplished either in computer hardware or software. Examples of hardware noise reduction are the use of shielded cable, analog filtering, and signal modulation. Examples of software noise reduction are digital filtering, ensemble average, boxcar average, and correlation methods.[16]

Applications[edit]

Analytical chemistry has applications including in forensic science, bioanalysis, clinical analysis, environmental analysis, and materials analysis. Analytical chemistry research is largely driven by performance (sensitivity, detection limit, selectivity, robustness, dynamic range, linear range, accuracy, precision, and speed), and cost (purchase, operation, training, time, and space). Among the main branches of contemporary analytical atomic spectrometry, the most widespread and universal are optical and mass spectrometry.[18] In the direct elemental analysis of solid samples, the new leaders are laser-induced breakdown and laser ablation mass spectrometry, and the related techniques with transfer of the laser ablation products into inductively coupled plasma. Advances in design of diode lasers and optical parametric oscillators promote developments in fluorescence and ionization spectrometry and also in absorption techniques where uses of optical cavities for increased effective absorption pathlength are expected to expand. The use of plasma- and laser-based methods is increasing. An interest towards absolute (standardless) analysis has revived, particularly in emission spectrometry.[citation needed]

Great effort is being put in shrinking the analysis techniques to chip size. Although there are few examples of such systems competitive with traditional analysis techniques, potential advantages include size/portability, speed, and cost. (micro total analysis system (µTAS) or lab-on-a-chip). Microscale chemistry reduces the amounts of chemicals used.

Many developments improve the analysis of biological systems. Examples of rapidly expanding fields in this area are genomics, DNA sequencing and related research in genetic fingerprinting and DNA microarray; proteomics, the analysis of protein concentrations and modifications, especially in response to various stressors, at various developmental stages, or in various parts of the body, metabolomics, which deals with metabolites; transcriptomics, including mRNA and associated fields; lipidomics - lipids and its associated fields; peptidomics - peptides and its associated fields; and metalomics, dealing with metal concentrations and especially with their binding to proteins and other molecules.[citation needed]

Analytical chemistry has played critical roles in the understanding of basic science to a variety of practical applications, such as biomedical applications, environmental monitoring, quality control of industrial manufacturing, forensic science and so on.[19]

The recent developments of computer automation and information technologies have extended analytical chemistry into a number of new biological fields. For example, automated DNA sequencing machines were the basis to complete human genome projects leading to the birth of genomics. Protein identification and peptide sequencing by mass spectrometry opened a new field of proteomics.

Analytical chemistry has been an indispensable area in the development of nanotechnology. Surface characterization instruments, electron microscopes and scanning probe microscopes enables scientists to visualize atomic structures with chemical characterizations.

See also[edit]

References[edit]

  1. ^Skoog, Douglas A.; West, Donald M.; Holler, F. James; Crouch, Stanley R. (2014). Fundamentals of Analytical Chemistry. Belmont: Brooks/Cole, Cengage Learning. p. 1. ISBN 0-495-55832-X. 
  2. ^Skoog, Douglas A.; Holler, F. James; Crouch, Stanley R. (2007). Principles of Instrumental Analysis. Belmont, CA: Brooks/Cole, Thomson. p. 1. ISBN 0-495-01201-7. 
  3. ^Arikawa, Yoshiko (2001). "Basic Education in Analytical Chemistry"(pdf). Analytical Sciences. The Japan Society for Analytical Chemistry. 17 (Supplement): i571–i573. Retrieved 10 January 2014. 
  4. ^Miller, K; Synovec, RE (2000). "Review of analytical measurements facilitated by drop formation technology". Talanta. 51 (5): 921–33. doi:10.1016/S0039-9140(99)00358-6. PMID 18967924. 
  5. ^Bartle, Keith D.; Myers, Peter (2002). "History of gas chromatography". TrAC Trends in Analytical Chemistry. 21 (9–10): 547. doi:10.1016/S0165-9936(02)00806-3. 
  6. ^Laitinen, H.A. (1989). "History of analytical chemistry in the U.S.A". Talanta. 36 (1–2): 1–9. doi:10.1016/0039-9140(89)80077-3. PMID 18964671. 
  7. ^Bard, A.J.; Faulkner, L.R. Electrochemical Methods: Fundamentals and Applications. New York: John Wiley & Sons, 2nd Edition, 2000.[page needed]
  8. ^Skoog, D.A.; West, D.M.; Holler, F.J. Fundamentals of Analytical Chemistry New York: Saunders College Publishing, 5th Edition, 1988.[page needed]
  9. ^Wilkins, C. (1983). "Hyphenated techniques for analysis of complex organic mixtures". Science. 222 (4621): 291–6. Bibcode:1983Sci...222..291W. doi:10.1126/science.6353577. PMID 6353577. 
  10. ^Holt, R. M.; Newman, M. J.; Pullen, F. S.; Richards, D. S.; Swanson, A. G. (1997). "High-performance Liquid Chromatography/NMR Spectrometry/Mass Spectrometry:Further Advances in Hyphenated Technology". Journal of Mass Spectrometry. 32 (1): 64–70. Bibcode:1997JMSp...32...64H. doi:10.1002/(SICI)1096-9888(199701)32:1<64::AID-JMS450>3.0.CO;2-7. PMID 9008869. 
  11. ^Ellis, Lyndon A; Roberts, David J (1997). "Chromatographic and hyphenated methods for elemental speciation analysis in environmental media". Journal of Chromatography A. 774 (1–2): 3–19. doi:10.1016/S0021-9673(97)00325-7. PMID 9253184. 
  12. ^Guetens, G; De Boeck, G; Wood, M; Maes, R.A.A; Eggermont, A.A.M; Highley, M.S; Van Oosterom, A.T; De Bruijn, E.A; Tjaden, U.R (2002). "Hyphenated techniques in anticancer drug monitoring". Journal of Chromatography A. 976 (1–2): 229–38. doi:10.1016/S0021-9673(02)01228-1. PMID 12462614. 
  13. ^Guetens, G; De Boeck, G; Highley, M.S; Wood, M; Maes, R.A.A; Eggermont, A.A.M; Hanauske, A; De Bruijn, E.A; Tjaden, U.R (2002). "Hyphenated techniques in anticancer drug monitoring". Journal of Chromatography A. 976 (1–2): 239–47. doi:10.1016/S0021-9673(02)01227-X. PMID 12462615. 
  14. ^Schermelleh, L.; Carlton, P. M.; Haase, S.; Shao, L.; Winoto, L.; Kner, P.; Burke, B.; Cardoso, M. C.; Agard, D. A.; Gustafsson, M. G. L.; Leonhardt, H.; Sedat, J. W. (2008). "Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy". Science. 320 (5881): 1332–6. Bibcode:2008Sci...320.1332S. doi:10.1126/science.1156947. PMC 2916659. PMID 18535242. 
  15. ^G.L. David - Analytical Chemistry
  16. ^ abcdCrouch, Stanley; Skoog, Douglas A. (2007). Principles of instrumental analysis. Australia: Thomson Brooks/Cole. ISBN 0-495-01201-7. [page needed]
  17. ^"Health Concerns associated with Energy Efficient Lighting and their Electromagnetic Emissions"(PDF). Trent University, Peterborough, ON, Canada. Retrieved 2011-11-12. 
  18. ^Bol'Shakov, Aleksandr A; Ganeev, Aleksandr A; Nemets, Valerii M (2006). "Prospects in analytical atomic spectrometry". Russian Chemical Reviews. 75 (4): 289. arXiv:physics/0607078. Bibcode:2006RuCRv..75..289B. doi:10.1070/RC2006v075n04ABEH001174. 
  19. ^"Analytical Chemistry - American Chemical Society". American Chemical Society. Retrieved 2017-05-26. 

Further reading[edit]

  • Skoog, D.A.; West, D.M.; Holler, F.J. Fundamentals of Analytical Chemistry New York: Saunders College Publishing, 5th Edition, 1988.
  • Bard, A.J.; Faulkner, L.R. Electrochemical Methods: Fundamentals and Applications. New York: John Wiley & Sons, 2nd Edition, 2000.
  • Bettencourt da Silva, R; Bulska, E; Godlewska-Zylkiewicz, B; Hedrich, M; Majcen, N; Magnusson, B; Marincic, S; Papadakis, I; Patriarca, M; Vassileva, E; Taylor, P; Analytical measurement: measurement uncertainty and statistics, 2012, ISBN 978-92-79-23070-7.

External links[edit]

Gustav Kirchhoff (left) and Robert Bunsen (right)
The presence of copper in this qualitative analysis is indicated by the bluish-green color of the flame
Block diagram of an analytical instrument showing the stimulus and measurement of response
Noise in a thermogravimetric analysis; lower noise in the middle of the plot results from less human activity (and environmental noise) at night

The main steps that are performed during a chemical analysis are the following: (1) sampling, (2) field sample pretreatment, (3) laboratory treatment, (4) laboratory assay, (5) calculations, and (6) results presentation. Each must be executed correctly in order for the analytical result to be accurate. Some analytical chemists distinguish between an analysis, which involves all the steps, and an assay, which is the laboratory portion of the analysis.

Sampling

During this initial step of analysis, a portion of a bulk material is removed in order to be assayed. The portion should be chosen so that it is representative of the bulk material. To assist in this, statistics is used as a guide to determine the sample size and the number of samples. When selecting a sampling program, it is important that the analyst has a detailed description of the information required from the analysis, an estimate of the accuracy to be achieved, and an estimate of the amount of time and money that can be spent on sampling. It is worthwhile to discuss with the users of the analytical results the type of data that is desired. Results may provide needless or insufficient information if the sampling procedure is either excessive or inadequate.

Generally the accuracy of an analysis is increased by obtaining multiple samples at varying locations (and times) within the bulk material. As an example, analysis of a lake for a chemical pollutant will likely yield inaccurate results if the lake is sampled only in the centre and at the surface. It is preferable to sample the lake at several locations around its periphery as well as at several depths near its centre. The homogeneity of the bulk material influences the number of samples needed. If the material is homogeneous, only a single sample is required. More samples are needed to obtain an accurate analytical result when the bulk material is heterogeneous. The disadvantages of taking a larger number of samples are the added time and expense. Few laboratories can afford massive sampling programs.

Sample preparation

After the sample has been collected, it may be necessary to chemically or physically treat it at the sampling site. Normally this treatment is done immediately after the sample has been collected. The nature of the treatment is dependent on the sample and the substances for which it is being analyzed. For example, natural water samples that are assayed for dissolved oxygen generally are placed in containers that are sealed, stored, and transported in a refrigerated compartment. Sealing prevents a change in oxygen concentration owing to exposure to the atmosphere, and refrigeration slows changes in oxygen levels caused by microscopic organisms within the sample. Similarly, samples that are to be assayed for trace levels of metallic pollutants are pretreated in order to prevent a decrease in the concentration of the pollutant that is caused by adsorption on the walls of the sample vessel. Metallic adsorption can be minimized by adding nitric acid to the sample and by washing the walls of the vessel with the acid.

After the samples arrive at the laboratory, additional operations might be required prior to performing the assay. In some cases, multiple samples simply are combined into a composite sample which is made homogeneous and then assayed. This process eliminates the need to assay each of the individual specimens. In other instances, the sample must be chemically or physically treated in order to place it in a form that can be assayed. For example, ore samples normally must be first dissolved in acidic solutions. Sometimes it is necessary to change the concentration of the analyte prior to performing the assay so that it will fall within the range of the analytical method. Once the specimen is prepared, enough laboratory assays are completed to allow the analyst to estimate the amount of random error. Typically a minimum of three assays are performed on each sample.

Evaluation of results

After the assays have been completed, quantitative results are mathematically manipulated, and both qualitative and quantitative results are presented in a meaningful manner. In most cases, two values are reported for quantitative analyses. The first value is an estimate of the correct value for the analysis, and the second value indicates the amount of random error in the analysis. The most common way of reporting the best value is to give the mean (average) of the results of the laboratory assays. In specific cases, however, it is better to report either the median (central value when the results are arranged in order of size) or the mode (the value obtained most often).

Accuracy is the degree of agreement between the experimental result and the true value. Precision is the degree of agreement among a series of measurements of the same quantity; it is a measure of the reproducibility of results rather than their correctness. Errors may be either systematic (determinant) or random (indeterminant). Systematic errors cause the results to vary from the correct value in a predictable manner and can often be identified and corrected. An example of a systematic error is improper calibration of an instrument. Random errors are the small fluctuations introduced in nearly all analyses. These errors can be minimized but not eliminated. They can be treated, however, using statistical methods. Statistics is used to estimate the random error that occurs during each step of an analysis, and, upon completion of the analysis, the estimates for the individual steps can be combined to obtain an estimate of the total experimental error.

The most frequently reported error estimate is the standard deviation of the results; however, other values, such as the variance, the range, the average deviation, or confidence limits at a specified probability level are sometimes reported. For the relatively small number of replicate samples that are used during chemical assays, the standard deviation (s) is calculated by using equation (1) where Σ represents summation, xi represents each of the individual analytical results, a is the average of the results, and N is the number of replicate assays.

The standard deviation is a popular estimate of the error in an analysis because it has statistical significance whenever the results are normally distributed. Most analytical results exhibit normal (Gaussian) behaviour, following the characteristic bell-shaped curve. If the results are normally distributed, 68.3 percent of the results can be expected to fall within the range of plus or minus one standard deviation of the mean as a result of random error. The units of standard deviation are identical to those of the individual analytical results.

The variance (V) is the square of the standard deviation and is useful because, in many cases, it is additive throughout the several steps of the chemical analysis. Consequently, an estimate of the total random error in the analysis can be obtained by adding the variances for each of the individual steps in the analysis. The standard deviation for the overall analysis can then be calculated by taking the square root of the sum of the variances.

A simple measure of variability is the range, given as the difference between the largest and the smallest results. It has no statistical significance, however, for small data sets. Another statistical term, the average deviation, is calculated by adding the differences, while ignoring the sign, between each result and the average of all the results, and then dividing the sum by the number of results. Confidence limits at a given probability level are values greater than and less than the average, between which the results are statistically expected to fall a given percentage of the time.