CHAPTER STATISTICAL ANALYSIS, II ERRORS, EXPRESSION OF RESULTS STATISTICAL ANALYSIS Statistical methods of analysis are widely used in the prese...

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STATISTICAL ANALYSIS Statistical methods of analysis are widely used in the presentation and reduction of the analytical data and in the evaluation of such data. The food analyst is usually required to measure the effect of certain environmental conditions on the composition of food, e.g., the effect of variety, maturity, and growing conditions on the composition of plant food. He is often asked to determine the efficacy of a particular treatment in preventing decay, and also has to evaluate the relative reliability of several available methods of analysis. A knowledge of statistical analyses is necessary: to provide a sound basis for the formulation of the experiment so that the effect of the primary factors and their interrelationship may be clearly and readily estimated; to provide means for evaluating the reliability of the results so that the significance of the effects observed may be determined; and to provide for adequate reduction of the data by means of tables or graphs, correlation diagrams, descriptive statistics, etc. The variability of the food product to be analyzed also requires the use of statistical methods in obtaining a representative and valid sample and in estimating the error of sampling. Investigations in food chemistry deal with data which involve both the errors of measurement and variability of the experimental material used. Measures of the precision of the mean result as well as statistical analysis of observational data are required. It is beyond the scope of this work to discuss the methods of statistical analysis, but it is believed desirable to present some definitions and ideas which would clarify the language of statistical analysis and make the descriptive statistics commonly used in the tabulation of experimental data intelligent. For the discussion of the principles and practices of statistical analysis the student is referred to the well-known works cited in the bibliography at the end of this chapter. In the tabulation of data the most commonly reported descriptive statistics are the arithmetic mean, average deviation from the mean, standard deviation, standard error of mean, and probable error of mean. The median or the mode is often used in measurement of central tendencies and correlation coefficient is used as a measure of the degree to which two variâtes tend to be associated in value. The probable error of the individual value is used also. Arithmetic Mean. The arithmetic mean, or mean, is that value of a set of values which best represents the sample as a whole. I t is the most commonly used average and is usually the most probable value of the un13



known, particularly for a large array of observed variants of a normal distribution. The arithmetic mean is the sum of all the measurements, or variâtes, divided by the number of measurements or observations. I t is denoted by x, m, a, and may be expressed Xl




XS +


' '

t +

Xi +

· · · +





Ν Ν where large 2 or S is the summation of all the Ν values of χ from 1 to n. The algebraic sum of a set of deviations from the mean is zero; and the sum of the squares of the deviation from the mean is a minimum. Median. In a normally distributed population the mean falls about the middle of the total range but when the distribution or the range of variation shifts, the mean shifts also and the mean may not be at all typical of populations which depart notably from the normal. In such cases, another measure of central tendency, the median, may be used to supplement the mean. The median is the middle item in an array of items ranged in order of magnitude, if there are an odd number of measurements. If the number is even, the median is approximately the average of the two middle items. Mode. The mode is the value of the measurement which occurs most frequently. I t is "la mode," the fashion, or the typical one. Sometimes, however, there is no measurement which occurs more frequently than any other. The position of the mode so defined will depend on the interval of classification chosen. The mode may be calculated from the equation Mode = Mean — 3 (Mean — Median) If the items .are plotted as a histogram (frequency diagram) and the graph fitted with a smooth curve, the abscissa of the maximum point of the curve is the mode. The position of the mode in certain cases of distribution, when it can be mathematically expressed by Pearson's Type I I I curve, y = ae~bt (b + t)c, can be calculated by the formula Mean minus mode σ

a3 ~~ ~ T ~

where σ is the standard deviation = μ is the "movement";

V μ2 =

\J ~^^(u

)w2 / /



u an observed value; and / is the frequency (number of times) it occurs in a distribution



aS — — is a measure of "skewness," or the extent to which the distribution bulges on one side more than the other, in comparison with the normal distribution, which is defined by the equation

where σ is the standard deviation of the population; Ν is the total number of variâtes; y is the frequency at any given point x, where χ represents the mean of the population. D i s p e r s i o n . The arithmetic mean supplies only a portion of the information about a sample since it does not describe the extent to which the values in the sample vary from the mean. Measures of the variability of the sample, or the dispersion of the values about the mean are necessary also. The most commonly used measures of dispersion are range, the average deviation, and the standard deviation. R a n g e . The simplest but least exact expression of dispersion is a statement of the range over which the values of a group vary. This may be stated explicitly as the difference between the highest and lowest values, or it may be stated implicitly by giving the maximum and minimum values together with the average. I t may be expressed also as the range coefficient, Kr


Range Mean

Obviously the range is not a sufficient measure of dispersion since it is dependent solely on the two extreme values and independent of the number of measurements. A v e r a g e Deviation. The deviation from the mean is, by convention, always found by subtracting the mean from the observation dx — Xn — χ The algebraic sum of the deviations from the mean is zero

2d =

2 x — nx =




but when the summation is made by disregarding the plus and minus signs, i.e., by using the absolute value of the deviation, one may obtain an average deviation which gives a dispersion measure that unlike the range is a function of all the values and not just the extremes. The average deviation D or δ is defined as


D =

η Standard Deviation. The most widely used average of variation is the root mean square average of the deviations from the arithmetic mean, known as the standard deviation from the mean. I t is not only a function of all the values of a group but is also more sensitive than the average deviation to the presence of large or small deviations. In a long series or a normally distributed sample about two-thirds of the deviation will be less than the standard deviation, σ or s, i.e., the interval χ-\- σ contains about two-thirds of the items. The standard deviation is obtained by taking the squares of the differences from the mean, finding their average, and then taking the square root of the average, i.e., I 2dx2

Ι $ ( χ — χ)2 _

_ \





where Xdx is the sum of the squares of the differences between the individual observations and the mean. Closely related to the standard deviation from the mean is the so-called standard error of the mean which is a measure of the deviations between means obtained from successive random samples taken from the same population. The standard error, S.E., or am is found to be σ crm



= r


where η is the number of observations. In dealing with small samples, the mean may not be a significant measure of the population, and the standard deviation of the sample is not likely to be equal to that of the population as a whole or the statistical ensemble. In dealing with small samples, in place of the true standard deviation of the observations, a greater value is ascribed to the standard deviation of the sample by dividing %(dx)2 by η— 1 instead of by n. The standard error is also given by BessePs formula as


This is equivalent to σ/yjn small samples as


where the standard deviation is calculated for


n—1 A closely related measure of the variability of the data is the coefficient of variation, Standard Deviation X

10 0

Elving et al. ( 1 9 4 8 ) use as indexes of precision for small numbers of items, Standard deviation Standard error Confidence range where χ di η c2

— — = —

* ~~


η— 1


am χ =






arithmetical average or mean deviation of an individual result from the mean number of items or results correction factor for the small number of items used n — 0.8

In a random distribution of observed values, the standard deviation of sample may be taken as a measure of the accuracy of sampling and of the significance of the result. The standard deviation of the average of a second sample will fall only approximately once out of three outside the range of variation mx db σι for the first, once out of 22 times outside mx ± 2σι, and once out of 3 7 0 times outside the range m± ± 3σ±. Since m ± 3 σ includes practically the entire area bounded by the normal frequency distribution curve, it is customary to take three times the standard deviation of mean as a measurement of accuracy. The difference between the averages of results of two series of observations is considered to be significant as a measure of the environmental factor studied when the quotient m± — m2 is greater than three.

V σ!



σ 22



P r o b a b l e Error. The probable error of the mean is an average measure of deviation which is as likely to be exceeded as not to be exceeded. In a normally distributed sample the interval m + Ρ·Ε. contains half the items. The probable error is approximately seven-tenths of the standard deviation. More accurately it is expressed as P.E. =



-' n— 1



Where the population is small this is the P.E. of the single value and the probable error of the mean is 1/V η times this. The probable error is a precision measure that is applicable to the ascertainment of the degree of certainty or trustworthiness of the result of an investigation whose data are subject to both errors of measurement and variability of experimental material. Coefficient of Correlation. The relation between the different characteristics of the members of a given population may be implied or explicitly stated. Where the data implies the relation between two variâtes, the correlation coefficient, r, is used as a measure of the degree in which two variâtes tend to be related or associated in value. The most commonly used coefficient of correlation is given by the expression r =

%{dadv) ησχσν

where ^{dxdy)

= the sum of the products of the differences between the individual measures and their respective means for each individual η = the number of individuals CTx, (Ty = the standard deviations of the two series This correlation coefficient may have all possible values from to — 1 . The value + 1 indicates that the one variable always increases or decreases by the same amount for unit increase or decrease of the other. A value of —1 indicates that an increase in one variable is always connected with a decrease in the other and vice versa. T h e greater the value of r, the more closely" it approaches 1.0, the closer is the relation between the variables. R e g r e s s i o n . Another and more powerful measure of relationship between variants is "regression." The statistical methods of regression are used not only to discover and evaluate such relationships but also to formulate the curves or graphs of closest fit. For discussion of the techniques used consult the sources of reference listed below. Another widely used method of cal-



culating the algebraic equation that best expresses the relation between two or more quantities is the method of least squares. The principle involved is that the equation be so calculated that the sum of the squares of the deviations of observed values of the variable from the corresponding values for the graph of best fit must be a minimum. The curve fitting by this procedure is rather laborious even for linear relation, y = ax -f- b, and is more so for the parabola of closest fit, y = ax2 + b% + -c Graphical R e p r e s e n t a t i o n a n d Graphical C o m p u t a t i o n . Graphical representation of statistical data is useful in statistical analysis. Various types of statistical graphs are employed: bar diagram, pie diagrams, pictograms, line diagrams, plots of variables on uniform Cartesian coordinates, and plots with nonuniform scales such as scales of squares, reciprocals, logarithmic, and semi-logarithmic. Graphs are also useful in analysis of experimental data, in establishing experimental details, and in correlation and computation of data. Graphical computation is a useful technique in the repeated solution of mathematical formulas. Simple equations may be solved by alignment charts consisting of straight-line scales, whereas more involved equations may necessitate the use of grids, curved scales, and combinations of Cartesian coordinate charts with alignment charts. Such charts are known as nomograms, and the principles involved in their preparation and use are known as nomography. Nomography may be used not only for the solution of explicit functions but also for the representation and use of implied relations such as those in tables of Balling degree vs. specific gravity, alcohol content vs. specific gravity, sugar content vs. reduced copper. Several of the useful standard reference works in this field are given in the bibliography. ERRORS The errors incident to the particular analysis of a given food are of two kinds: errors due to sampling caused by the variability in composition of the food and errors in the determination itself. The latter are either errors due to variation in technique, in reagents used, etc., which may be called the error of operation; errors of standardization if the actual determination is performed by means of instruments whose indication must be translated by reference to some primary standard; and errors inherent in the particular determination selected either as a result of uncontrollable factors or because of other inadequacies. The error of a determination is defined strictly as the discrepancy between an observed value and the "true value" based on the average of a finite number of observations. A distinction usually is made between error and the percentage error based on the average of a number of détermina-



tions. The former is more correctly referred to as the deviation of the observation. The average deviation, average difference between a single observed value and the average of all values, is a measure of the consistency of the results; but because of the possibility of the occurrence of constant errors a low average deviation is not necessarily an indication of accuracy. It merely determines the reproducibility or the precision of the method, and not its correctness or accuracy. Sometimes, by compensation of errors, an inherently inaccurate procedure may yield the correct result. A common source of error in oxidimetric titrations is the variation in method of mixing the reductant and oxidant which under certain conditions markedly changes the extent of induced oxidation by oxygen, Bray and Ramsey (1933). Manov and Kirk (1937), for example, observed constant irregularities in the current micromethods for chromate. Results could often be reproduced to within ± 0 . 3 per cent, but the absolute error was usually —5 per cent or more, owing to the counterbalancing of unknown and often large errors. The most commonly used measure of precision is the probable error, r> defined as

An approximate formula for computing the probable error from the average deviation, known as Peter's formula, is r — 0.8454 2|<£r

The experimental errors which may affect the accuracy of a result are usually classified as systematic and erratic or accidental. Systematic errors are those whose sources are known and which may be eliminated or for which corrections may be made. Observations may be influenced by the instruments used, the external conditions at the time of observation and by the personal errors of the observer. The latter will depend on the experience of the observer and his psychological and physiological limitations and idiosyncracies. The personal error may be estimated only for a careful experienced worker when in his normal state of health or when free from fatigue. Erratic errors are errors which are beyond the control of the experimenter. They follow the law of chance and there is equal probability that any given observation may be greater or less than the true value. Among these accidental errors are included those due to changes in the personal error of the observer, the undetected residual instrument errors, and the overrunning of an end point in a titration. The constant error, an unknown



factor which affects the accuracy of the result in one direction only, is usually not included in erratic errors. The difficulty of evaluating these various errors and of determining the kind of errors involved makes the determination of the probable error in any given observation very complicated. The observer must be aware of the systematic errors, he must know what erratic errors are involved and must judge from the reproducibility of his results the probable error in any one isolated observation. When the results of several observations are estimated independently, then it may be shown by the laws of small numbers that if a result R is to be calculated from measurements A and B, and the percentage error in each is small, the percentage errors for various cases are as follows: R =

A Χ Β, ^

X 100 = > / ( j X





X 100


( j X



= > J ( j X 1 0 0 ) 2+

R = A^B^XIQ0 r

1 0 θ ) 2+


I a2 + b2 Α - Β


10 0

where r, a, and b are the actual errors respectively in R, A and B. In general the resultant error Ε due to errors βι, e2, es, etc. is given by the expression Ε =


2 x

+ e22 + es2 + . . .

In expressing the results of analyses there is often a tendency to state the value in terms of more significant figures than are justified by the precision of the particular method of analysis or the variability of the sample. This lends a fictitious air of accuracy to the result. It is customary to retain only one uncertain figure in stating the numerical value of a measured quantity. In general the accuracy of a determination will depend on the completeness with which interfering substances are removed, the error involved in transforming the substance into the form suitable for analysis, and the error involved in the actual analysis. Gravimetric processes are subject to errors resulting from loss of the compound to be weighed either from solubility or manipulation, to errors of occlusion or other difficulties that yield an impure product or one of inconstant composition, and to errors in weighing. Volumetric processes involve errors in preparation and measurement of standard solution used, in the determination of the end point of the reaction upon which the process depends, and errors in measuring



volumes. Since weight can be more accurately determined than volume, and if other factors are comparable, the volumetric process is less accurate than the gravimetric. Actually, however, many volumetric processes are more accurate than the corresponding gravimetric processes, and they are on the whole more sensitive. The accuracy of volumetric processes may be improved by using weight burets instead of volume burets or by using more dilute solutions and more sensitive indicators. The macro-procedures are usually more accurate than the corresponding micro-procedures. For a more detailed discussion of the factors influencing the accuracy of analytical procedures consult the general references cited in the terminal bibliography. A common systematic error constantly (but often unconsciously) made in the calculation of analytical results is the use of atomic weights carefully corrected for buoyancy of air for results of weighings which are not reduced to vacuo. The error involved is actually greater than is commonly supposed, Schoorl (1930), certainly far greater than the error of 27 parts in 1 million introduced by the commonly accepted assumption that 1 cc. is 0.001 1. (actually the volume occupied by 1 kg. of water at 4° C , the international liter, is 1000.027 c c ) . Another and very common source of error is that due to incomplete drainage of volumetric apparatus, Jones and Ferrell (1939), which sometimes may exceed tolerance in the volume of the glassware used. EXPRESSION OF RESULTS The analyses of foods are reported in arbitrary units and upon various bases. The composition may be expressed in terms of the fresh or green weight or upon a moisture free basis. I t may be expressed in terms of the food as purchased (A.P.) or in terms of the edible portion ( E . P . ) , i.e., the part most commonly eaten which is free from "refuse" (the peel, pits, stems that are discarded in preparation). It may be expressed in terms of percentage by weight, or percentage by volume; the composition of liquids, beverages, often is expressed in grams per 100 ml. When the constituent is present in very small amounts, it is usually expressed in terms of parts per million (p.p.m.), milligrams per kilogram or per liter, or as in the case of the vitamins as micrograms per 100 g. or 100 ml. Spray residues are stated as grains per pound. The mineral content may be expressed in terms of either the ash or the fresh basis. The composition may be given, not in terms of the constituent actually determined or even that present, but in terms of an arbitrarily selected unit. Thus the mineral content may be expressed in terms of the highest valency oxide of the particular element in terms of the carbon dioxide-free ash.



Where several similar constituents are present, the composition is given in terms of that which predominates. Thus, the total free acid is expressed as the total titratable acid, as citric, malic, tartaric, lactic, or acetic, depending upon which is considered to predominate. The total organic nitrogen present is assumed to be derived entirely from proteins, usually from proteins averaging 16% in nitrogen content, so that protein in nearly all cases is total nitrogen times 6.25. Where the type of proteins present and their nitrogen content are known, the appropriate factors may be used; many of these have been published by Jones (1931). All the substances capable of reducing alkaline copper (or in some cases alkaline ferricyanide solutions) tartrate solutions are assumed to be reducing sugars and are expressed as invert or dextrose. The sucrose and starch content are even more difficult to interpret because they depend so much upon the specificity of the particular method of analysis used. Extraction procedures are used in many instances, and compounds possessing similar solubilities in particular solvents used are grouped together. Thus fat determined as ether extract includes not only true fats but various other ether soluble substances such as fatty acids, sterols, lecithin, plant pigments, and waxes. The common practice of stating composition as percentages of the dry matter necessarily implies that the water content be determined as loss in weight in drying the material under specified conditions. Sometimes, as in analysis of dried fruit, it is useful to report the composition in terms of an assumed moisture content, e.g. 20% moisture. Although there is some variation in the terms used for reporting analytical results, there has been a great unification of terms for the analyses of given foods. For the most part the recommendations made by the Association of Official Agricultural Chemists in the current edition of the "Official and Tentative Methods of Analysis" are followed save in some particular case. The development of the present practice was based upon a resolution adopted at the twenty-first meeting of the A.O.A.C. in 1904, Davidson (1905). The customary and alternative usage is given in the appropriate place elsewhere in the text. Food analysts are concerned also with the determination of effect of processing, treatment as well as maturity and growing conditions on the composition of foods. Here they are faced with the same difficulty as, e.g., are plant physiologists. The expression of composition as percentages of dry matter may fail to denote changes in the absolute amount of constituents. In this respect the amount of constituent per plant, or multiple thereof, is advantageous. Chibnall (1923), however, stresses the possibility that variations of less abundant constituents may be masked by increases of inert



wall material. The basis of dry matter plus bound water content has been suggested also, Newton and Gortner (1922). Pitman (1935) found the oil content of olives expressed on the fresh basis to be a better criterion of olive maturity than oil content on the dry basis. He found also that the oil content per olive decreased after pickling about to the same extent as did the oil content expressed as per cent oil on the wet or green basis. On the dry basis, apparently because of loss of substances soluble in water and in lye which counterbalanced actual loss in oil, the average oil content on the dry basis before pickling was about the same as that after pickling. In judging the maturity of canned olives from their oil content, the wet basis was preferred by Cruess et al. (1939) to the dry basis, although for following changes in olive constituents during pickling they preferred expression of the analytical results in terms of grams per two hundred olives. D . Appleman and L. B. Noda (1941) found that the oil content of the California Fuerte avocadoes increased more regularly with maturity when expressed per fruit rather than as per cent by weight. The usability of analytical data showing the effect of maturity or environmental conditions depends on the correct choice of a basis for measurement which should remain constant. In the exact determination of the diurnal variation of the carbohydrate content of leaves, Denny (1932) found neither the fresh weight nor the leaf area as satisfactory as the residual dry weight of all constituents minus total carbohydrates, for the latter show the greatest variation. Kertesz (1933) has stressed the necessity for the determination of absolute amounts of a given constituent per plant unit in following its appearance or disappearance with metabolic changes. By the treatment of peas in groups of equal numbers he demonstrated the fallacy of a common belief that sucrose is converted to starch when canning of the harvested crop is delayed [see, however, Bisson and Jones ( 1 9 3 4 ) ] . Bisson et al. (1932-1936) introduced the use of a reference element which does not undergo change in weight during storage as a basis for the calculation of the absolute amounts of the various components for the determination of changes in stored material. Elements such as magnesium, calcium, and phosphorus which do not undergo change in weight during storage may be chosen as reference elements. In peas the phosphorus content was particularly useful. The application of this method to the determination of changes in composition of shelled peas stored at 25° C. is illustrated as follows. "In the check 49.9 grams of dry matter (column 2 of Table 1) was associated with 0.292 gram of phosphorus (column 3 ) . The average percentage of phosphorus in the eight samples of the check was 0.586, with a









and Jones,


C. S.,




Α., Determining






























Total sugars








M a t e r i a l b y U s e of a R e f e r e n c e













4.66 4.88



13 4.31 10.62



Crude fiber

of dry


of 49.9 grams









initial weight phosphorus





4 49.9





Days in storage


Actual weight of phosphorus

corrected to a constant and 0.292 gram of



of constituents,

Corrected dry weight Weight out of of dry storage matter in from 49.9 g. Loss of sample of initial dry weight as stored dry matter in storage




of stored samples with phosphorus as the basis of calculation.

Absolute weight (grams) of constituents in shelled peas stored at 25° C , calculated on the basis of constant initial weight


Actual dry weight out of storage







departure of =1=0.006 per cent. Assuming that the same ratio was present initially in all the storage samples, the peas stored for different lengths of time had a greater weight of total solids before storage than did the check, because they contained more phosphorus. For instance, to get the true initial dry weight of the samples stored for one day, one need only divide the weight of the phosphorus (0.294 gram) by that present in the check (0.292 gram) and multiply the quotient by 49.9, the actual weight of the check. This gives a weight of 50.2 grams of dry matter (column 4 ) , the amount originally associated with 0.294 gram of phosphorus. "If there was a greater weight of total solids initially in the stored samples than in the check, then the weight present after storage must also be too high (column 2 ) . The lot taken out of storage after one day is larger than it should be because it contained 0.294 gram of phosphorus, which is more than the amount in the initial sample. If this storage lot is to be comparable with the check, one must make a correction, dividing its average dry weight (column 2) by the weight of the phosphorus in it (column 3) and multiplying by the weight of the phosphorus in the check. The same calculation is made for each of the storage lots. The results (column 5) give the average dry weight after storage of each lot containing the same weight of this element as is present in the check. These corrected sample weights after storage are those that would result if the fresh samples at time of storage had contained exactly the same weights of solids. By multiplying each of these corrected dry weights (column 5) in turn by the percentage of the various constituents, the true weight of each constituent is obtained (columns 7 to 13). These values can be compared to show the true changes in the amounts of the various constituents when stored for different lengths of time." (From Bisson and Jones, 1934). For additional information on the expression of results consult the general references listed in the terminal bibliography. REFERENCES Statistical





Allcock, H. J., and Jones, H. R., "The Nomogram. The Theory and Practical Construction of Computation Charts," Sir Isaac Pitman and Sons, Ltd., London, 1938. Banister, H., "Elementary Applications of Statistical Methods," Blackie and Son, Ltd., London, 1929. Bond, W. N., "Probability and Random Errors," E. Arnold & Co., London, 1935. Buros, Ο. K., "The Second Yearbook of Research and Statistical Methodology," Gryphon Press, Highland Park, N e w Jersey, 1941. Camp, B. C , "The Mathematical Part of Elementary Statistics," D . C. Heath and Company, N e w York, 1931. Demiag, W. E., "Statistical Adjustment of Data," John Wiley and Sons, N e w York, 1943.



Fisher, R. Α., "Statistical Methods for Research Workers," 6th ed., Oliver and B o y d , London, 1938. Hoil, P. G., "Introduction to Mathematical Statistics, John Wiley and Sons, N e w York, 1947. Goulden, C. H., "Methods of Statistical Analysis," John Wiley and Sons, N e w York, 1939. Kelley, T. L., "Statistical Method," The Macmillan Company, N e w York, 1923. Levens, A. S., "Nomography," John Wiley and Sons, N e w York, 1948. Lipka, J., "Graphical and Mechanical Computation," John Wiley and Sons, N e w York, 1918. Lovitt, W. V., and Holtzcaw, H. F., "Statistics," Prentice-Hall, Inc., N e w York, 1929. Merriman, M., "A Textbook on the M e t h o d of Least Squares," 8th ed., John Wiley and Sons, N e w York, 1909. Snedecor, G. W., "Statistical Methods," 4th ed., Collegiate Press, Inc., Ames, Iowa, 1946. Whittaker, E. T., and Robinson, G., "Calculus of Observations. A Treatise on Numerical Mathematics," Blackie and Son, Ltd., London, 1924. Errors



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