This Maple worksheet accompanies the paper: Di Nardo E. (2010) A new approach to Sheppard\342\200\231s corrections. In press Mathematical Methods in Statistics. (http://arxiv.org/abs/1004.4989) A new approach to Sheppard's corrections 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E. Di Nardo* elvira.dinardo@unibas.it http://www.unibas.it/utenti/dinardo/home.html; Tel: +39 0971205890, Fax: +39 0971205896 G. Guarino** giuseppe.guarino@rete.basilicata.it * Dipartimento di Matematica e Informatica, Universit\303\240 degli Studi della Basilicata, Viale dell'Ateneo Lucano n.10, 85100 Potenza, Italy **Medical School, Universit\303\240 del Sacro Cuore (Rome branch), Largo Agostino Gemelli n.8, 00168 Roma, Italy

Abstract: in the real world, continuous variables are observed and recorded in finite precision through a rounding or coarsening operation, i.e. a grouping rule. A compromise between the desire to know and the cost of knowing is then a necessary consequence. Attention has been paid in the literature to the computation of moments when data are grouped into classes. The moments computed by means of the resulting grouped frequency distribution are looked upon as a first approximation to the moments of the parent distribution, but they suffer from the error committed in grouping. A good correction procedure is given by Sheppard's corrections that are nowadays still employed. Sheppard's corrections are usually referred to continuos parent distribution. But grouping includes also censoring or splitting data into categories during collection or publication, and so it does not only involve continuous variables. A very simple closed-form formula for Sheppard's corrections has been recovered by the classical umbral calculus (see [5]) as well as a more general closed-form formula for discrete parent distributions (see [2]). No attention was paid in the literature to multivariate generalizations of Sheppard's corrections, probably due to the complexity of the resulting formulae (see [1]). Via the umbral calculus, the generalization to the multivariate case turns to be straightforward. All these new formulae are particularly suited to be implemented in MAPLE. The theoretical background of these formulae can be found in Di Nardo E. (2010) (see [3]) Application Areas/Subject: combinatorics & algebraic methods in statistics. Keywords: raw moment, grouped moment, Sheppard's correction, umbral calculus. See Also: background on umbral calculus in [4]

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

Suppose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYsLUklbXN1YkdGJDYlRistRiM2Ji1JI21uR0YkNiRRIjFGJ0Y5LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRlU2JUYrLUYjNiYtRlo2JFEiMkYnRjlGZ25Gam5GOUZdb0Zgby1GNjYtUSMuLkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZlb0Zgby1GVTYlRistRiM2Ji1GLDYlUSJqRidGL0YyRmduRmpuRjlGXW9GZ25Gam5GOUY5RmduRmpuRjk= a multivariate random vector. The raw multivariate moment of X of order 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 is denoted by 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. The moments calculated from the grouped frequencies are denoted by 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. Assume 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 are not-zero width window for each component and 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 the numbers of consecutive values grouped in a frequency class of width LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQUZERj8=. The procedure raw2grp gives raw moments 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 in terms of grouped moments 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, by using formula (31) of the paper [3]. In particular, set the variable t = 0 when Sheppard's corrections are required for continuous parent distribution. Note: that sequence f1 in the procedure refers to formula (14) and sequence f2 refers to formula (17). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZhcS1JI21pR0YkNiVRKHJhdzJncnBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni9RJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlVGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZZLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEiVkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlgvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUYsNiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRmBwLyUqbGluZWJyZWFrR1ExZmlyc3Rwcm9jbmV3bGluZUYnLUY2Ni9RJmxvY2FsRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlotRlxwNiZGXnBGYXBGZHAvRmdwUSVhdXRvRictRiw2JVEibkYnRi9GMkZdby1GLDYlUSJNRidGL0YyRl1vLUYsNiVRJGVGMUYnRi9GMkZdby1GLDYlUSRlRjJGJ0YvRjItRjY2LVEiO0YnRjlGO0Zgb0ZARkJGREZGRkhGWEZNRltwRmBxRjUtRiw2JVElbm9wc0YnRi9GMi1GZm42JC1GIzYlRmpuRmZvRjlGOUZcckZbcEZjcUY1LUYsNiVRJ2V4cGFuZEYnRi9GMi1GZm42JC1GIzYnRmhvLUYjNictRiw2JVEkbXVsRidGL0YyLUY2Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGO0Y+RkBGQkZERkZGSEZYRlotRmZuNiQtRiM2Ky1JJW1zdXBHRiQ2JS1GZm42JC1GIzYrLUklbXN1YkdGJDYlLUYsNiVRJyYjOTU2O0YnL0YwRj1GOS1GIzYlLUYsNiVRImlGJ0YvRjJGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZjdS1GYXQ2JS1GLDYlUSJoRidGL0YyRmd0Rlx1LUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZYRlotRmF0NiUtRiw2JVEjZjFGJ0ZmdEY5Rmd0Rlx1Rl91LUYsNiVRJWBpZmBGJ0YvRjItRmZuNiQtRiM2LEZjby1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRIjBGJ0Y5Rl1vRlx3Rl1vLUkmbWZyYWNHRiQ2KC1GIzYnRmV1Rmp1LUZhdDYlLUYsNiVRI2YyRidGZnRGOUZndEZcdUZmb0Y5LUYjNiUtRmF0NiUtRiw2JVEibUYnRi9GMkZndEZcdUYvRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZ4LyUpYmV2ZWxsZWRHRj1GaG9GZm9GOUY5RjlGOS1GIzYlLUZhdDYlRmpuRmd0Rlx1Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHRl51Rl1vRml0Rml2LUZddzYkUSIxRidGOS1GNjYtUSMuLkYnRjlGO0Y+RkBGQkZERkZGSEZidUZaRmBxRmZvRjlGOUZmb0Y5RmhvRmZvRjlGOUZcckZbcEZjcUY1LUYsNiVRJGFkZEYnRi9GMi1GZm42JC1GIzYrLUYsNiVRInhGJ0ZmdEY5Rmp1LUZhdDYlLUZhdDYlLUYsNiVRImdGJ0ZmdEY5LUYjNictRiw2JVEkc2VxRidGL0YyLUZmbjYkLUYjNiotRiw2JVEnZGVncmVlRidGL0YyLUZmbjYkLUYjNiYtRiw2JUZgekYvRjJGXW9GYHRGOUY5Rl1vRml0Rml2LUYjNiZGYXlGZHlGYHFGOUZob0Y5RjlGaG9GL0YyRlx1RmhvRlx1Rl1vRmhbbEZpdkZjcUZmb0Y5RjlGXHJGW3BGY3FGNS1GLDYlUSVldmFsRidGL0YyLUZmbjYkLUYjNidGY3FGXW8tRmZuNiYtRiM2J0Zoby1GIzYnRmp6RmJzLUZmbjYkLUYjNi5GYHRGaG9GaXZGYXlGXW9GaXRGaXZGYXlGZHlGYHFGZm9GOUY5RmZvRjlGaG9GZm9GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmZvRjlGOUZcckZbcEZmcUY1Rmp6LUZmbjYkLUYjNilGaG8tRiM2J0ZqekZicy1GZm42JC1GIzYxLUkobXN1YnN1cEdGJDYoLUYsNiVGYXZGL0YyRmd0LUYjNiUtRiw2JVEiakYnRi9GMkYvRjJGX3lGXHUvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRml2LUZmbjYkLUYjNigtRmpzNiUtRl13NiRRIjJGJ0Y5LUYjNidGYXktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSEZidUZkdUZkXmxGL0YyRl95LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGWEZaLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSEZidUZkdUZoX2xGYXlGOUY5Rmp1LUYsNiVRKmJlcm5vdWxsaUYnRmZ0RjktRmZuNiQtRiM2JUZkXmxGZm9GOUY5Rl1vRmRebEZpdkZheUZkeS1GLDYlUSRtYXhGJ0ZmdEY5LUZmbjYkLUYjNidGaG8tRiM2Jy1GLDYlUSNvcEYnRi9GMkZic0ZickZmb0Y5RmhvRmZvRjlGOUZmb0Y5RjlGZm9GOUZdby1GIzYoRml0Rml2LUYjNidGYXlGZHlGYHFGZm9GOUZob0Zmb0Y5RmhvRmZvRjlGOUZcckZbcC1GNjYvUSNpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxRmNvRml2Rlx3RlxxLUY2Ni9RJXRoZW5GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWi1GXHA2JkZecEZhcEZkcC9GZ3BRNmluY3JlYXNlaW5kZW50bmV3bGluZUYnRmlxRjUtRiw2JVElTlVMTEYnRi9GMi1GXHA2JkZecEZhcEZkcC9GZ3BRNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnLUY2Ni9RJWVsc2VGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZbYmxGaXFGNUZqei1GZm42JC1GIzYpRmhvLUYjNidGanpGYnMtRmZuNiQtRiM2MC1GXl5sNigtRiw2JUZpd0YvRjJGZ3RGYl5sRl95Rlx1RmdebEZpdkZidi1GZm42JC1GIzYtLUYsNiVRJWlyZW1GJ0YvRjItRmZuNiQtRiM2J0ZkXmxGXW9GYF9sRmZvRjlGOUZpdkZheUZdb0Zcd0Zdby1GYXc2KC1GanM2JS1GZm42JC1GYXc2KEZheS1GIzYlRmBfbEYvRjJGYXhGZHhGZ3hGaXhGOUZiXmxGX3ktRiM2J0ZkXmxGX3VGYXlGL0YyRmF4RmR4Rmd4Rml4RmhvRmZvRjlGOUZdb0ZoX2xGZF5sRml2RmF5RmR5RmVgbEZoYGxGZm9GOUY5RmZvRjlGXW9GYWFsRmhvRmZvRjlGOUZoby1GXHA2JkZecC9GYnBRJjAuMGVtRidGZHBGXnEtRlxwNiZGXnBGYGVsRmRwL0ZncFEobmV3bGluZUYnRmJlbEZeZWxGXmVsRl5lbEZiYmxGaF9sRmhfbEZoX2xGaF9sLUY2Ni9RJGVuZEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRmhfbEZlYWxGXHJGW3BGYmVsRmhfbEZoX2xGaF9sRmhfbEZoX2xGaF9sRlxcbC1GZm42JC1GIzYnRmNxRl1vLUZmbjYmLUYjNitGZnFGXW9GaXFGXW8tRmF0NiUtRiw2JUZnekYvRjItRiM2J0Zcdy1GNjYtUSIkRidGOUY7Rj5GQEZCRkRGRkZIRlhGWkZgcUYvRjJGXHVGaXZGYXlGZm9GOUY5Rl1dbEZgXWxGZm9GOUY5RmhvRmJibEZiZWxGXmVsLUY2NjBGaGVsRlJGZm9GVEZWRjtGPkZARkJGREZGRkhGWEZaLUY2Ni5Gal9sRmZvRjlGO0Y+RkBGQkZERkZGSEZYRlotRjY2MEZRRlJGZm9GVEZWRjtGPkZARkJGREZGRkhGWEZaLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRmZvRjk= JSFH

continuous parent distributions 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

LComSSJnRzYiNiQiIiNGJyIiIiooI0YoIiM3RigpJkkiaEdGJTYjRidGJ0YoJkYkNiRGJyIiIUYoISIiKihGKkYoKSZGLjYjRihGJ0YoJkYkNiRGMkYnRihGMyooI0YoIiRXIkYoRjVGKEYsRihGKA==

discrete parent distributions 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

LDQqKiMiIiIiIzdGJSkmSSJoRzYiNiNGJSIiI0YlKSZJIm1HRipGK0YsISIiJkkiZ0dGKjYkIiIhRixGJUYlJkYyNiRGLEYsRiUqKiNGJSIkVyJGJUYnRiVGLUYwKSZGKTYjRixGLEYlRjAqKEYkRiVGJ0YlRjFGJUYwKihGOEYlRidGJUY6RiVGJSooRiRGJUY6RiUmRjI2JEYsRjRGJUYwKipGOEYlRidGJUY6RiUpJkYvRjxGLEYwRjAqKkYkRiVGOkYlRkNGMEZARiVGJSosRjhGJUYnRiVGLUYwRjpGJUZDRjBGJQ==

JSFH

Suppose 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 a multivariate random vector. The raw multivariate moment of X of order 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 is denoted by 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. The moments calculated from the grouped frequencies are denoted by 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. Assume 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 are not-zero width window for each component and 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 the number of consecutive values grouped in a frequency class of width LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQUZERj8=. The procedure grp2raw gives grouped moments LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiNRIUYnLUYjNigtRiw2JS1GLzYlUSJ0RidGMkY1LUYjNiQtSSNtbkdGJDYkUSIxRicvRjZRJ25vcm1hbEYnRkovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZKLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRlUvJSpzeW1tZXRyaWNHRlUvJShsYXJnZW9wR0ZVLyUubW92YWJsZWxpbWl0c0dGVS8lJ2FjY2VudEdGVS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUZQNi1RIy4uRidGSkZTL0ZXRlVGWEZaRmZuRmhuRmpuL0Zdb1EsMC4yMjIyMjIyZW1GJy9GYG9GXm9GTy1GLDYlRkEtRiM2JC1GLzYlUSJqRidGMkY1RkpGTEZKRjpGSkZMRko= in terms of raw moments 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, by using formula (32) of the paper [3]. In particular, set the variable t = 0 when Sheppard's corrections are required for continuous parent distribution. Note: that sequence f1 in the procedure refers to formula (14) and sequence f2 refers to formula (17). 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

continuous parent distributions 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

LComSSJyRzYiNiQiIiNGJyIiIiooI0YoIiM3RigpJkkiaEdGJTYjRidGJ0YoJkYkNiRGJyIiIUYoRigqKEYqRigpJkYuNiNGKEYnRigmRiQ2JEYyRidGKEYoKigjRigiJFciRihGNEYoRixGKEYo

discrete parent distributions 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

LDQqKiMiIiIiJFciRiUpJkkiaEc2IjYjRiUiIiNGJSkmSSJtR0YqRitGLCEiIikmRik2I0YsRixGJUYwKiojRiUiIzdGJUYnRiVGLUYwJkkickdGKjYkIiIhRixGJUYwKipGNUYlRjFGJSkmRi9GM0YsRjAmRjg2JEYsRjpGJUYwKihGJEYlRidGJUYxRiVGJSooRjVGJUYxRiVGPkYlRiUqKEY1RiVGJ0YlRjdGJUYlKipGJEYlRidGJUYxRiVGPEYwRjAmRjg2JEYsRixGJSosRiRGJUYnRiVGLUYwRjFGJUY8RjBGJQ==

The procedure raw2grp gives raw moments LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiNRIUYnLUYjNigtRiw2JS1GLzYlUSJ0RidGMkY1LUYjNiQtSSNtbkdGJDYkUSIxRicvRjZRJ25vcm1hbEYnRkovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZKLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRlUvJSpzeW1tZXRyaWNHRlUvJShsYXJnZW9wR0ZVLyUubW92YWJsZWxpbWl0c0dGVS8lJ2FjY2VudEdGVS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUZQNi1RIy4uRidGSkZTL0ZXRlVGWEZaRmZuRmhuRmpuL0Zdb1EsMC4yMjIyMjIyZW1GJy9GYG9GXm9GTy1GLDYlRkEtRiM2JC1GLzYlUSJqRidGMkY1RkpGTEZKRjpGSkZMRko= in terms of grouped moments 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 If the output is evaluated using 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 = grp2raw(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]) you obtain the raw moment LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmYWdhaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLkYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUY2Ni1RIn5GJ0YyRjlGO0Y9Rj9GQUZDRkVGR0ZKRjI= continuous parent distributions 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

LComSSJnRzYiNiQiIiNGJyIiIiooI0YoIiM3RigpJkkiaEdGJTYjRidGJ0YoJkYkNiRGJyIiIUYoISIiKihGKkYoKSZGLjYjRihGJ0YoJkYkNiRGMkYnRihGMyooI0YoIiRXIkYoRjVGKEYsRihGKA==

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

Jkkickc2IjYkIiIjRiY=

discrete parent distributions 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

LDQmSSJnRzYiNiQiIiNGJyIiIioqI0YoIiM3RigpJkkiaEdGJTYjRidGJ0YoKSZJIm1HRiVGL0YnISIiJkYkNiRGJyIiIUYoRigqKiNGKCIkVyJGKCkmRi42I0YoRidGKCkmRjJGPEYnRjNGLEYoRjMqKEYqRihGOkYoJkYkNiRGNkYnRihGMyooRjhGKEY6RihGLEYoRigqKEYqRihGLEYoRjRGKEYzKipGOEYoRjpGKEYsRihGMEYzRjMqKkYqRihGOkYoRj1GM0ZARihGKCosRjhGKEY6RihGPUYzRixGKEYwRjNGKA==

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

Jkkickc2IjYkIiIjRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The procedure grp2grp gives grouped moments 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 in terms of raw moments 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 If the output is evaluated using 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 = raw2grp(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]) you obtain the grouped moments LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmYWdhaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLkYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUY2Ni1RIn5GJ0YyRjlGO0Y9Rj9GQUZDRkVGR0ZKRjI= continuous parent distributions 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

LComSSJyRzYiNiQiIiNGJyIiIiooI0YoIiM3RigpJkkiaEdGJTYjRidGJ0YoJkYkNiRGJyIiIUYoRigqKEYqRigpJkYuNiNGKEYnRigmRiQ2JEYyRidGKEYoKigjRigiJFciRihGNEYoRixGKEYo

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

JkkiZ0c2IjYkIiIjRiY=

discrete parent distributions 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

LDQqKiMiIiIiJFciRiUpJkkiaEc2IjYjRiUiIiNGJSkmSSJtR0YqRitGLCEiIikmRik2I0YsRixGJUYwKihGJEYlRidGJUYxRiVGJSooI0YlIiM3RiVGMUYlJkkickdGKjYkRiwiIiFGJUYlKihGNkYlRidGJSZGOTYkRjtGLEYlRiUqKkYkRiVGJ0YlRjFGJSkmRi9GM0YsRjBGMCZGOTYkRixGLEYlKipGNkYlRidGJUYtRjBGPUYlRjAqLEYkRiVGJ0YlRi1GMEYxRiVGQEYwRiUqKkY2RiVGMUYlRkBGMEY4RiVGMA==

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

JkkiZ0c2IjYkIiIjRiY=

JSFH

We have shown how the corrections of moments resulting from grouping into classes may be summarized in few closed-form formulae. Once more, this algorithm shows how the classical umbral calculus should be taken into account for managing sequence of numbers related to random variables, since many calculations are reduced. For example, the reader interested in recovering corrections for cumulants and factorial moments, by using the classical umbral calculus, can refer to [4].