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I'm just curious about what the meaning of $i$ and $j$ subscript in statistic formula. I think $i$ means individual and $j$ means higher level unit in a multilevel frame work. How did statisticians come up with these two letters? What are they short for?

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    $\begingroup$ There's no generally accepted meaning. It depends on the context and the paper. For instance, in one econometrics course, professor used i for column vectors and j for row vectors. $\endgroup$ – Aksakal Oct 14 '15 at 19:42
  • $\begingroup$ That was an unkind professor. $\endgroup$ – Cliff AB Oct 14 '15 at 23:59
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The $i$ and $j$ are mathematical, rather than statistical, convention. The $i$ is because it is the first letter in the word index, and then $j$ comes after $i$. They have the benefit of being small, clear, and unobtrusive ($x_{ij}$ looks pretty good).

This is usually seen in the context of matrix entries, where $i$ indexes the rows, and $j$ indexes the columns. The matrix convention is followed far and wide in general branches of mathematics and science using linear algebraic machinery.

This is what I was told in my first linear algebra course in school, I know of no definitive reference.

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    $\begingroup$ Hmmm, re: "... clear...", I've been in a number of classrooms where the $i$ and $j$ were essentially indistinguishable (including in my own handwriting...). $\endgroup$ – gung - Reinstate Monica Oct 14 '15 at 19:53
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    $\begingroup$ @gung The professor who taught ANOVA to us had handwriting that made i,j and . indistinguishable. That was not conducive to learning. $\endgroup$ – Peter Flom - Reinstate Monica Oct 14 '15 at 20:16
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    $\begingroup$ Mathematical convention has changed over the centuries. The use of $i$ and $j$ for indexes is by no means universal. One convention that is documented was the determination that all variables beginning with the letters "i" through "n" would automatically be considered INtegers in FORTRAN (and therefore were frequently used as INdexes). Given that the designers of FORTRAN spoke English, this makes sense. As an explanation, it makes less sense in the mathematical community of the early 20th century where the leaders tended to write in French (les Entiers) or German (die Zahlen). $\endgroup$ – whuber Oct 14 '15 at 22:36
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    $\begingroup$ I just checked some old (stats-related) papers; K.Pearson here (1901) uses $u$ and $v$. Fisher here (1925) uses $k$ and $j$, whereas here (also 1925) he uses $i$ and $j$. To my recollection I've seen him use $i$ and $j$ in other papers from the 20's. Not much of a sample (only tried 3), and obtained haphazardly, but I thought it was interesting. One $i,j$ use pre-FORTRAN, anyway. $\endgroup$ – Glen_b -Reinstate Monica Oct 15 '15 at 0:24
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    $\begingroup$ @Glen Yes, Fortran was building on what had become conventional in some communities. But in the mid-19th century, for instance, many people did not even use subscripts. Papers would refer to "$A, B, \ldots, Z$" for instance, rather than "$A_i, i=1, 2, \ldots$". Although Fisher is only a sample of one or two, it was surely his work on ANOVA and his stats text that would have established conventional notation for a very long time. $\endgroup$ – whuber Oct 15 '15 at 0:42

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