## 5. Data series

Although you sometimes have only one value for a variable, you typically encounter several values. Where these values have some sort of order, they are usually stored in a series. Individual values in the series are denoted using a subscript, which can be thought of as a label or tag that identifies a particular position in the series. If the variable is one of several repeat measurements, so that there are several data series, the same subscripts will label the corresponding observations for each variable.

a_{0} |
The first element or observation of a series of variable a (typically the starting value) |

a_{3} |
Element 3 of a series of variable a |

a_{i} |
Element 'i' of a series of variable a |

a_{i} . x_{i} |
Element 'i' of a series of variable a, mulitplied by the corresponding value for the variable x (that is the values of a and x come from the same observation or sample) |

a_{0 … i} |
The series of elements from a_{o} to a_{i} |

a_{i + 1} |
The next element in the series of variable a above ai |

a_{3}^{2} |
The square of element three of a series of variable a |

a_{3}^{-1} |
The reciprocal of element three of a series of variable a |

a_{3}^{0.5} |
The square root of element three of a series of variable a |

a_{i}^{x} |
The x^{th} power of the i^{th} element of a series of variable a |

b . a_{i}^{x} |
The x^{th} power of the i^{th} element of variable a, then multiplied by b |

(b . a^{i})^{x} |
The i^{th} element of a series of variable a multiplied by b, with the result raised to power x |

Sometimes it is useful to order measurements of a variable in a two-way classification, which can be visualized as rows and columns in a table. The elements are then identified by a double subscript like a_{i,j}.

Particular notation is used in statistical calculations, where typically various totals of squares and products are used to derive values that contribute to the calculation of statistics. Subscripts are normally used in statistics to denote something other than position in a series. The upper-case Greek letter sigma (written Σ) is used to denote a sum (ie all values added together). The following table includes some expressions that you are likely to encounter (variables in these examples are x and y).

Σx | The sum of all of the values for the variable x |

Σx^{2} |
The sum of all the squares of all the values for the variable x |

(Σx)^{2} |
The square of the sum of all the values for the variable x |

Σxy | The sum of all the products of all the values of the variable x and the corresponding values of the variable y |