Summation_Notation
Summation Notation in Statistics
Introduction to Variables
In statistics, a sample is represented by an algebraic variable.
Individual observations are denoted by subscripts, such as x1, x2, x3, etc.
Example: Length of a sample of fossils is represented as x.
Set of measurements: x = {6.5, 6.4, 6.2, 6.6, 6.5} where n (sample size) = 5.
Individual measurements: x1 = 6.5, x2 = 6.4, x3 = 6.2, x4 = 6.6, x5 = 6.5.
A generic measurement can be indicated by xi.
Summation Notation
Summation notation allows for the compact representation of adding long series of numbers.
Denoted by the symbol Σ (sigma).
General format: ( \sum_{i=1}^{n} x_i )
i: summation index variable.
1: lower limit of the sum.
n: upper limit of the sum.
xi: summand (the value being summed).
Evaluation process:
Vary i from lower to upper limits.
Evaluate summand for each i.
Add the results.
Example summation formulas:
a) ( \sum_{i=1}^{3} i = 1 + 2 + 3 = 6 )
b) ( \sum_{i=1}^{3} i^2 = 1^2 + 2^2 + 3^2 = 14 )
c) ( \sum_{i=1}^{3} x_i = x_1 + x_2 + x_3 )
d) ( \sum_{i=1}^{3} x_i^2 = x_1^2 + x_2^2 + x_3^2 )
e) ( \sum_{i=2}^{5} x_i = x_2 + x_3 + x_4 + x_5 )
Mean Calculation Using Summation Notation
Mean (average) is computed by adding values and dividing by the number of values.
Compact representation using summation notation: ( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i )
Example applied to previous fossil lengths:
Set: x = {6.5, 6.4, 6.2, 6.6, 6.5}, n = 5
Calculation: ( \bar{x} = \frac{1}{5} \sum_{i=1}^{5} x_i = \frac{1}{5} (6.5 + 6.4 + 6.2 + 6.6 + 6.5) = 6.4 )
Note: The result should reflect the precision of the data.
Understanding the Σ Operator
The Σ operator acts as a grouping symbol.
Everything to the right of Σ is part of the summand unless parentheses are added.
Example of grouping:
( \sum_{i=1}^{n} x_i + 5 = \sum_{i=1}^{n} (x_i + 5) )
Important distinction:
( \sum_{i=1}^{n} x_i + 5 ) expands to include 5 in each term.
( 5 + \sum_{i=1}^{n} x_i ) means 5 is added only once.
Expansion of Summation Expressions
Example expression: ( \sum_{i=1}^{5} (x_i - 3)^2 )
Expansion: ( (x_1 - 3)^2 + (x_2 - 3)^2 + (x_3 - 3)^2 + (x_4 - 3)^2 + (x_5 - 3)^2 ) .