Variables, Notation & Summation

Variables, Notation & Summation

Basic Summation (Σ) Notation FORMULA

How do we calculate SUMMATION NOTATION?

• Look at the lower limit (i = 1):
  Start with the first value (X₁).

• Look at the upper limit (n):
  End at the last value (Xₙ).

• Add up all values in order:

  X1+X2+X3+⋯+Xn

Formula of basic summation

Three different summation rules:

• Multiplying by a Constant

• Summing a Constant

• Sum of Multiple Variables

Multiplication of each value by a Constant FORMULA (rule 1)

• Multiplication of each value by a constant

How to calculate multiplication of a constant?

Steps:

1. Multiply each value Xi​ by the constant c.

2. Add the new values together.

3. OR: Add all original Xi first, then multiply the total sum by c.

Adding/ summing up a constant repeatedly formula (rule 2)

How to calculate the summed up constant repeatedly (rule 2)

• If you're adding the same number (like 5) multiple times:

• Multiply the number by n (the number of times it appears).

Formula of Adding/ summing up a constant repeatedly (rule 2)

Summing multiple variables (Rule 3) Formula

How to calculate “Summing multiple variables” (Rule 3)

• Add up all X values, all Y values, all Z values separately.

• Add the three sums together.

• the formula of multiple variables which are summed up

Double Summation (ΣΣ) - Multiple people & variables (FORMULA)

How to calculate the double summation?

• Inner sum (j):
  Add all variables for one person.

• Outer sum (i):
  Repeat for each person, adding their totals.

• Final step:
  Add up all persons' totals for the final sum.

The formula of double summation

Summation 

• means I add up all the values of the variable, from the first value (i = 1) to the last (n).

How to Explain in the Exam:

• Clearly state:
  “Summation means I add up all the values of the variable, from the first value (i = 1) to the last (n).”

• Describe rule used:
  “If multiplying by a constant, I can either multiply first or after summing.”

• For double summation:
  “I sum across variables first, then across cases (people).”