Question

summation i=1-n (xi-3) =0, summation i= (xi+3) =66 then find x bar and n

Answer 1

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1 Sigma Notation

1.1 Understanding Sigma Notation

The symbol Σ (capital sigma) is often used as shorthand notation to indicate the sum of

a number of similar terms. Sigma notation is used extensively in statistics.

For example, suppose we weigh five children. We will denote their weights by x1, x2, x3,

x4 and x5.

The sum of their weights x1 + x2 + x3 + x4 + x5 is written more compactly as -

5

j=1

xj .

The symbol Σ means ‘add up’. Underneath Σ we see j = 1 and on top of it 5. This

means that j is replaced by whole numbers starting at the bottom number, 1, until the

top number,5, is reached.

Thus

-

5

j=2

xj = x2 + x3 + x4 + x5,

and

-

4

j=2

xj = x2 + x3 + x4.

So the notation -

n

j=1

xj tells us:

a. to add the scores xj ,

b. where to start: x1,

c. where to stop: xn (where n is some number).

Now take the weights of the children to be x1 = 10kg, x2 = 12kg, x3 = 14kg, x4 = 8kg

and x5 = 11kg. Then the total weight (in kilograms) is

-

5

i=1

xi = x1 + x2 + x3 + x4 + x5

= 10 + 12 + 14 + 8 + 11

= 55.

Notice that we have used i instead of j in the formula above. The j is what we call a

dummy variable - any letter can be used, ie,

-

n

j=1

xj = -

n

i=1

xi.

Now let us find -

4

i=1

2xi where x1 = 2, x2 = 3, x3 = −2 and x4 = 1.