How to calculate deviation, variance and standard deviation

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Dispersion and variation Dispersion and variation are words that are used to describe the spread of values in a given set of data. Before you proceed, I recommend you check this post out, it's a prerequisite. Consider the set of examination marks of ten students in the  image below The marks are the same. They do not vary. No dispersion. Check the image below You can see that the marks are not the same. They vary. Their mean is 60. Some of the marks are greater than 60 while others are less. This set of marks has a greater variation or dispersion than those in table 1 Also, consider this table which shows a third set of marks of ten students. Although the mean is also 60. Don't know how to calculate mean? click here . However, it is clear that the marks are more varied than those in table 2. Thus, the marks here have a greater dispersion than the other marks. Let's talk about range. Range The range of a set of numbers is the difference between the ...
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The Simplest Method Of Solving Indices And Laws Of Indices

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INDICES

An index (power) is the small floating number that appears after a number or letter. It indicates a number has been multiplied by itself multiple times.

For instance

• 3 x 3 x 3 x 3 x 3 (3 multiplied by itself)
 = 3⁵ (multiplied itself 5 times)

• 8 x 8 x 8 x 8
 = 8⁴

• a x a x a
 = a³

NB: An index can be any number, be it fraction or counting number

When working with indexed numbers, certain operations can be carried out. i.e you can multiply as well as divide. All these follow a certain principle called LAWS OF INDICES


LAWS OF INDICES



NB: In multiplication, when the base numbers are the same, just add the index numbers together. Same goes to division, instead of adding, you subtract.

But a° = 1 i.e any number with the power 0 is 1
E.g 2⁰ is 1; -3⁰ is also 1


Examples on multiplication of index form

First of, let's do a quick recap.
We know 3 x 3 x 3 x 3 x 3 x 3 = 3⁶ and vice versa

(a) 2³ x 2⁵   (b) (3³ x 5) x (4² x 3²)  (c) 3° x 2³

                             Solution
 
(a) 2³ x 2⁵

    Here we are multiplying; remember the law
    a² x a³ 
   = 2²+³
   = 2⁵

(b) (3³ x 5) x (4² x 3²)

 First thing here is to open the brackets

     3³ x 5 x 4² x 3²

It's advisable to collect like terms here and rearrange.

  = 3³ x 3² x 4² x 5
  = 3³+² x 4² x 5
  = 3⁵ x 4² x 5

NB: 3, 4 and 5 are of different bases. There's nothing left to do. That's our final answer.

(c) 3° x 2³
 
Remember our law a° = 1
Therefore 3° x 2³
               = 1 x 2³
               = 2³

Examples on division of index form


We already know the law



Evaluate each of these

(a) 4⁴ ÷ 4²   (b) 5⁸ ÷ (5² x 5³)   (c) 3⁹ ÷ 3² 
                          (d) 3⁷ ÷9

                               Solution

(a) 4⁴ ÷ 4²
      = 4⁴-²
      = 4²


(b) 5⁸ ÷ (5² x 5³)

First things first, open the bracket

(5² x 5³) 

This is multiplication, so we add the indexes.

5² x 5³ 
= 5²+³
= 5⁵

Recall the question is 5⁸ ÷ (5² x 5³)

Therefore 5⁸ ÷ 5⁵
= 5⁸-⁵
= 5³


(c) 3⁹ ÷ 3²

= 3⁹-²
= 3⁷


(d) 3⁷ ÷ 9

Okay, this is interesting. You might be wondering how we can solve this using the laws of indices because the two terms are of different bases. The best thing to do is cover the whole number (9) to index form. Best way to do that is to travel down the memory lane.
We find the factors of 9
   

9 in index form is 3²

Therefore 3⁷ ÷ 9 will be 3⁷ ÷ 3²

= 3⁷-²
= 3⁵



That's all for now, please comment and share your thoughts. Any question will surely be answered.

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