MEDIAN

How to Find the Median Value

It's the middle of a sorted list of numbers.

Median Value

The Median is the "middle" of a sorted list of numbers.

How to Find the Median Value

To find the Median, place the numbers in value order and find the middle.

Example 1 : find the Median of 12, 3 and 5

Put them in order:

3, 5, 12

The middle is 5, so the median is 5.

Example 2 :

3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

There are fifteen numbers. Our middle is the eighth number:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

The median value of this set of numbers is 23.

(It doesn't matter that some numbers are the same in the list.)

Two Numbers in the Middle

BUT, with an even amount of numbers things are slightly different.

In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.

Example 3 :

3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don't have just one middle number, we have a pair of middle numbers:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

In this example the middle numbers are 21 and 23.
To find the value halfway between them, add them together and divide by 2:

21 + 23 = 44
then 44 ÷ 2 = 22

So the Median in this example is 22.

(Note that 22 was not in the list of numbers ... but that is OK because half the numbers in the list are less, and half the numbers are greater.)

EXERCISE :

Find the median of the following numbers:

a) 4, 2, 7, 4, 15, 16, 18

b) 5, 2, 2, 3, 4, 11, 12, 12, 14

INEQUALITIES 3

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example 1 :

Solve: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 - 3 < 7 - 3    

x < 4

And that is our solution: x < 4
In other words, x can be any value less than 4.

What did we do?

 And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality.

Example 2: 

Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example 3: 

12 < x + 5
If we subtract 5 from both sides, we get:

12 - 5 < x + 5 - 5    

7 < x

That is a solution!
But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7

 Note: "x" can be on the right, but people usually like to see it on the left hand side.

Exercise :

Solve x + 5 < 12

INEQUALITIES 2

Examples:

How to Solve

Solving inequalities is very like solving equations ... we do most of the same things ...
... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points".

Some things we do will change the direction!

< would become >

> would become <

≤ would become ≥

≥ would become ≤

Safe Things To Do

These are things we can do without affecting the direction of the inequality:

Add (or subtract) a number from both sides
Multiply (or divide) both sides by a positive number
Simplify a side

Example 1: 

3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things will change the direction of the inequality ("<" becomes ">" for example):

Multiply (or divide) both sides by a negative number
Swapping left and right hand sides

Example 2: 

2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality:

12 > 2y+7

Example 3: 

2x+6 < 12

Ans = x < 3

Exercise :

Solve the following

(a) 3x < 12

(b) 2y +4 > 16

PROBABILITY 2

Probability line EXAMPLES:
We can show probability on a Probability Line:

Probability is always between 0 and 1.

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide.

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.
But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.