Mean Median Mode

Today we learned a lot about Mean, Median and Mode. We also talked a little on Range, IQR, Variance, and Standard Deviation. These homework problems have been the hardest to do, but not because I don't understand but because it is just a lot of work. A lot of times people get scared when they see that there is a lot do where in actually it is pretty easy to do. So to help you in this I will do a problem where I can show you how easy it really is. First we must have a set group of numbers: 90, 60, 70, 80, 40, 30, 60, 10, 50, 20  With these number the very first thing I want to do is rearrange them from least to greatest: 10, 20, 30, 40, 50, 60, 60, 70, 80, 90 (This will make it much easier later on) Lets first start of with Mean, which means average. To do that we add up all the numbers and divide by how many there are, in this case there are 10 set of numbers.

10+20+30+40+50+60+60+70+80+90=510
510/10=51  Mean=51

We will now do the Median. We will do that by seeing which number is in the middle. If there are an even amount of number then you take the middle two and divide by two. If not just take the middle number at that is your answer. Since we have a even amount of numbers (10) we will take 50+60 and take that answer and divide it by two

50+60=110
110/2=55  Median=55

Next we will do the Mode. All you have to do is find which number is repeated most often, if there are a couple number that repeat the same amount of times, them put both of them, but if there isn't any you say there is no mode. In this case 60 is repeated most often and so the Mode=60

Now let us find the Range. To do that you will need get the largest and smallest number, in our example 90 and 10. All you have to do is subtract 10 from 90 and that is your Range.

90-10=80  Range=80

To help us find the IQR here is a website that I found that uses pictures just is case you don't understand my explanation. It gives the examples if there were an even amount of numbers and odd amount of numbers.   How To Find IQR

IQR means Interquartile Range. To do that will will need to find the upper and lower quartile. The median of our example is 55. So to find the lower quartile we need to take all the numbers from our group of numbers that are lower than 55, which are 50, 40, 30, 20, 10. We then take the Median of that set of number which is 30. So our Lower Quartile is 30. We now must get our higher quartile where we use all the numbers higher than 55, which are 60, 60, 70, 80, 90. The Median of this set of numbers is 70. So our Upper Quartile is 70. To find the IQR we need to subtract our lower quartile from our upper quartile.

70-30=40    IQR=40

We now must find the Variance. These next two part the Variance and Standard Deviation are probably the longest so don't get frustrated, here we go. We must first subtract the Mean from each of our numbers, our Mean was 51 so subtract 51 from all the numbers we started with

10-51= -41
20-51=-31
30-51=-21
40-51=-11
50-51=-1
60-51=9
60-51=9
70-51=19
80-51=29
90-51=39

We now must square them so they will all become positive numbers.

-41x-41=1681
-31x-31=961
-21x-21=441
-11x-11=121
-1x-1=1
9x9=81
9x9=81
19x19=361
29x29=841
39x39=1521

We will now add the sums of the squared numbers, and then divide by 10 because that is how many there are

1681+961+441+121+1+81+81+361+841+1521=6090
6090/10=609   Variance=609

Now with Standard Deviation all you have you do is take the square root of your answer  for the Variance which was 609

Square root of 609=24.67792536

Standard Deviation=24.8 (Rounded to the nearest tenth)

As you can see it is pretty lengthy but not difficult at all. I hope that this helped you out.

Tree Diagram

Have any of you heard of a tree diagram? Or what it is used for? Well we learned about them today and well they were confusing at first, but actually really fun after I understood how they worked. Here is an example of a Tree Diagram. They are used to help calculate probabilities and helps you understand how it works. So I hope this picture doesn't scare you right off the back because it is actually easier than it looks. 

Lets do one that will go along with this graph so you can have a visual aid. Suppose there is vending bouncy ball machine. In this machine there are 3 Red Balls, 2 Green Balls, and 1 White Ball. There is this mother named Sally who has two kids, twins to be exact and she wants to get them each a ball. What is the probability of getting a red, and then a green ball in that order?

So in the FIRST STAGE

On the line going up to red we would put 3/6
On the line that goes to the white one we would put 1/6
Finally on the line that goes to the green we would write 2/6

Now if you were to add up all the numerators you should get 6
1+2+3=6

Now the SECOND STAGE would be written like this, since there is one ball being taken out each time we have to take that into account.

Red to Red = 2/5
Red to White = 1/5
Red to Green = 2/5

Add the numerators up and you get 5

White to Red = 3/5
White to White = 0/5
White to Green = 2/5

Add the numerators up and you get 5

Green to Red = 3/5
Green to White = 1/5
Green to Green = 1/5

Add the numerators up and you get 5

Now for the THIRD STAGE you have

RR 6/30 + GG 2/30 = 8/30 = 4/15

The probability of Sally getting a red and then a green ball is 4/15

Whats the Difference Between Probability and Odds?

So far this semester this was one of the hardest things for me to understand was the difference between probability and odds. After looking at my test that was over this I clearly can see I was not entirely grasping this concept. So to help you all out I am going to try to explain it so you can have a better understanding than I did. To just make sure we are on the same page, I didn't understand it because I was taught wrong, it is because it was a difficult concept for me. So I want to share what I know to help you out. Here are a couple of sites that might be of some use to use while learning the differences between probability and odds.
                          - Changing Probability to Odds 
                          - Differences Between Odds and Probability
                          - Finding the Odds
                          - Probability of Events

Lets start by simple definition of each

Probability: Is the amount of successes divided by the total.

Odds in Favor: Is the amount of successes divided by the amount of failures.

Odds Against: Is the amount of failures divided by the amount of successes.

An example of probability would be if I was to say I have a 60% chance of getting a banana out of a basket that had only 6 bananas and 4 apples.  I would have a 6/10 = 3/5 chance

Odds in Favor: Would be 3:2 to get that I need to make sure that when the two numbers add up that they equal the total. So I subtract the successes from the total to give me the failures.

5-3=2 Since 3 is the successes and 2 is the failures and Odds in Favor is the amount of successes over amount of failures we get this...

3:2

Odds Against: Would be 2:3, and since we know that Odds Against is amount of failures over amount of successes it write it like this...

2:3

Making a circle graph from a bar graph

Today in class we talked about bar and circle graphs. Doing a bar graph isn't that difficult to make. All you really need is a ruler and a pen or pencil and some coloring utensils. Then according to the information you have, categorize it in any order you need and make sure that your bars don't touch each other. If they touch each other it would mean that all the information would relate. A bar graph is easily read, but does not display trends over time.

With circle graphs it is very difficult to make them accurate, because you need a protractor, legend and several other things and depending on your students comprehension it could be difficult for them. But here is a good way to make a circle graph by using a bar graph. We actually did this in class and it was a lot of fun. First we made our bar graph exact with a ruler and made sure the size was appropriate to the information we were using, (you want to make it big enough but not real big so you can eventually cut it out and make your circle graph). we then cut the individual bars out and taped them end to end to make a circle. We then (depending on how big our circle was) laid it down on several different blank white sheets of paper that we taped together and traced the circle. Next we marked on the outline of the circle where each category was. From there we found the center of the circle by measuring end to end (diameter), then taking that measurement and divided it by 2 to get the radius. We measured the radius a couple of times to other sides of the circle to make sure it would be the same measurement. Once we did that we made lines from the marks we made where the color taped bar/circle graph to the center of the circle and then colored it in accordingly. We then had a perfect circle graphed that was made from a bar graph. Here are some pictures from my experience.

 Comparing Graphs Here is a great website I found that helped me a little more to understand the differences between the types of graphs.

This is the Bar Graph cut out and taped together

This is the Bar Graph taped in a circle and then the Circle Graph made from the Bar Graph

Rock-Paper-Scissors

Have you ever played Rock, Paper, Scissors? Have you ever felt that your opponent knew what you where going to choose, therefore your opponent always choosing the thing that will make you lose? Is this a fair game? What are the chances of you winning? Losing? Or even tying? Well before we go into how this all works here is a video that should help you understand a little more about how competitive this game really is...

Ok so now that we have that out of the way, lets learn how we can actually find out if this game is a fair or rigged game. Today in class we actually found out if it was a fair game or not, and here is how we did it.
First we got into groups of two and played  45 games of Rock-Paper-Scissors and kept track who won and lost with rock paper and scissors, and we also kept track of the ones we tied. Afterwards we added up (individually) how many times I won, my partner won, and how many times we tied. In my example I won 19/45, Tied 10/45, Lost 16/45 times. We then were asked to explain if it was fair or not. We both decided it was because There was not a huge margin between the wins. So theoretically it is a fair game. Here is the reason why....

If you were to make a 3x3 square and mark the first column and row with Paper, then the second column and row with Scissors, then finally the last column and row with Rock. Now Player A will be the column side and Player B will the row side. Fill out your 3x3 square with all the possible outcomes and mark where there would be a tie with T, if player A would win mark it with an A, likewise for player B but with B.
What is the probability of player A winning? = 3/9 = 1/3
What is the probability of player B winning? = 3/9 = 1/3
What is the probability of a tie? = 3/9 = 1/3

Here is a link to that will also provide another way of seeing if the game is fair or not.
         Is Rock-Paper-Scissors Fair?

You now can see that Rock-Paper-Scissors is a fair game.

Theoretical Probability vs Experimental Probability

What is the difference between Theoretical and Experimental Probability? Are they not the same? Well I didn't even know there were two kinds of probability. In class we have been talking a lot about probability, and you will notice a lot of my recent posts have something to do a little about probability in one shape or another, but I feel it is important to talk about all the differences of probability because there is so much to talk about.

Lets start with the definitions of each.

Theoretical Probability: is calculating the possible outcomes rather than actually doing them. An example would be calculating the probability of a coin toss. Will it be heads or tails? You can determine the Theoretical Probability by calculating if it will be heads or tails.

Experimental Probability: Is actually doing an experiment and calculating the results/outcomes. An example would be flipping an actual coin and seeing it is heads or tails.

Here are a several sites that I have found that might help you a little more is mine doesn't make sense.

         - Theoretical vs Experimental Probability
         - Experimental vs Theoretical Probability
         - Experimental Probability
         - Theoretical Probability

In a nut shell the main difference between the two is that one is actually experimented on rather than only calculating numbers.