Introduction to Statistics

Class Activity

For my final post, I'd like to talk about a recent class activity that was done in the area of statistical data, such as mean, median, mode, and standard deviation. Once again, we gathered data from the students within my class about the total dollars spent on textbooks for the semester. Using TI-73 calculators, we were able to run a statistical analysis of the data.

Originally, to find the mean, the costs of the textbooks would all have to be added and the sum would then be divided by the number of students in the classroom.

To find the median, the costs would be arranged from least to greatest and the number in the middle would be defined as the 'median'.

The most common cost of the textbooks would be considered as the mode. The common numbers, meaning the number was used twice or more.

The calculation for standard deviation is shown in the video below.

With Replacement

Suppose you have 2 bags with the exact same sets of marbles inside. There are 4 red, 5 blue and 7 yellow marbles in each bag. From the first bag, you reach in and take out a marble.  You record the color and then put the marble back into the bag.  You then try a second time, and from that bag you do the exact same thing except, after you select the first marble and record it's color, you do not put the marble back into the bag,  You then select a second marble, just like the first time. This trial involves a process called "with replacement".  You put the object back into the bag so that the number of marbles to choose from is the same for both draws.

You may notice that after going into detail with this topic of 'replacement', you will be able to notice the difference between With Replacement and Without Replacement. The post I wrote of Without Replacement shows an actual equation. And, this one I was able to write it out and I wanted to explain to the best of my knowledge. I have now learned the difference from the two.

Pictographs

In my previous post of pie charts and dot plots, I mentioned a lesson that was done in class. I will give a few details as to what we did in that lesson. The instructor handed everyone a small bag of M&M's and we each counted how many brown, green, orange, red, blue and yellow pieces of candy were in a bag. As a class we added up all our amounts and we put the data into a pictograph, dot plot, bar graph, and pie chart.

We, the students had to create a pictograph based on our own data from our own individual M&M bags. That was the first step in the lesson, and we discussed whether pictographs were indeed accurate and easy to read as compared to another graph; such as a bar graph or a pie chart. After doing a little research, I learned that students should be learning about graphs in Kindergarten, and as they move on to 1st grade they should be familiar with graphs and be able to read them.

Pie Charts and Dot Plot

I found pie charts and dot plot graphs to be very useful. One of the lessons my instructor gave was using graphs and how important they are to show an amount broken into parts of percentages. Pie charts are broken into different sectors, and each visually represents an item in a data set to match the amount of the item as a percentage or fraction of the total data set.
In class, we did a lesson that dealt with frequencies of M&M's. This lesson gave me a better understanding that a dot plot helps create a pie chart with a more accurate percentage of what the data will be divided into within the sectors.

Without Replacement

A player is dealt 2 cards from a standard deck of 52 cards. What is the probability of getting a pair of aces?   Think about it....obviously this is an experiment "without replacement" because the player was given 2 cards. To calculate the probability of a pair of aces you use the rules for compound events: P(ace on first card) = 4/52 (remember,there are 4 aces in the deck) P(ace on the second card) = 3/51! (the first card drawn was an ace!) So the probability of getting 2 aces is: P(ace,ace) = (4/52)(3/51) = 12/2652 = 1/221!

In class I had trouble understanding the difference between Without Replacement and With Replacement. I will explain my learning experiences in another blog about understanding With Replacement. For now I would like to share what, and how I learned to deal with probability problems that deal with Without Replacement. 

So, I found this equation and I found it to be simple and helpful in understanding how exactly Without Replacement works. Using a standard deck of cards is the best way, in my opinion to understand the concept. The equation asked for the probability of getting a pair of aces. By drawing the first card, which is an ace and remember there are 4 aces in a deck, the probability is 4/52. Without Replacement comes into play by not putting that first drawn card back in the deck and simply drawing the next one from a deck of 53 cards.

Tree Diagrams

During the beginning of the course, we begin with Probabilities. When we were given the assignment to use a Tree Diagram, I was thrown off guard because I didn't know much about it. I found out that calculating probabilities isn't an easy task. Yet there are ways of solving a given problem, either with addition or multiplication.
After searching through a couple of sites, I found a particular site that gave me more information on what and how a Tree Diagram works. There is a Branch, Probability and an Outcome. Using these three, you calculate the probabilities by multiplying the first and second stages, and you would add all the outcomes together to reach the sum of 1