We have a review lesson today to go over what we have learned this semester so far: probabilities, complex numbers, and standard deviation.

We will have our midterm exam next week. There is no specific homework today, but students should go over their class notes and homework to prepare for the test.

We continue study of standard deviation. We first review the formula for ungrouped data and grouped data.

We apply the standard deviation to help analyze two data sets. While means can give us the picture of average, standard deviation helps us to determine how consistent the data are. For example, if we look at two sets of data representing two basketball players’ scores, the lower the standard deviation, the more consistent the player is.

Homework:

Print these pages: Page 1, Page 2, Page 3, Page 4

Workbook Page 2, #7, 8, 9, 10, 12, 14, 16.

We study standard deviation. In statistics, we have learned using mean to find the average, and we have learned to use range and interquartile range to see the spread of the data. But both range and interquartile range are not very good as the tool to represent the spread of the data.

A better way is the measure called standard deviation:

Standard Deviation SD = sqrt(sum of (Xi – AvgX)^2/N)

where N is the total number of data, AvgX is the mean of the data. Xi – AvgX is called the deviation of Xi from the mean AvgX for each i = 1, 2, …, N

For a set of grouped data in the form of a frequency table, we have

Mean AvgX = sum of fx / sum of f

where x is the class mark of each class and f is the frequency of the corresponding class.

Standard Deviation SD = sqrt(sum of f(Xi – AvgX)^2 / sum of f)

[Sorry, I can’t type using the summation notation sigma, which is what we learn in the class. This will simplify the writing.]

Here is a written proof of going from the definition to a formulae often used:

sqrt(∑(Xi – AvgX)^2 / N) = ∑Xi^2 / N – (∑Xi / N)^2

I showed the proof in the classroom, but students may not get it.

Homework:

Page 1  and Page 2: #1, #2, #3, #4, #5, #6.

We continue our study of complex number. Today we talk about representing complex numbers in Complex Plane and how to calculate the magnitude of a complex number. We also look at various properties of a complex number with its conjugate.

Homework: Page 1, Page 2, Page 3,

3.2.1 – 3.2.7, 3,3.1 – 3.3.1 – 3.3.7

In the next two lessons, we study complex numbers.

Today we talk about the arithmetic of complex numbers, addition, subtraction, multiplication and division.

Homework: Page 1, Page 2,

3.1.1 – 3.1.7

Two events are called to be independent events if the occurrence or non-occurrence of one event does not affect the probability of the other event.

If A and B are independent events, the probability of both events A and B occurring is the product of their individual probabilities:

P(A and B) = P(A) * P(B)

With this, we can simplify tree diagrams when dealing with problems with multiple same items by simply marking their probabilities.

When two (or more) events are dependent to each other, we cannot use the product rule, but we can still use the tree diagrams to help us finding probabilities.

Homework:

Print these pages: Page 1, Page 2, Page 3

Workbook Page 12, #6 – #10, #16 – #20.

We start on probability. Students should have learned some basic concepts before about simple probability, the concept of a sample space with events consisting of some outcomes. In this chapter ,we study probabilities with multiple stages.

When a random experiment involves two stages, we can use a rectangular grid, called a probability diagram, to represent the sample space to help us find probabilities.

If a random experiment has two or more stages, we can use a tree diagram to represent the process, which should help us see all the possible outcomes and figure out the outcomes associated with a particular event.

We then talk about mutually exclusive events. In a sample space, two events are mutually exclusive if they cannot occur at the same time. If A and B are two mutually exclusive events, the the probability of A or B occurring is: P(A or B) = P(A) + P(B).

Homework:

Print these pages: Page 1, Page 2

Workbook Page 11, #1 – #5, #21.