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 the study of logarithm. Today we go over logarithmic identities.
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 start a new chapter on Exponents and Logarithms.
Today we cover Exponential Functions Basics and Introduction to Logarithms.
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:
We study Cauchy-Schwarz Inequality and solving problems for finding Maxima and Minima.
This concludes Chapter 12. Next week we will start a new chapter.
We continue our study of complex number. Today we talk about representaing 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
We study Arithmetic Mean and Geometric Mean with 2 variables and more variables.
For any n nonnegative real numbers a1, a2, …, an,
(a1 + a2 + … + an) / n >= n-th root of (a1a2…an)
