Dr. Hong Zhang

 Tables and Graphs  Populations and Samples  Mean, Median, and Standard Deviation  Standard Error & 95% Confidence Interval (CI)  Error Bars  Comparing Means of Two Data Sets  Linear Regression (LR)  Coefficient of Correlation

 Statistics is a huge field, I’ve simplified considerably here. For example: ◦ Mean, Median, and Standard Deviation  There are alternative formulas ◦ Standard Error and the 95% Confidence Interval  There are other ways to calculate CIs (e.g., z statistic instead of t; difference between two means, rather than single mean…) ◦ Error Bars  Don’t go beyond the interpretations I give here! ◦ Comparing Means of Two Data Sets  We just cover the t test for two means when the variances are unknown but equal, there are other tests ◦ Linear Regression  We only look at simple LR and only calculate the intercept, slope and R2. There is much more to LR!

 All of the possible outcomes of experiment or observation ◦ US population ◦ Cars in market  A large population may be impractical and costly to study. It might be impossible to collect data from every member of the population. ◦ Weight and height of every US citizen ◦ Quality of every car in market

 A part of population that we actually measure or observe and to draw outcome or conclusion ◦ 1000 US citizens ◦ 100 cars  We use samples to estimate population properties ◦ Use 1000 US citizens to estimate the height of entire US population ◦ Use 100 cars to estimate quality of all Toyota Corolla cars under 3 years old

 Sample should fully represent the entire population. ◦ Good  Randomly select 1000 names from a phone book to represent the region  Randomly select 100 cars from DMV record ◦ Bad  Use a college campus to represent the country  Use cars in dealers lot to represent cars in market  Reporters randomly stop 3 persons on street for opinions

 Sum of values divided by number of samples, also called Average  Example: ◦ Data: 3, 8, 5, 10, 4, 6 ◦ Sum = = 36 ◦ Number of samples (data points) = 6 ◦ Mean = 36 / 6 = 6  Exercise ◦ Mean of height of the entire class ◦ Average commute time of the students

 Bill Gates comes to give a presentation to 100 of students in Rowan Auditorium.  Suppose the personal wealth of Bill Gates is $50 billion.  The personal wealth of each student is $0.  What is the mean of the personal wealth for the entire population in the room?

 V alue of the middle item of data arranged in increasing or decreasing order of magnitude  Example: ◦ Data: 3, 8, 5, 10, 4, 6 ◦ Rearrange: 3, 4, 5, 6, 8, 10 ◦ The middle two are 5 & 6, the average of the two is 5.5 ◦ The mean of the data set is 5.5  Exercise: ◦ Medium height of the class ◦ Medium commute time of the class ◦ Medium personal wealth in the room with Bill Gates.

Data Points: 3, 8, 5, 10, 4, 6

 Standard deviation of mean ◦ Sample size n ◦ taken from population with standard deviation s ◦ Estimate of mean depends on sample selected ◦ As n , variance of mean estimate goes down, i.e., estimate of population mean improves ◦ As n , mean estimate distribution approaches normal, regardless of population distribution

μ: Mean, n: Sample size, x i : Data point

For n > 30 For n < 30

S=s 2

 Data:

 Flip a coin, chances of upside up and downside up are equal. (It’s also called binomial dist.) up dow n 50%

 Normal distribution ◦ Women’s shoe size sold by a shoe store.

 Chemical distribution of a well mixed compound

 where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately , and e is approximately

NσConfidence IntervalsError per million sigma

 Rank k has a frequency roughly proportional to 1/k, or more accurately P n =a/n b  Developed by George Kingsley Zipf  Occurs naturally in many situations ◦ City population ◦ Colors in images ◦ Call center ◦ Website traffic

Rank Word Freq % Freq Theoretical Zipf Distribution 1 the of and to a in that is was he for it with as his on be at by I

 If a distribution gives us a straight line on a log-log scale, then we can say that it is a Zipf Distribution.

 Count the vehicles in Rowan Parking lots ◦ Distribution of colors ◦ Distribution of cars and trucks ◦ Distribution of last letter (digit) of license number  Select a parking lot  Design a strategy to count  Design a method to record data  Design a method to represent result  Write a one page report per group

 White:2  Black:1  Red:2  Blue:2  Silver:4  Gold: 1  Beige: 1

Voltage (V)Height (in) Result for Pressure Transducer Calibration

Time (s) Voltage (V)

Time (s) Voltage (V)log(Voltage)

Concentration (Mol/ft 3 ) Reaction Rate (Mol/s)

ConcentrationReaction Rate log (concentration) log (reaction rate)

Table 1: Average Turbidity and Color of Water Treated by Portable Water Filters Consistent Format, Title, Units, Big Fonts Differentiate Headings, Number Columns

11 Figure 1: Turbidity of Pond Water, Treated and Untreated Consistent Format, Title, Units Good Axis Titles, Big Fonts