Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions.

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1 Lecture 1 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a process consisting of 3 parts.

2 Lecture 1 2 First part: collecting data Identify the research objective: the group that it is to be study is called population. A member of the population is called individual. Collect the information needed to answer the questions posed: typically look at a subset of the population called sample. Example: what is a placebo, experimental group, treatment, control group, double-blind experiment, single-blind experiment

3 Lecture 1 3 Second part: Organize and summarize the information This step is called descriptive statistics, or exploratory data analysis. Uses tables, charts, graphs, etc to describe the data collected. Third part: Draw conclusions from the information This part is called inferential statistics. We can not learn everything about the population just by looking at a sample!!! But we might be able to say something with a certain level of confidence.

4 Lecture 1 4 Types of data Definition 2. The characteristics, that we decided we are interested to study, of the individual within the population are called variables. Variables can be classified into two groups: Definition 3. Qualitative or categorical variables allow for classification of individuals based on some attribute or characteristics. Quantitative variables provide numerical measures of individuals. Arithmetic operations can be performed on the values of a quantitative variable and provide meaningful results. Examples: The distribution of a variable tells us what values it takes and how often it takes these values.

5 Lecture 1 5 Quantitative variables can be classified into two types: Definition 4. A discrete variable is a quantitative variable whose possible values could be counted: 0,1,2,3,4,5. Examples: A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable. Examples: The list of observations a variable assumes is called data. Data could be classified in the same categories as variables.

6 Lecture 1 6 Organizing Categorial Data We are interested in the number of individuals that occur in each category. Definition 5. A frequancy distribution lists the number of occurances (or the count) for each category of data. The relative frequency is the proportion or percent of observations within each category and is found using the formula Relative frequency = frequency sum of all frequencies A relative frequency distribution lists the relative frequency of each category of data. Examples:

7 Lecture 1 7 Definition 6. A bar graph is constructed by labeling each category of data on the horizontal axis and the freequency or relative frequency of the category on the vertical axis. A rectangle of equal width is drawn for each category. The height of the rectangle is equal to the category s frequency or relative frequency. A Pareto chart is a bar graph whose bars are drawn in decreasing order of frequency or relative frequency. Definition 7. A side-by-side bar graph is used when we want to compare two sets of data. Carefull!: We should use relative frequencies when drawing a side-by-side chart!!! (Why?) Examples:

8 Lecture 1 8 Definition 8. A pie chart is a circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category. Remark: The size of the angle of the sectors of the pie chart is given by percetange 360

9 Lecture 1 9 Organizing Quantitative Data 1. Discrete data Discrete data could be organized in tables and histograms. Tables To create a table we will use the values of the discrete variable. Histograms Definition 9. A histogram is constructed by drawing rectangle for each class of data. The height of each rectangle is the frequency or relative frequency of the class. The width of each rectangle should be the same and the rectangles should touch each other.

10 Lecture Continuous data Continuous data could be organized in tables, histograms, stem-and-leaf plots, frequency polygons, ogives, cumulative and relative frequency tables, and time series plots. Tables We create categories of data by using intervals of numbers called classes. Each class has a lower class limit and an upper class limit. The difference between them is the class width. All of the classes have to have equal size unless we deal with a table that is open ended (the last class does not have an upper class limit).

11 Lecture 1 11 Frequency and relative frequency distributions We use the same techniques as for tables.

12 Lecture 1 12 Histograms of continuous data On the x axis we put the lower class limit of each class. For each class we draw a rectangle with the width equal to the width of the class and the height equal to the frequency or relative frequency of the class.

13 Lecture 1 13 Stem-and-Leaf Plots 1) Draw the stem of the graph by putting down the digits to the left of the rightmost digit. The leaf of the graph will be the rightmost digit. Modify the method of choosing the stem if a different class width is desired. 2) Write the stems in a vertical column in increasing order. Draw a vertical line to the right of the stems. 3) Write each leaf corresponding to the stems to the right of the vertical line. The leaves must be written in ascending order. Example: Final exam scores from fall , 82, 80, 77, 77, 76, 74, 63, 63, 54.

14 Lecture 1 14 Distribution shape In any graph of data, look for the overall pattern and for striking deviations from the pattern. Several types of distributions we will deal with are: a) Uniform distribution b) Bell-shaped (symmetric) c) Skewed right (the right tale, or the larger values, is much longer than the left tail d) Skewed left.

15 Lecture 1 15 Definition 10. A time series plot is obtain by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Lines are drawn connecting the points. Homework due Thursday, September 3rd: problem 1.28 a) and b)

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