The rise of statistics Statistics is the science of collecting, organizing and interpreting data. The goal of statistics is to gain understanding from data. Historically, the ideas and methods of statistics developed gradually as society became interested in collecting and using data for a variety of applications. The discipline of statistics took shape in the twentieth century when methods for producing and understanding data grew in number and sophistication. week1

Elements of Statistics - Introduction Data are numerical facts with context and we need to understand the context if we are to make sense of the numbers. A set of data contains some information about a group of individuals. Individuals are the objects upon which we collect data. Individuals can be people, animals, plots of land and many other things. A population is a set of individuals that we are interested in studying. A variable is any characteristic of an individual. A sample is a subset of the individuals of a population. week1

Questions to ask when planning a statistical study Why? What purpose do the data have? Do we hope to answer some specific questions? Do we want to draw conclusions about individuals other than the ones we actually have data for? Who? What individuals do the data describe? How many individuals appear in the data? What? How many variables do the data contain? Exact definitions of these variables. What are the units of measurements in which each variable is recorded? Weights for example, might be recorded in pounds, or in kg. week1

Collecting Data Generally, data can be obtained in four different ways. Published source. Designed experiment. Survey. Observational study. week1

Types of Variables A categorical variable places an individual into one of several groups or categories, e.g. gender, college major. A quantitative variable takes numerical values for which arithmetic operations are defined, e.g. height, weight. The distribution of a variable tells us what values it takes and how often it takes these values. Example 1.4 on page 12. week1

Displaying Distributions with Graphs Statistical tools and ideas help us examine data in order to describe their main features. This examination is called exploratory data analysis. Two basic strategies for exploration of data set: Begin by examining each variable by itself. Then move on to study the relationships among the variables. Begin with graphs. Then add numerical summaries of specified aspects of the data. week1

Graphs for categorical variables A class is one of the values of a categorical variable. It is usually the label for the categories such as “male” and “female”. The class frequency (count) is the number of observations in the data set falling in a particular class. The class relative frequency is the class frequency divided by the total number of observation in the data set. The distribution of a categorical variable lists the categories and gives either the count or the percent of individuals who fall in each category. week1

Example : The of marital status for all Americans age 18+. Count (millions) Percent Never married Married Widowed Divorced 43.9 116.7 13.4 17.6 22.9 60.9 7.0 9.2 week1

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Measurement - example We want to compare the “size” of several statistics books. Describe three possible numerical variables that describes the “size” of a book. In what units would you measure each variable? What measuring instrument does each require? Describe a variable that is appropriate for estimating how long it would take to read the book? week1

Describing Quantitative Data The pattern of variation of a variable is called its distribution. The distribution of a variable is best displayed graphically. There are three main graphical methods for describing summarizing and detecting patterns in quantitative data: Dot plot Stem-and-leaf plot Histogram week1

Stem-and-Leaf Plots To make a stem-and-leaf plot: Separate each observation into a stem consisting of all but the final (rightmost) digit and a leaf, the final digit. Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. Write each leaf in the row to the right of its stem, in increasing order out from the stem. Examples 2.2 on page 42-43. week1

Example Here are the scores of a basketball player (say player A) 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22 Make a stem-and-leaf plot of these data. Describe the main features of the distribution. Solution : Min. = 22, Max = 60 … MINITAB command: Graph > Stem-and-Leaf week1

Exercise Write down the min, max, and the median of the data sets summarized by the following MINITAB stemplot. Stem-and-leaf of Fuel Use N = 15 Leaf Unit = 1.0 1 0 5 5 0 6677 (4) 0 8899 6 1 001 3 1 23 1 1 1 2 1 week1

Exercise Write down the min., max, and the median of the data sets summarized by the following MINITAB stemplot. Stem-and-leaf of weight N = 12 Leaf Unit = 0.010 3 107 479 (4) 108 0779 5 109 5 110 5 111 0 4 112 56 2 113 79 week1

Examining a distribution In any graph of data, look for the overall pattern and for striking deviations from that pattern. Overall pattern of a distribution can be described by its shape, centre, and spread. An important kind of deviation is an outlier, an individual value that falls outside the overall pattern. Some other things to look for in describing shape are: Does the distribution have one or several major peaks, called modes? A distribution with one major peak is called unimodal. Is it approximately symmetric or skewed in one direction. week1

Exercise Describe the shape of the distributions summarized by the following stemplot. Stem-and-leaf of sta220 marks N = 42 Leaf Unit = 1.0 1 6 7 3 7 44 11 7 77888999 (11) 8 00011233444 20 8 555556666778 8 9 000001 2 9 7 1 10 0 week1

Exercise Describe the shape of the distributions summarized by the following stemplots. Stem-and-leaf of C1 N= 50 Leaf Unit = 0.10 18 0 000111122233334444 (17) 0 55555566667889999 15 1 0011444 8 1 5669 4 2 03 2 2 2 3 1 1 3 1 4 2 . week1

Exam Question Forty students wrote a Statistics examination having a maximum of 50 marks. The mark distribution is given in the following stem-and-leaf plot: Stem Leaf 0 28 1 2245 2 01333388888 3 011156679 4 22444466788 5 000 State whether the following statements are true or false.  (a) The distribution is right skewed.  (b) The median of the distribution is 31.  (c) The median of the distribution is 28. (d) The mode of the distribution is 48.   (e) More than 18% of the students scored 45 or more on the examination. week1

Histograms A histogram breaks the range of values of a variable into intervals and displays only the count or percent of the observations that fall into each interval. We can choose a convenient number of intervals. Histograms do not display the actual values observed. (only counts in each interval). Example: Here is some data on the number of days lost due to illness of a group of employees: 47, 1, 55, 30, 1, 3, 7, 14, 7, 66, 34, 6, 10, 5, 12, 5, 3, 9, 18, 45, 5, 8, 44, 42, 46, 6, 4, 24, 24, 34, 11, 2, 3, 13, 5, 5, 3, 4, 4, 1 week1

The main steps in constructing a histogram Determine the Range of the data (largest and smallest values) In our example the data ranges from a min. of 1 day to a max. of 66 days. Decide on the number of intervals (or classes) , and the width of each class (usually equal). Count the number of observations in each class. These counts are called class frequencies. 4. Draw the histogram. week1

Class No. of employees (Frequency) Cumulative. Frequency Relative frequency 0-10 10-20 20-30 30-40 40-50 50-60 60-70 23 5 3 2 1 28 31 33 38 39 40 0.575 0.125 0.075 0.050 0.025 Total 1.000 A table with the first two columns above is called frequency table or frequency distribution. A table with the first column and the third column is called relative frequency distribution. week1

MINITAB command: Graph > Histogram week1

Comments The above histogram suggests that the distribution is skewed to the right. No gaps or outliers. Since this data set is not very large (40 observations) we can also use a dot plot or a stem-and-leaf plot to represent the data. MINITAB commands for dotplot: Graph > Character Graphs > Dotplot week1

Stem-and-leaf of days lost N = 40 Leaf Unit = 1.0    (22) 0 1112333344455555667789 18 1 012348 12 2 44 10 3 044 7 4 24567 2 5 5 1 6 6 week1

Question Which type of display Uses the actual numbers as building blocks for the display? Gives the most flexibility for setting the class width and number of classes? Is most convenient for very large data sets? Is quickest to construct? Keeps the most detail re the actual data? week1

Review Questions The purpose of a frequency distribution is to ____. present scores and their frequency of occurrence present data in a more meaningful way than single scores provide more information than a graph all of the above a and b Which of the following indicates the proportion of the total number of scores which occurred in each interval? Relative frequency distribution Cumulative frequency distribution Cumulative percentage distribution None of the above week1

Relative frequency distribution Cumulative frequency distribution 3. Which of the following indicates the number of scores which fell below the upper limit of each interval? Relative frequency distribution Cumulative frequency distribution Cumulative percentage distribution None of the above 4. Which of the following is not a symmetrical distribution? A bell-shaped distribution A J-shaped distribution A U-shaped distribution An inverted U-shaped distribution week1