Descriptive Statistics Methods of organizing and summarizing any data/information.

Transcription

1 Introductory Statistics, 10 th ed. by Neil A. Weiss Chapter 1 The Nature of Statistics 1.1 Statistics Basics There are lies, damn lies, and statistics - Mark Twain Descriptive Statistics Methods of organizing and summarizing any data/information. Here are some fun descriptive statistics: There are 40,000 pounds of Fritos bagged every hour 4 million bags of Cheetos are bagged every day Cheetos uses 12 million pounds of cheddar each year Over 400 million M&M s are produced in the U.S. every day (Have trouble wrapping your brain around huge numbers? See the Mega Penny Project on Ms. V s homepage!) Population ALL individuals or items under consideration in a statistical study. Sample A part/subset of the population from which information is obtained. Inferential Statistics Methods of using data collected from a sample to draw conclusions about the population from which the sample was obtained. Two sub-classifications of statistical studies are: Observational study Researchers observe and collect existing characteristics and take measurements. Designed experiment Data doesn t exist until researchers set up and implement a study. After the study produces results, measurements are collected.

2 1.2 Simple Random Sampling Census - A collection of data from EVERY element of a population. The U.S. Census cost the taxpayers $13 billion in about $42 per capita. The cost has doubled every decade since 1970!!!!!! Simple random sampling (random sampling) - The process of selecting individuals/subjects from a population to be included in a sample. We MUST collect a representative sample, that is, it must mimic the characteristics of the population. NOTE: With random sampling, each element of the population must have an equally likely chance of being selected. Said another way, every sample of the same size (denoted by n) has an equally likely chance of being selected. 99% of the time, we rely on sampling. A valid form of sampling MUST be used to obtain meaningful results! IMPORTANT TO NOTE: In our text, the author assumes that students aren t learning this subject using technology, so he describes how to obtain results by hand. We will use StatCrunch whenever possible. There are two types of random sampling: - Sampling with replacement A member of the population can be selected more than once. Once selected, the item/individual is placed back in the population. - Sampling without replacement A member of the population can be selected at most once. Once selected, the item/individual does not get put back in the population. Unless specified otherwise, we will assume that sampling is done without replacement. Disregard random-number tables on page 11. We will use StatCrunch to sample.

3 Bias When some source works its way into the sample such that the sample isn t representative of the population. Bias can occur in different ways: - If the way the sample was chosen favors one part of the population over any other. (sampling bias) - When the opinions of individuals who were selected for a survey but didn t respond have different opinions from those who did respond. (nonresponse bias) When the response selections don t accurately reflect the true feelings of the respondent. (response bias) StatCrunch: Data, sample data 1.3 Other Sampling Designs More interesting descriptive statistics: One in 20 men is at least partially color blind Color blindness is 10 times more likely in men than in women All babies are color blind at birth All human fetuses start out female Systematic Random Sampling A starting point is chosen; then every kth element of the population is selected. The steps for systematic sampling are on page 17. Cluster sample The population is divided into geographic groups, or clusters. A number of clusters are randomly selected, then all individuals/subjects within the selected clusters are sampled. (random sampling can occur within each cluster, too) Steps for cluster sampling are on page 18. Stratified sample The population is first divided into at least two groups (stratum) that share the same characteristic. Then random samples are selected from each stratum. - The researcher may want to sample proportional to the size of each stratum, (stratified sampling with proportional allocation) but the sample sizes don t have to be equal nor proportional.

4 There are also: Convenience sample Obtaining the easiest available data or participant. Voluntary response sample - The subjects self-select to end up in the sample. 1.4 Experimental Designs Experimental units - The individuals/items on which an experiment is performed. In the case that the experimental units are human, they are called subjects. Response variable The characteristic (numeric or otherwise) that is to be measured or observed. Factor A variable whose effect on the response variable is of interest. Level The possible values/subsets of a factor. Treatments Can be a factor or a combination of factors. Control A group that gets no, or a fake (placebo) treatment. It is the group to which the other treatments are compared. Chapter 2 Organizing Data 2.1 Variables and Data Yet more fun descriptive statistics! The average dog costs $18,000 throughout its lifetime The average dog is walked 575 miles per year One in 10 dog owners let their dog sleep on their bed

5 Data The responses/measurements/observations/values of a variable that are collected. Variable Measurement/characteristic that has the capability of change from subject to subject or item to item. There are two main types of variables: o Qualitative variables Based on an attribute, classification or category. Non-numeric. o Quantitative variables Always numeric and represents counts or measurements. Additionally, there are two types of quantitative variables: o Discrete Has a finite number of possible outcomes. (usually counts) o Continuous Has an infinite number of possible outcomes. (usually measurements) 2.2 Organizing Qualitative Data The human brain does a good job of recalling information which has been summarized visually, as in tables and graphs. Frequency distribution A table consisting of values of a variable (qualitative in this case) and their corresponding frequencies. Relative frequency distribution A table consisting of values of a variable (again, qualitative in this case) and their corresponding relative frequencies which are the percentages of data that correspond to each variable. The Expression builder in StatCrunch is very useful for tedious calculations. To construct a relative frequency distribution from an existing frequency distribution in StatCrunch, go to Data, Compute, Expression. Then Build. The function you build will show up in the bar on top. In the Columns column, click Frequency and Add Column at the bottom of the column. Then click the / (divide) symbol in the calculator-type buttons on the left. Then in the Functions column, scroll down and find Sum. The functions are in alphabetical order. Add Function. Then click Frequency and Add Column again. Your expression at the top should look like this: FREQUENCY/sum(FREQUENCY). The word frequency may or may not be in capital letters depending on the problem. Then Okay and Compute. A column of the relative frequencies will be added to your spreadsheet. Ta da!!

6 The two best graphs for qualitative data are bar graphs and pie charts. The terms graph, chart and diagram usually mean the same thing. A bar graph - Depicts categories/classes on the horizontal axis, frequencies or relative frequencies on the vertical axis. (or vice-versa) The bars heights (or lengths) indicate the amount of data in each category. Bars are separated for comparison and both axes are clearly labeled. Pie chart - A circle with pie-shaped wedges that represent categories/classes. The size of each wedge is proportional to the frequency/relative frequency in that category/class. Qualitative frequency or relative frequency distributions in StatCrunch: Stat, Tables, Frequency (for raw data) Bar graphs in StatCrunch: Graph, Bar plot Graph, Chart (for summarized data) Pie charts in StatCrunch: Graph, Pie chart 2.3 Organizing Quantitative Data Classes Categories (sometimes called bins) that contain the possible numbers or intervals of numbers of the data. Frequencies/relative frequencies are the same as in 2.2. For quantitative discrete data, the classes are made up of single values. There are two similar ways to organize data using interval classes. One is called Limit Grouping. Generally used when the data is discrete. Lower class limits The smallest value that can belong to a class.

7 Upper class limits The largest value that can belong to a class. Class width - The difference between two consecutive lower (or upper) class limits. Class mark The average of the lower and upper class limits of a class. The second is called Cutpoint Grouping. Generally used when the data is continuous. Lower class cutpoint The smallest value that can belong to a class. Upper class cutpoint The smallest number that can belong to the next-higher class. Said another way, the same number as the lower class cutpoint of the next-higher class. Class width - The difference between the cutpoints of a class. Class midpoint The average of the two cutpoints of a class. Important about graphing quantitative data: o Each observation belongs to exactly one class o Include all classes, even if the frequency is zero o Use the same width for all classes (exception: an open-ended first or last class) o Use between 5 and 20 classes Frequency/relative frequency histogram - Depicts classes on the horizontal axis, frequencies or relative frequencies on the vertical axis. Bars indicate both the width of the classes (width of each bar) and the amount of data in each class (heights of each bar). Bars are always drawn touching when the data is numeric. Dot plot Used for discrete data. Consists of a horizontal axis with the range of data values labeled. Dots are drawn to represent data values with duplicate values stacked. Stem and leaf plot Each data value is split into two parts, the stem and the leaf which are then drawn about a vertical line. Stems are written on the left side and only once for every data value with that same stem. Leaves are written in lines on the right side after their corresponding stem.

8 Discrete frequency distributions in StatCrunch (for raw data) Stat, Tables, Frequency For continuous data, you may need to bin the data first to assign it to classes. Data, Bin Histograms in StatCrunch Graph, Histogram If data is summarized, Graph, Chart Stem and leaf plots in StatCrunch Graph, Stem and leaf Dotplots in StatCrunch Graph, Dotplot 2.4 Distribution Shapes The shape of the graph of numeric data is incredibly important, as it gives us insight on how the data can be analyzed. First, Population data Values of the variable under consideration for the entire population. A population distribution is one that comes from a population. Sample data Values of the variable under consideration for a sample, or subset from a population. A sample distribution is one that comes from a sample. You will get tired of your instructor saying this. If a sample is collected randomly, it will mimic the characteristics of the population from which it came. Symmetric o Bell-shaped o Triangular o Uniform

9 Skewness o Right skewed o Left skewed Modality o Unimodal o Bimodal o Multimodal 2.5 Misleading Graphs Read this section.