Introduction To Statistics

Chapter 1 Introduction to Statistics 1-1 Overview 1-2 Types of Data 1-3 Critical Thinking 1-4 Design of Experiments

Created by Tom Wegleitner, Centreville, Virginia Section 1-1 Overview

Overview A common goal of surveys and other data collecting tools is to collect data from a smaller part of a larger group so we can learn something about the larger group. In this section we will look at some of ways to describe data.

Data observations (such as measurements, genders, survey responses) that have been collected. Definitions

Statistics a collection of methods for planning experiments, obtaining data, and then then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions based on the data. Definitions

Definitions Population the complete collection of all elements (scores, people, measurements, and so on) to be studied. The collection is complete in the sense that it includes all subjects to be studied.

Census the collection of data from every member of the population. Sample a sub-collection of elements drawn from a population. Definitions

Key Concepts Sample data must be collected in an appropriate way, such as through a process of random selection. If sample data are not collected in an appropriate way, the data may be so completely useless that no amount of statistical torturing can salvage them.

Created by Tom Wegleitner, Centreville, Virginia Section 1-2 Types of Data

Parameter a numerical measurement describing some characteristic of a population Definitions population parameter

Definitions Statistic a numerical measurement describing some characteristic of a sample . sample statistic

Definitions Quantitative data numbers representing counts or measurements. Example: weights of supermodels.

Definitions Qualitative (or categorical or attribute) data can be separated into different categories that are distinguished by some nonnumeric characteristics. Example: genders (male/female) of professional athletes.

Working with Quantitative Data Quantitative data can further be distinguished between discrete and continuous types.

Discrete data result when the number of possible values is either a finite number or a ‘countable’ number of possible values. 0, 1, 2, 3, . . . Example: The number of eggs that hens lay. Definitions

Continuous (numerical) data result from infinitely many possible values that correspond to some continuous scale that covers a range of values without gaps, interruptions, or jumps. Definitions 2 3 Example: The amount of milk that a cow produces; e.g. 2.343115 gallons per day.

Levels of Measurement Another way to classify data is to use use levels of measurement. Four of these levels are discussed in the following slides.

nominal level of measurement characterized by data that consist of names, labels, or categories only. The data cannot be arranged in an ordering scheme (such as low to high) Example: survey responses yes, no, undecided Definitions

ordinal level of measurement involves data that may be arranged in some order, but differences between data values either cannot be determined or are meaningless Example: Course grades A, B, C, D, or F Definitions

interval level of measurement like the ordinal level, with the additional property that the difference between any two data values is meaningful. However, there is no natural zero starting point (where none of the quantity is present) Example: Years 1000, 2000, 1776, and 1492 Definitions

ratio level of measurement the interval level modified to include the natural zero starting point (where zero indicates that none of the quantity is present). For values at this level, differences and ratios are meaningful. Example: Prices of college textbooks ($0 represents no cost) Definitions

Nominal - categories only Ordinal - categories with some order Interval - differences but no natural starting point Ratio - differences and a natural starting point Summary - Levels of Measurement

Recap Basic definitions and terms describing data Parameters versus statistics Types of data (quantitative and qualitative) Levels of measurement In Sections 1-1 and 1-2 we have looked at:

Created by Tom Wegleitner, Centreville, Virginia Section 1-3 Critical Thinking

Success in Statistics Success in the introductory statistics course typically requires more common sense than mathematical expertise. This section is designed to illustrate how common sense is used when we think critically about data and statistics.

Misuses of Statistics Bad Samples

Definitions Voluntary response sample (or self-selected survey) one in which the respondents themselves decide whether to be included. In this case, valid conclusions can be made only about the specific group of people who agree to participate.

Misuses of Statistics Misleading Graphs Bad Samples Small Samples

To correctly interpret a graph, we should analyze the numerical information given in the graph instead of being mislead by its general shape.

Misuses of Statistics Bad Samples Small Samples Misleading Graphs Pictographs

Figure 1-2 Double the length, width, and height of a cube, and the volume increases by a factor of eight

Misuses of Statistics Bad Samples Small Samples Misleading Graphs Pictographs Distorted Percentages Loaded Questions

97% yes: “Should the President have the line item veto to eliminate waste?” 57% yes: “Should the President have the line item veto, or not?”

Bad Samples Small Samples Misleading Graphs Pictographs Distorted Percentages Loaded Questions Order of Questions Refusals Correlation & Causality Self Interest Study Precise Numbers Partial Pictures Deliberate Distortions Misuses of Statistics

Recap Reviewed 13 misuses of statistics. Illustrated how common sense can play a big role in interpreting data and statistics In this section we have:

Created by Tom Wegleitner, Centreville, Virginia Section 1-4 Design of Experiments

Major Points If sample data are not collected in an appropriate way, the data may be so completely useless that no amount of statistical tutoring can salvage them. Randomness typically plays a critical role in determining which data to collect.

Observational Study observing and measuring specific characteristics without attempting to modify the subjects being studied Definitions

Experiment apply some treatment and then observe its effects on the subjects Definitions

Cross Sectional Study Data are observed, measured, and collected at one point in time. Retrospective (or Case Control) Study Data are collected from the past by going back in time. Prospective (or Longitudinal or Cohort) Study Data are collected in the future from groups (called cohorts ) sharing common factors. Definitions

Confounding occurs in an experiment when the experimenter is not able to distinguish between the effects of different factors Try to plan the experiment so confounding does not occur! Definitions

Controlling Effects of Variables Blinding subject does not know he or she is receiving a treatment or placebo Blocks groups of subjects with similar characteristics Completely Randomized Experimental Design subjects are put into blocks through a process of random selection Rigorously Controlled Design subjects are very carefully chosen

Replication repetition of an experiment when there are enough subjects to recognize the differences in different treatments Replication and Sample Size Sample Size use a sample size that is large enough to see the true nature of any effects and obtain that sample using an appropriate method, such as one based on randomness

Random Sample members of the population are selected in such a way that each individual member has an equal chance of being selected Definitions Simple Random Sample (of size n ) subjects selected in such a way that every possible sample of the same size n has the same chance of being chosen

Random Sampling selection so that each has an equal chance of being selected

Systematic Sampling Select some starting point and then select every K th element in the population

Convenience Sampling use results that are easy to get

Stratified Sampling subdivide the population into at least two different subgroups that share the same characteristics, then draw a sample from each subgroup (or stratum)

Cluster Sampling divide the population into sections (or clusters); randomly select some of those clusters; choose all members from selected clusters

Random Systematic Convenience Stratified Cluster Methods of Sampling

Sampling Error the difference between a sample result and the true population result; such an error results from chance sample fluctuations Nonsampling Error sample data that are incorrectly collected, recorded, or analyzed (such as by selecting a biased sample, using a defective instrument, or copying the data incorrectly) Definitions

Recap In this section we have looked at: Types of studies and experiments Controlling the effects of variables Randomization Types of sampling Sampling Errors

Introduction To Statistics

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Introduction To Statistics