Statistics Branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. Practice or science of collecting and analyzing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample. Statistics is a discipline which is concerned with: designing experiments and other data collection, summarizing information to aid understanding, drawing conclusions from data, and estimating the present or predicting the future. Dave D.: At heart, the study of ways to model uncertainty in data quantitatively and systematically, to help us infer characteristics of populations of things of interest to us in the natural world and attach degrees of likelihood to those inferences.

Population vs. Sample Population (four definitions from four sources on the Web) Subjects of a particular study—everything or everyone who is the subject of a statistical observation. Members of a population share some feature in common Any complete group with at least one characteristic in common. Needs to be clearly identified at the beginning of the study. Total set of observations that can be made. All members of a defined group that we are studying or collecting information about for data-driven decisions. Sample A part or subset of a population.

Why Sample? Can’t always observe characteristic(s) of entire population, either because it’s not practical (affordable), or because it’s theoretically impossible (e.g., population has infinite size) Problems with Samples (Errors) Bias Type of error in which measurements are systematically shifted in one direction from the actual values. Math (statistical methods) no help here—need better measurements or sampling method Sampling error Type of error that is random (measurements are just as likely to be higher than the true value as lower, by the same amount on average) Math can be a big help here

Distributions Listing or function showing all the possible values (or intervals) of the data and how often they occur [in a population or sample]. Frequency distribution Summary (table, or a graph) that shows how often different values (or range of values) of a characteristic of a population (or sample) occur in the population (or sample). Probability distribution Summary (table or graph) that shows how likely it is (i.e., the probability) that you’ll observe a particular value (or range of values) of a characteristic of a population (or sample) if you pick a member of the population at random.

Characteristics (Descriptors) of a Distribution (These are also characteristics or descriptors of the population or sample) Characteristic/Descriptor ”Moments” 1. Location (central tendency) 1 2. Dispersion (width, spread, variability) 2 3. Skew(ness) (asymmetry; left- or right-leaning) 3 4. Kurtosis (“tailedness”; sharpness of the peak) 4

Parameters vs. Statistics (Quantitative) measures of characteristics of a population Mathematical convention: Greek letters Statistics (Quantitative) measures of characteristics of a sample Use it to estimate the corresponding population parameter Mathematical convention: Roman letters

Good Characteristics of a Statistic Unbiased Equally likely to be higher or lower than the parameter we are trying to estimate Efficient Converges rapidily toward the population parameter as we increase the sample size (so don’t need a large sample to get a good estimate) Consistent Converges smoothly toward the population parameter as we increase the sample size (no abrupt jumps in the quality of the estimate as sample size increases) Robust (resistant) Not distorted by outliers (exceptionally large or small, possibly but not always spurious, observations)

Etc.