2. Chapter 1 Overview and Descriptive Statistics 1111.1 - Populations, Samples and Processes 1111.2 - Pictorial and Tabular Methods in Descriptive Statistics 1111.3 - Measures of Location 1111.4 - Measures of Variability Note that these are textbook chapters, although Lecture Notes may be referenced.

5. Examples: Toasting time, Temperature settings, etc. of a population of toasters… 4 What is “random variation” in the distribution of a population? POPULATION 1: Little to no variation In engineering situations such as this, we try to maintain “quality control”… i.e., “tight tolerance levels,” high precision, low variability. But what about a population of, say, people? (e.g., product manufacturing)

6. Density POPULATION 1: Little to no variation 5 Most individual values ≈ population mean value Example: Body Temperature (  F) Very little variation about the mean! 98.6  F (e.g., clones) What is “random variation” in the distribution of a population?

7. Example: Body Temperature (  F) Examples: Gender, Race, Age, Height, Annual Income,… POPULATION 2: Much variation (more common) Density 6 Much more variation about the mean! What is “random variation” in the distribution of a population?

8. Click on image for full.pdf article Links in article to access datasets Example

9. How is this accomplished? Hospital records, etc. “sampling frame” Women in U.S. who have given birth POPULATION “Random Variable” X = Age at first birth mean μ = ??? That is, the Population Distribution of X ~ N( , ).). Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. Study Question: How can we estimate “mean age at first birth” of women in the U.S.? {x 1, x 2, x 3, x 4, …, x 400 }  and  are “population characteristics” i.e., “parameters” (fixed, unknown) standard deviation σ

10. Population Distribution That is, the Population Distribution of X ~ N( ,  ). Women in U.S. who have given birth POPULATION “Random Variable” X = Age at first birth mean {x 1, x 2, x 3, x 4, …, x 400 } FORMUL A Study Question: How can we estimate “mean age at first birth” of women in the U.S.? Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. mean μ = ??? is an example of a “sample characteristic” = “statistic.” (numerical info culled from a sample) This is called a “point estimate“ of  from the one sample. Can it be improved, and if so, how? Choose a bigger sample, which should reduce “variability.” Average the sample means of many samples, not just one. (introduces “sampling variability”) “Sampling Distribution” ~ ??? How big??? ????????? ???  and  are “population characteristics” i.e., “parameters” (fixed, unknown) Other possible parameters: standard deviation median minimum maximum standard deviation σ

11. mean Without knowing every value in the population, it is not possible to determine the exact value of  with 100% “certainty.”

12. Population Distribution That is, the Population Distribution of X ~ N( ,  ). Women in U.S. who have given birth POPULATION “Random Variable” X = Age at first birth mean {x 1, x 2, x 3, x 4, …, x 400 } FORMUL A Study Question: How can we estimate “mean age at first birth” of women in the U.S.? Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. mean μ = ??? For concreteness, suppose  = 1.5  and  are “population characteristics” i.e., “parameters” (fixed, unknown) standard deviation σ

13. 95% CONFIDENCE INTERVAL FOR µ 25.74725.453 BASED ON OUR SAMPLE DATA, the true value of μ is between 25.453 and 25.747, with 95% “confidence” (…akin to “probability”). Without knowing every value in the population, it is not possible to determine the exact value of  with 100% “certainty.” This is called an “interval estimate“ of  from the sample. μ Used in “Statistical Inference” via “Hypothesis Testing”… mean

14. Women in U.S. who have given birth POPULATION “Random Variable” X = Age at first birth mean {x 1, x 2, x 3, x 4, …, x n } FORMUL A Study Question: How can we estimate “mean age at first birth” of women in the U.S.? Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. mean μ = ??? Population Distribution That is, the Population Distribution of X ~ N( ,  ).  and  are “population characteristics” i.e., “parameters” (fixed, unknown) standard deviation σ Arithmetic Mean Arithmetic Mean Geometric Mean Geometric Mean Harmonic Mean Harmonic Mean Each of these gives an estimate of  for a particular sample. Any general sample estimator for  is denoted by the symbol Likewise for and

15. Women in U.S. who have given birth POPULATION “Random Variable” X = Age at first birth mean {x 1, x 2, x 3, x 4, …, x n } FORMUL A Study Question: How can we estimate “mean age at first birth” of women in the U.S.? Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. mean μ = ??? Population Distribution That is, the Population Distribution of X ~ N( ,  ). “PARAMETER ESTIMATION”  and  are “population characteristics” i.e., “parameters” (fixed, unknown) standard deviation σ Extending these ideas to other parameters of a population gives rise to the general theory of…

16.  and  are “population characteristics” i.e., “parameters” (fixed, unknown) standard deviation σ Population Distribution That is, the Population Distribution of X ~ N( ,  ). How is… “Random Variable” X (age, income level, …) … distributed? Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. mean μ = ??? composed of “units” (people, rocks, toasters,...) What do we want to know about this population? Suppose we know that X follows a known “probability distribution” in the population… but with parameters unknown vals. Population Distribution That is, the Population Distribution of X ~ Dist(  1,  2,…). Ideal properties… Unbiased estimator of  Unbiased estimator of  Minimum Variance among all such unbiased estimators Minimum Variance among all such unbiased estimators i.e., “MVUE” heavily skewed tail SAMPLE For a particular , want to define a corresponding “parameter estimator”  To make certain calculations simpler, we assume that populations are “arbitrarily large” (or indeed, infinite). POPULATION

17. 1010½11 Quantitative [measurement]  length  mass  temperature  pulse rate  # puppies  shoe size 16 “Random Variable” X = any numerical value that can be assigned to each unit of a population “Random” refers to the notion that this value is unknown until actually observed (usually as part of an outcome of an experiment to test a specific hypothesis). Contrast this with the idea of a “nonrandom” variable with no empirical error, e.g., X = # cards in a deck = 52. There are two general types......... Quantitative and Qualitative How is… “Random Variable” X (age, income level, …) … distributed? What do we want to know about this population? composed of “units” (people, rocks, toasters,...)  To make certain calculations simpler, we assume that populations are “arbitrarily large” (or indeed, infinite). POPULATION

18. Quantitative [measurement ]  length  mass  temperature  pulse rate  # puppies  shoe size 17 “Random Variable” X = any numerical value that can be assigned to each unit of a population “Random” refers to the notion that this value is unknown until actually observed (usually as part of an outcome of an experiment to test a specific hypothesis). Contrast this with the idea of a “nonrandom” variable with no empirical error, e.g., X = # cards in a deck = 52. There are two general types......... Quantitative and Qualitative How is… “Random Variable” X (age, income level, …) … distributed? What do we want to know about this population? composed of “units” (people, rocks, toasters,...)  To make certain calculations simpler, we assume that populations are “arbitrarily large” (or indeed, infinite). POPULATION CONTINUOUS (can take their values at any point in a continuous interval) DISCRETE (only take their values in disconnected jumps)

19. Qualitative [categorical]  video game levels (1, 2, 3,...)  income level (low, mid, high)  zip code  PIN #  color (Red, Green, Blue) ORDINAL, RANKED 18 “Random Variable” X = any numerical value that can be assigned to each unit of a population “Random” refers to the notion that this value is unknown until actually observed (usually as part of an outcome of an experiment to test a specific hypothesis). Contrast this with the idea of a “nonrandom” variable with no empirical error, e.g., X = # cards in a deck = 52. There are two general types......... Quantitative and Qualitative How is… “Random Variable” X (age, income level, …) … distributed? What do we want to know about this population? composed of “units” (people, rocks, toasters,...)  To make certain calculations simpler, we assume that populations are “arbitrarily large” (or indeed, infinite). POPULATION IMPORTANT SPECIAL CASE: Binary (or Dichotomous) “Pregnant?” (Yes / No) Coin toss (Heads / Tails) Treatment (Drug / Placebo) 1 2 3 NOMINAL 1 2 3 (ordered labels) (unordered labels)

20. Random Variable Define a new parameter  = P(Success) Point estimator Suppose we intend to select a random sample of size n from this population of Success and Failures… … in such a way that the “Success or Failure” outcome of any selected individual conveys no information about the “Success or Failure” outcome of any other selected individual. That is, the “Success or Failure” outcomes between any two individuals are independent. (Think of tossing a coin n times.) POPULATION Then a natural estimator for  could be (0, 1, 2, …, n) Let X = “Number of Successes in the sample.” the sample proportion of Success Ex: Ex: n = 500 tosses, X= 285 Heads 