Mathematical Statistics

Transcription

1 Mathematical Statistics 1

2 Statistics: converting data into useful information (1) Statistics is an applied field with a wide range of practical applications. (2) You don t have to be a math guru to learn from real, interesting data. (3) Data is messy, and statistical tools are imperfect. But, when you understand the strengths and weaknesses of these tools, you can use them to learn about the real world. Quote from OpenIntro Statistics 2 nd edition

3 The Big Picture

4 Exploratory Data Analysis Summarize and describe, using graphs and numerical measures, the distribution of categorical and quantitative variables in context. Use these graphs and measures to compare one distribution with another.

5 Lab 1: Maternal Smoking and Infant Health Is there a significant difference in weight between babies born to mothers who smoked during pregnancy and mothers who did not?

6 Lab 1: Maternal Smoking and Infant Health By Friday: Read Introduction, Data, and Background (p 1-8) Come to class ready to work. We will do the Investigations together Today: Some definitions and theory

7 Raw Data Raw data are for numbers and category labels that have been collected but have not yet been processed in any way. Example list of questions and raw data for a student:

8 Raw Data Getting the data for Lab 1: birthwt <- read.csv(" statlabs/data/babiesi.data", header=true, sep="") smk <- birthwt$bwt[birthwt$smoke==1] nosmk <- birthwt$bwt[birthwt$smoke==0]

9 Raw Data An observation is an individual entity in a study. A variable is a characteristic that may differ among individuals. Sample data are collected from a subset of a larger population.

10 Types of Variables Raw data from categorical variables consist of group or category names that don t necessarily have a logical ordering. Examples: eye color, country of residence. Categorical variables for which the categories have a logical ordering are called ordinal variables. Examples: highest educational degree earned, tee shirt size (S, M, L, XL). Raw data from quantitative variables consist of numerical values taken on each individual. Examples: height, number of siblings.

11 Asking the Right Questions One Categorical Variable Example: What percentage of college students favor the legalization of marijuana, and what percentage of college students oppose legalization of marijuana? Ask: How many and what percentage of individuals fall into each category?

12 Asking the Right Questions One Quantitative Variable Example: What is the average body temperature for adults, and how much variability is there in body temperature measurements? Ask: What are the interesting summary measures, like the average or the range of values?

13 Numerical Summaries for Categorical Variables Frequency and Relative Frequency A frequency distribution for a categorical variable is a listing of all categories along with their frequencies (counts). A relative frequency distribution is a listing of all categories along with their relative frequencies (given as proportions or percentages, for example).

14 Visual Summaries for Categorical Variables Pie Charts: OK for summarizing a single categorical variable if not too many categories. Bar Graphs: useful for summarizing one or two categorical variables and particularly useful for making comparisons when there are two categorical variables.

15 Example Humans Are Not Good Randomizers Survey of n = 190 college students. Randomly pick a number between 1 and 10. Hard to compare frequencies Results: Most chose 7, very few chose 1 or 10.

16 Finding Information in Quantitative Data Long list of numbers needs to be organized to obtain answers to questions of interest.

17 Pictures for Quantitative Data Histograms: similar to bar graphs, used for any number of data values. Stem-and-leaf plots and dotplots: present all individual values, useful for small to moderate sized data sets. Boxplot or box-and-whisker plot: useful summary for comparing two or more groups.

18 Interpreting Histograms and Stemplots Values are centered around 20 cm. Two possible low outliers. Apart from outliers, spans range from about 16 to 23 cm.

19 Creating a Histogram Step 1: Decide how many equally spaced (same width) intervals to use for the horizontal axis. Step 2: Decide to use frequencies (count) or relative frequencies (proportion) on the vertical axis. Step 3: Draw equally spaced intervals on horizontal axis covering entire range of the data. Determine frequency or relative frequency of data values in each interval and draw a bar with corresponding height. Decide rule to use for values that fall on the border between two intervals.

20 Example Ages of Death of First Ladies Two different histograms

21 Describing Shape Symmetric, bell-shaped Symmetric, not bell-shaped Skewed Right: values trail off to right Skewed Left: values trail off to left

22 Numerical Summaries of Quantitative Data Notation for Raw Data: n = number of individuals in a data set x 1, x 2, x,, x 3 n represent individual raw data values Example: A data set consists of handspan values in centimeters for six females; the values are 21, 19, 20, 20, 22, and 19. Then, n = 6 x 1 = 21, x 2 = 19, x 3 = 20, x 4 = 20, x 5 = 22, and x 6 = 19

23 Describing the Location of a Data Set Mean: the numerical average Median: the middle value (if n odd) or the average of the middle two values (n even) Symmetric: mean = median Skewed Left: mean < median Skewed Right: mean > median 23

24 Determining the Mean and Median The Mean where xi The Median xi x = n means add together all the values If n is odd: M = middle of ordered values. Count (n + 1)/2 down from top of ordered list. If n is even: M = average of middle two ordered values. Average values that are (n/2) and (n/2) + 1 down from top of ordered list.

25 Example Will Normal Rainfall Get Rid of Those Odors? Data: Average rainfall (inches) for Davis, California for 47 years Mean = inches Median = inches In , a company with odor problem blamed it on excessive rain. That year rainfall was inches. More rain occurred in 4 other years.

26 The Influence of Outliers on the Mean and Median Larger influence on mean than median. High outliers will increase the mean. Low outliers will decrease the mean. If ages at death are: 76, 78, 80, 82, and 84 then mean = median = 80 years. If ages at death are: 46, 78, 80, 82, and 84 then median = 80 but mean = 74 years.

27 Explanatory and Response Variables Many questions about the relationship between two variables. It is useful to identify one variable as the explanatory variable and the other variable as the response variable. In general, the value of the explanatory variable for an individual is suspected to partially explain the value of the response variable for that individual.

28 Our research question is about the relationship between a categorical variable (mother's smoking status -- smoke ) and quantitative variable (newborn's birth weight -- bwt ). C --> Q Explanatory variable: smoke (C) Response variable: bwt (Q) The variable smoke has two categories: yes and no. To answer the research question, we should compare the distribution of smoke within the yes category with the distribution of smoke within the no category.

29 Boxplots: Picturing Location and Spread for Group Comparisons Box covers the middle 50% of the data Line within box marks the median value Possible outliers are marked with asterisk Apart from outliers, lines extending from box reach to min and max values.

30 Box plots are OK for a general sense of how they compare. Another way to compare is to use quantile-quantile plots. (1) Is each distribution roughly normal? (2) Are the two distributions roughly the same as each other?