Chapter 3 Statistics for Describing, Exploring, and Comparing Data

Transcription

1 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Overview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing 3-5 Exploratory Data Analysis (EDA) 1 Slide 1 Section 3-1 Overview Created by Tom Wegleitner, Centreville, Virginia Slide 2

2 Overview Descriptive Statistics summarize or describe the important characteristics of a known set of data Inferential Statistics use sample data to make inferences (or generalizations) about a population 2 Slide 3 Section 3-2 Measures of Center Created by Tom Wegleitner, Centreville, Virginia Slide 4

3 Key Concept When describing, exploring, and comparing data sets, these characteristics are usually extremely important: Center Variation Distribution Outliers Changes over time 3 Slide 5 Measures of Center: the value at the center or middle of the data. Arithmetic Mean Median Mode Midrange GENERAL ROUNDING RULE: Always round to one more place than the data when the final answer is computed. Slide 6

4 Notation N: population size n: sample size µ: population mean x: sample mean Xi : a value of the variable. : the sum of a set of values. 4 Slide 7 Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values Mean of a population Mean of a sample Slide 8

5 Example Mean I : 5, 10, 15, 20, 25, 22 x = Σ x n 5 Slide 9 Example Mean II : 5.0, 10.1, 15.3, 20.2, 57.3 x = Σ x n Slide 10

6 Example Monitoring Lead in Air Listed below are measured amounts of Lead (in µg/m 3 ) in the air. The air quality standard for lead, established by the EPA, is 1.5 µg/m 3. The measurements shown below were recorded at Building 5 of the World Trade Center site on different days immediately following the terrorist attacks of September 11, Find the mean for this sample of measured levels of lead in the air. 6 Slide 11 Median- It is the middle value in an ordered list of data. It is the value with the same number of data values above and below it. Used for data sets with outliers. In absence of outliers, use the mean. It is often denoted by ~ x (pronounced x-tilde ) It is not affected by an outlier. Slide 12

7 Median- to calculate the median, place the data in increasing order and find a value in the center of the ordered list. When the sample size is even the median is the value in the middle. When the sample size is odd the median is the mean of the two values in the middle. 7 Slide 13 Median Example 1 Example Data: Ordered Data: Median: Median Example 2 Example Data: Ordered Data: Median: Slide 14

8 Mode For ungrouped data, the mode is the value that occurs Most often in a data set. For grouped data, the mode is the value the class with the highest frequency and is called the modal class. Bi-modal the two classes with the largest frequency OR Two data values each with the largest frequency. Multi modal more than two data values or more than two classes with the same frequency. No symbol for mode. The only measure of center that can be used with nominal data. 8 Slide 15 Mode - Examples a b c Slide 16

9 Midrange (MR) The value midway between the maximum and minimum values in the original data set. ( The value in the middle of the range.) Midrange = 9 Slide 17 Find the midrange value for the sample of lead in air Midrange = maximum value + minimum value 2 Slide 18

10 Table 3-1 Comparison of Ages of Best Actresses and Best Actors. Best Actresses Best Actors Mean Median Mode and 42 Midrange Slide 19 Critical Thinking: Example 1 Identify a major reason why the mean and median are not statistics that accurately measure the center. 1) Zip codes: 12601, 90210, 02116, 76117, ) Ranks of stress level from different jobs: 2, 3, 1, 7, 9 3) Surveyed respondents are coded as 1 (for Democrat), 2 (for Republican), 3 (for Libertarian), 4 (for independent) Slide 20

11 Critical Thinking: Example 2 Mean Salary of Teachers For each of the 50 states, a researcher obtains the mean salary of secondary school teachers: $37,200 $49,400 $40, $37800 The mean of the 50 amounts is $42,210. Is this the mean salary of all secondary school teachers in the US? Why or why not? 11 Slide 21 The mean of a population is not necessarily the mean of the means from different subsets. Slide 22