Statistical Graphs and Measures of Center in Sierra College Math 13 - Prof. John Burke, Study notes of Statistics

Download Statistical Graphs and Measures of Center in Sierra College Math 13 - Prof. John Burke and more Study notes Statistics in PDF only on Docsity! 1 Sierra College – Math 13 Spring 2009 – Class 5/32 Today: Sections 2-4; 3-1/3-2 Assignment: 2-4 {1, 5, 7, 9, 11, 13, 15} 3-2 {1, 3, 5, 7, 9, 11, 13, 15, 17} Next: Sections 3-3/3-4 Instructor: John Burke E-mail: john_burke@mindspring.com Web Page: http://math.sierracollege.edu/Staff/JohnBurke/ Telephone: 916 337-0425 Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official) 2 2-4 Statistical Graphics The main objective in using graphical representations of data is to better understand the data through: – Description (center, shape, etc.), – Exploring (relative strengths), and – Comparing (data from different populations) 3 Frequency Polygon A frequency polygon uses line segments connected to points located directly above class midpoint values. The heights of the points correspond to the class frequencies, and the line segments are extended to the right and left so that the graph begins and ends on the horizontal axis. 13 10 7 4 1 112 – 14 29 – 11 156 – 8 143 – 5 200 – 2 FrequencyRating Class midpoints 4 Ogive An ogive is a line graph that depicts the cumulative frequencies, just as the cumulative frequency distribution lists cumulative frequencies. 52112 – 14 14.5 11.5 8.5 5.5 2.5 -0.5 2 15 14 20 Frequency 51 49 34 20 Cumulative Frequency 9 – 11 6 – 8 3 – 5 0 – 2 Rating 5 Dotplot A dotplot consists of a graph in which each data value is plotted as a point (or dot) along a scale of values. Dots representing equal values are stacked. 6 Stem-and-Leaf Plot A stem-and-leaf plot represents data by separating each value into two parts: the stem (such as the leftmost digit(s)) and the leaf (such as the rightmost digit). The stem-and-leaf plot shows the distribution, but maintains all the information in the original list. Raw Data (Test Grades) 67 72 85 75 89 89 88 90 99 100 10 9 8 7 6 Stem 0 0 9 5 8 9 9 2 5 7 Leaves 13 3-2 Measures of Center Definition: A measure of center is a value at the center, or middle, of a data set. We will consider the following measures of center ( the 4 M’s): • Mean • Median • Mode • Midrange 14 Mean The arithmetic mean (or just mean or average) of a set of values is the measure of center found by adding the values and dividing by the total number of values. xmean x n ∑ = = Note: The mean is particularly sensitive to outliers. Called “x-bar” 15 Mean - Example 6.726.443.603.46 26.76.726.443.603.46 Mean = Σx/4 = 5.055 Mean = Σx/5 = 9.384 Notice the effect of the outlier. 16 Round-Off Rule In general, carry one more decimal place than is present in the original set of values. Round only the final answer, not intermediate values! For example, the mean of 2, 3 and 5 is 3.3333333, which we round to 3.3. 17 Mean - Example 71 36221425251 8327271626 24528510227 10866577504 2433621522 41 0002103010 3104051402 0530415024 5313304304 0200001302 Table 2-1: Qwerty Keyboard Word Ratings Table 2-2: Dvorak Keyboard Word Ratings Mean = 4.4 Mean = 1.7 18 Median The median of a data set is the measure of center that is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. The median is sometimes denoted as (pronounced “x-tilde”).x To find the median, first sort the values. Note: The median is largely unaffected by outliers. If the number of values is odd, the median is the number located in the exact middle of the list. If the number of values is even, the median is found by computing the mean of the two middle numbers. 19 Median - Example 6.726.443.603.46 26.76.726.443.603.46 No exact middle – even number of values Median = (3.60 + 6.44)/2 = 5.020 Mean = Σx/4 = 5.055 Exact middle – median = 6.440 Mean = Σx/5 = 9.384 Notice the effect of the outlier. 20 Mode The mode of a set of data, often denoted M, is the value that occurs most frequently. When two values occur with the same greatest frequency, each is a mode and the data set is bimodal. When more than two values occur with the same greatest frequency, the data set is said to be multimodal. When no value is repeated, there is no mode. Mode is the only measure of central tendency that can be used with nominal data. 21 Mode - Example 5, 5, 5, 3, 1, 5, 1, 4, 3, 5 Mode is 5 1, 2 ,2, 2, 3, 4, 5, 6, 6, 6, 7, 9 Bimodal – 2 and 6 1, 2, 3, 6, 7, 8, 9 No mode