Transcript Slide 1
Organizing & Visualizing Data MATH 102 Contemporary Math S. Rook Overview • Section 15.1 in the textbook: – Populations & samples – Frequency tables & relative frequency tables – Histograms – Stem-and-Leaf displays Populations & Samples Populations & Samples • Statistics is the area of mathematics where we collect data (information), analyze the data, and predict outcomes based on the data • Best case is when we can collect data from a population – i.e. ALL individuals or objects that satisfy certain conditions – e.g. Polling a class of 60 students to see who they voted for Class President Populations & Samples (Continued) • Often infeasible to survey a population if it is extremely large so we survey a sample – i.e. A subset of the population – e.g. Polling a sample of 5,000 prospective voters to see who they voted for President out of all 1.2 million prospective voters in a state • An important aim of statistics involves taking data acquired from a sample and using it to make inferences about the larger population – Thus extremely important to obtain a sample that has the same makeup of the population Populations & Samples (Example) Ex 1: Identify whether the situation describes a population or a sample: a) By using the 10 students in the front row, the instructor determined that the average of the exam was 80 for the entire class of 30. b) The full union voted 98 – 2 on the motion to strike. Frequency Tables & Relative Frequency Tables Frequency Tables & Relative Frequency Tables • Collected data is often arranged in a visual manner so as to make it more understandable • A frequency table lists options and the number of objects that satisfy that option • A relative frequency table lists options and the proportion of objects that satisfy that option – Proportion is calculated like probability – The sum of all proportions MUST add to 1 Frequency Tables & Relative Frequency Tables (Example) Ex 2: Construct a frequency table for the data and then extend the table to include relative frequency: a) 10 rolls of a single six-sided die produced the following results: 3, 5, 5, 6, 2, 4, 1, 5, 2, 1 b) A parking lot of 12 cars was observed and the colors of the cars, (B)lue, (b)lack, (S)ilver, and (R)ed were recorded: S, b, B, R, R, R, b, b, S, R, B, B Histograms Histogram • Another visual way to depict a collection of data is through a histogram (i.e. bar graph) – A graphical way to represent a frequency table • Just like a Cartesian Plane a histogram has a horizontal axis and a vertical axis – Options are listed on the horizontal axis – Frequencies are listed on the vertical axis • Sometimes data is too spread out to see a pattern – Group data into classes or bins • The class width is the distance between the beginning of one class and the beginning of the next class Histograms (Example) Ex 3: Construct a histogram to visually represent the heights of athletes in a certain sport. Use classes of width 2 starting at 64.5: a) 69, 71, 74, 74, 78, 68, 68, 71, 75, 69, 76, 68, 69, 74, 65, 73 b) 72, 77, 71, 80, 69, 73, 69, 75, 66, 68, 77, 74, 76, 73, 72, 75 Stem-and-Leaf Displays Stem-and-Leaf Display • Another commonly used visual aid for displaying data is a stem-and-leaf display • View each data point as having a stem (usually first digit) and a leaf (last digit) • To create a stem-and-leaf display: – List stems in the left column and draw a vertical bar – Write leaves with the same stem to the right in ascending order • Data with the same stem must be ordered Stem-and-Leaf Displays (Example) Ex 4: The following data are the ages of students in a class. Represent the data by using a stem-and-leaf display: a) 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33, 42, 18, 22, 39, 21, 26, 18, 43 b) 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21, 29, 22, 24, 33, 47, 23, 22, 18, 33 Summary • After studying these slides, you should know how to do the following: – Differentiate between samples and populations – Visually depict data by constructing a: • Frequency table and/or relative frequency table • Histogram • Stem-and-leaf display • Additional Practice: – See problems in Section 15.1 • Next Lesson: – Measures of Central Tendency (Section 15.2)