Chapter 5: Statistical Reasoning Section 5.1 Exploring Data Measures of central tendency (Mean, Median and Mode) attempt to describe a set of data by identifying the central position within a set of data or typical value. Measures of central tendency are summary statistics. Measures of central tendency are not always sufficient to represent or compare sets of data. You can draw inferences from numerical data by examining how the data is distributed around the mean or the median When analyzing two sets of data, it is important to look at both similarities and differences in the data. Section 5.1: Exploring Data Range: the difference between the maximum value and the minimum value in a data set. Ex. 25, 24, 26, 37 Median: a measure of central tendency represented by the middle value of an ordered data set. Ex. 1) 3, 3, 4, 5, 6 2) 3, 3, 5, 6 Outlier: a value in a data set that is very different from other values in the set Ex. 12, 26, 27, 29 Mean: a measure of central tendency determined by adding the values together and dividing by the number of values Ex. 12, 13, 12, 14, 16 STEPS: Put numbers in numerical order (smallest to largest) middle number is median if even number of data points, find the middle pair of numbers and find the half way value. Mode: a measure of central tendency represented by the value that occurs the most in a set of data Ex. 2, 3, 4, 3, 2, 3, 5
Example 1 Determine the range, median, mode, mean, and outliers from the following set of data: 3, 7, 8, 4, 5, 7, 6, 5, 11, 3, 2 The class performed better on Test # l because the range is smaller, with the mean, median and mode all being equal.
Line plots easy to see mode and range Example 2 Create a line plot with the following data: 2, 3, 4, 8, 6, 11, 3, 3, 5 5 above 5 below https://www.eduplace.com/math/mw/background/5/06a/te_5_06a_overview.html Section 5.2: Frequency Tables, Histograms, & Frequency Polygons Frequency Distribution a set of intervals (table or graph), usually of equal width, into which raw data is organized Histogram: the graph of a frequency distribution, in which equal intervals of values are marked on a horizontal axis and the frequencies associated with these intervals are indicated by the areas of the rectangles drawn for these intervals Ex. 1, 21, 13, 8, 9, 10, 15, 24, 16, 14, 7 Interval Frequency 1 10 5 11 20 4 21 30 2
Frequency Polygon: the graph of a frequency distribution, produced by joining the midpoints of the intervals using straight lines Example 1: Creating a Frequency Polygon The test scores for a math class are shown below: 68 77 91 66 52 58 79 94 81 60 73 57 44 58 71 78 80 54 87 43 61 90 41 76 55 75 49 a) Determine the range. b) Determine a reasonable interval size and the number of intervals. c) Produce a frequency table for the grouped data. d) Produce a histogram and a frequency polygon for the grouped data.
YOUR TURN The speeds of 24 motorists ticketed for exceeding a 60 km/h limit are listed below: 75 72 66 80 75 70 71 82 69 70 72 78 90 75 76 80 75 96 81 77 76 84 74 79 b) Construct a histogram and a frequency polygon for the grouped data. a) Construct a frequency table for the grouped data. c) How many motorists exceeded the speed limit by 15 km/h or less? Example 2 Create a line plot with the following data: 2, 3, 4, 8, 6, 11, 3, 3, 5 d) How many motorists exceeded the speed limit by over 20 km/h?
Section 5.3: Standard Deviation The measures of central tendency (mean, median, mode) are great tools to have, but they often do not tell the whole story. It is important to consider the spread of data. This will help you to gauge the consistency. This is where standard deviation comes in. First, some symbols to be aware of: x = individual data Ʃ = sum μ= mean σ = standard deviation N = # of data points Example Find the deviation of each number in the following data set: 10, 8, 9, 12 1. Find the mean 2. Set up a table like the one below 3. Subtract the mean from each data point Deviation Number Mean Deviation 10 8 9 13 squared Deviation the difference between a data value and the mean for the same set of data; (x μ) How far away # is data point Mean from the mean Tells how spread out or close data is to mean