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3.3 Displaying Numerical Data with Frequency Distributions and Histograms and Describing Data A. Histograms for Discrete Numerical Data When to Use - Discrete numerical data. Works well even for large data sets How to Construct 1. Draw a horizontal scale, and the possible values of the variable 2. Draw a vertical scale, and mark it with either frequency or relative frequency 3. Above each possible value, draw a rectangle centered at that value. The height of each rectangle is determined by the corresponding frequency or relative frequency. What to Look For CENTER SHAPE SPREAD DEVIATIONS Example Ungrouped Discrete Data Promiscuous Raccoons 1 3 2 1 1 4 2 4 1 1 1 3 1 1 1 1 2 2 1 1 4 1 1 2 1 1 1 1 3 Frequency Distribution DISCRETE DATA: Frequency Histogram Relative Frequency Histogram Fig. 3-16, p. 99 Example Grouped Discrete Data Relative Freq Histogram B. Histograms for Continuous Numerical Data Histogram for Continuous Data When the Class Interval Widths are Equal When to Use - Continuous numerical Data. Works well even for large data sets How to Construct 1. Mark the boundaries of the class intervals on the horizontal axis 2. Use either frequency or relative frequency on the vertical axis 3. Draw a rectangle for each class directly above the corresponding interval. The height of each rectangle is the frequency or relative frequency of the corresponding class interval What to Look For CENTER SPREAD SHAPE DEVIATIONS Enrollment at Public Universities - % of College Students Enrolled in Public Institutions 86 96 66 86 80 78 62 81 77 81 77 76 73 69 76 90 78 82 70 83 46 80 78 93 66 91 76 86 58 81 91 57 81 88 71 86 84 57 44 82 79 67 86 75 55 75 80 80 85 69 The difficulty in constructing tabular or graphical displays with continuous data is that there are no natural categories. Also we need to figure out were to place values that fall on a boundary between classes 1) 2) The quantity intervals number of observations is often used as an estimate of an appropriate number of Example - Bicycle Accidents Ogive – Cumulative Relative Frequency Graph C. Class Intervals of Unequal Widths Constructing a Histogram for Continuous Data When Class Interval Widths are Unequal In this case, frequencies or relative frequencies should not be used on the vertical axis. Instead, the height of each rectangle, called the density for the class, is given by density = rectangle height = relative frequency of class interval class interval width The vertical axis is called the density scale. The use of the density scale to construct the histogram ensures that the area of each rectangle is the histogram will be proportional to the corresponding relative frequency. U.S. Census data for San Luis Obispo County, CA, were used to construct the following frequency distribution for commute time (in minutes) of working adults. Commute Time Width 0 to < 5 Freq 5200 Relative Freq .0518 5 to < 10 18200 .1813 10 to < 15 19600 .1952 15 to < 20 15400 .1534 20 to < 25 13800 .1375 25 to < 30 5700 .0568 30 to < 35 10200 .1016 35 to < 40 2000 .0199 40 to < 45 2000 .0199 45 to < 60 4000 .0398 60 to < 90 2100 .0209 90 to < 120 2200 .0219 Density x D. Do Sample Histograms Resemble Population Histograms?