Variables. Lecture 13 Sections Wed, Sep 16, Hampden-Sydney College. Displaying Distributions - Quantitative.
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1 - - Lecture 13 Sections Hampden-Sydney College Wed, Sep 16, 2009
2 Outline Even-numbered
3 - Exercise 4.7, p According to the National Center for Health Statistics, in the year 1940, the percentage of people age 65 who were expected to survive to the age of 90 was 7%. For the year 1960, that percentage was 14%. For the year 1980, that percentage was 25%. Based on projections, the percentages for the years 2000 and 2050 are 26% and 42%, respective. Present a bar graph to display the percentage of people age 65 who are expected to live to the age of 90 for the years 1940, 1960, 1980, and projections for the years 2000 and 2050.
4 - Solution The bar graph. Percentage 50% 40% 30% 20% 10% Year
5 - Solution As a time-series plot. Percentage 50% 40% 30% 20% 10% Year
6 - Solution An improved time-series plot. Percentage 50% 40% 30% 20% 10% Year
7 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
8 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
9 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
10 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
11 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
12 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
13 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
14 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
15 - Now we wish to create displays of quantitative data. With quantitative data, what is it that we wish to convey through our display? Some possibilities: Are the data clustered around the middle? Or are they spread out? Where is the middle? Are they any outliers? What are the extremes? Are the values shifted to the left or right?
16 - To obtain such information, we need a graph that plots the value of the variable on one axis and its frequency on the other axis.
17 Example - The following data represent rainfall amounts at the Richmond airport in September for the years (30 years) How might we display these data graphically?
18 - Definition ( plot) A frequency plot is a display of quantitative data in which each data point is represented by an X over that value on a horizontal scale.
19 Drawing - Draw a horizontal line. Choose a resolution, e.g., 0.1. Keeping in mind the minimum and maximum values, mark reference points on the scale, as on a ruler. Mark at regular intervals. For each data value, draw an X over that value on the scale.
20 Example - Make a frequency plot of the rainfall data
21 What information is conveyed by this frequency plot?
22 - Definition (Symmetric) The distribution is symmetric if the left side is a mirror image of the right side. Definition (Unimodal) The distribution is unimodal if it has a single peak, showing the most common values. Definition (Bimodal) The distribution is bimodal if it has two peaks.
23 - Definition (Uniform) The distribution is uniform if all values have equal frequency. Definition (Skewed) The distribution is skewed if it is stretched out more on one side than the other.
24 What properties does this distribution have?
25 What properties does this distribution have?
26 - Definition (leaf display) A stem-and-leaf display is a display of quantitative data in which the numerical representation of each data value is split into a stem and a leaf. The stem is the part to the left of the division point. The leaf is the first digit to the right of the division point.
27 - For example, the value 1.23 could be split as 1 23 stem = 1, leaf = 2, or 12 3 stem = 12, leaf = 3. Indeed, it could even be split as 123 stem = 0, leaf = 1, or 123 stem = 123, leaf = 0.
28 - The stem consists of the digits to the left of the division point. The leaf consists of the first digit to the right of the division point. A note should be added indicating how to interpret the numbers. Note: 12 3 means 1.23.
29 Example - Draw a stem-and-leaf display of the rainfall data
30 Splitting the Numbers - We choose where to split the numbers in order to avoid Too many stems, each with too few leaves. Too few stems, each with too many leaves.
31 Splitting the Numbers - We choose where to split the numbers in order to avoid Too many stems, each with too few leaves. Too few stems, each with too many leaves
32 Example - Draw a stem-and-leaf display of the rainfall data, in centimeters
33 Example - We may split the values after the 10 s digit: Stem Note: 1 2 means
34 Example - Or we may split the values at the decimal point: Stem Note: 12 3 means
35 Example - Which is better? Is either one particularly good?
36 - We can obtain a good compromise (in this example) by splitting the stems. Each stems appears twice. The first time for leaves 0-4. The second time for leaves 5-9.
37 - Display with split stems: Stem Note: 1 2 means
38 - Read Section , pages , Let s Do It! 4.10, Page 241, exercise 18. Page 248, exercises 22-27, 29.
39 Even-numbered - Page 241, Exercise (a) (b) The distribution is symmetric and unimodal. The middle is near 7. There do not appear to be any outliers.
40 Even-numbered - Page 248, 22, 24, Symmetric, unimodal:] Characteristic No. of heads out of 12 tosses of a fair coin. Characteristic Amount of money a family spends on college Characteristic Housing prices Characteristic No. of years a person lives Characteristic Length of time sitting at a particular red light 4.24 (a) Stem
41 Even-numbered - Page 248, 22, 24, (b) Unimodal, symmetric with one outlier. (c) Connecticut. (d) Stem They are both unimodal and symmetric, and with similar middles has more large outliers; has more small outliers.
42 Even-numbered - Page 248, 22, 24, (a) Calories (my choice). Stem (b) Hard to say, because there is not much data. I would probably say it is uniform because I see no clear shape. Maybe it is skewed right. (c) Best: Orange Roughy; worst: Salmon.
43 Even-numbered - Page 248, 22, 24, (d) Fat: Stem Unimodal and skewed right. (e) No. Yes.
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