HS 67 Lectue Notes Picturing Distributions with Graphs Thursday, November 22, 2018 Chapter 1 Picturing Distributions with Graphs 11/22/2018 11/22/2018 Chapter 1 1 Chapter 1

What is Statistics? Statistics is the science that involves the extraction of information from data. It is a mistake to think of statistics as merely mathematical computations! 11/22/2018 11/22/2018 Chapter 1 2

Data Structure Individuals = observations ≡ individual units (e.g., people, institutions) upon which measurements are made rows in the data table Variables ≡ characteristic that are measured columns in the data table 11/22/2018 11/22/2018 Chapter 1 3

Types of Variables Categorical variables: named (“nominal”) categories HS 67 Lectue Notes Thursday, November 22, 2018 Types of Variables Categorical variables: named (“nominal”) categories Counts (frequencies), percentages Quantitative variables: numerical scales arithmetic operations such as means and standard deviations 11/22/2018 11/22/2018 Chapter 1 4 Chapter 1

Types of Variables Willet et al. (1995). Weight, weight change, and coronary heart disease in women. JAMA, 273(6). Objective: to determine the effect weight gain on coronary heart disease (CHD) risk in women Unit of observation: women between 30- to 55-years of age initially free of CHD. Explanatory variable: body mass index (BMI) at age 18 Follow-up for 14 years (cohort study) Response variable: fatal and nonfatal CHD 11/22/2018 11/22/2018 Chapter 1 5

Types of Variables Willet et al. (1995) Explanatory variable BMI at age 18 Weight / Height2 QUANTITATIVE CHD occurrence Yes or no CATEGORICAL Response variable 11/22/2018 11/22/2018 Chapter 1 6

Distributions Distributions tell us how often a variable takes on various values (!) Picture distributions with graphs Categorical variables: pie charts, bar graphs Quantitative variables: stemplots, histograms, (next chapter: boxplots) 11/22/2018 11/22/2018 Chapter 1 7

Types of Solid Waste (Categorical) Material Weight (million tons) Percent Food scraps 25.9 11.2 % Glass 12.8 5.5 % Metals 18.0 7.8 % Paper, paperboard 86.7 37.4 % Plastics 24.7 10.7 % Rubber, leather, textiles 15.8 6.8 % Wood 12.7 Yard trimmings 27.7 11.9 % Other 7.5 3.2 % Total 231.9 100.0 % 11/22/2018 11/22/2018 Chapter 1 8

Types of Solid Waste (Categorical) Bar charts: bars do not touch (compare histograms) Pie charts: Use Excel or Applet Percentages must add to 100% 11/22/2018 11/22/2018 Chapter 1 9

Body Weight (Quantitative) n = 53 students 11/22/2018 11/22/2018 Chapter 1 10

Body Weight (Quantitative) Histogram HS 67 Lectue Notes Thursday, November 22, 2018 Body Weight (Quantitative) Histogram Create class interval frequency table Approx 4 to 12 non-overlapping class-intervals Tally frequencies and proportions Wt. interval Frequency % 100 – 119 7 13.5% 120 – 139 12 23.1 140 – 159 13.5 160 – 179 8 15.4 180 – 199 200 – 219 4 7.7 220 – 239 1 1.9 240 – 259 0.0 260 – 279 TOTAL 52 100% 11/22/2018 11/22/2018 Chapter 1 11 Chapter 1 11

Body Weight (Quantitative) Histogram HS 67 Lectue Notes Thursday, November 22, 2018 Body Weight (Quantitative) Histogram Draw histogram (frequencies and/or proportions) and label axes Number of students 100 120 140 160 180 200 220 240 260 280 Weight (pounds) 11/22/2018 11/22/2018 Chapter 1 12 Chapter 1

Body Weight: Stem-and-Leaf Separate each value into stem value (first one or two significant digits) and leaf value (next significant digit) 11/22/2018 11/22/2018 Chapter 1 13

Include “axis multiplier” Write leaf values next to stem HS 67 Lectue Notes Thursday, November 22, 2018 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Pounds(×10) Draw “stem” Label stem Include “axis multiplier” Write leaf values next to stem 192 5 152 2 135 2 The student should construct a stem & leaf plot here using the first two digits as the stem and the last digit as the leaf. The shape of the stem & leaf plot should look similar to the bar graph shown on an upcoming slide. 11/22/2018 11/22/2018 Chapter 1 14 Chapter 1

Plot one leaf per data point Then sort leaves in rank order HS 67 Lectue Notes Thursday, November 22, 2018 10 0166 11 009 12 0034578 13 00359 14 08 15 00257 16 555 17 000255 18 000055567 19 245 20 3 21 025 22 0 23 24 25 26 0 Pounds(×10) Plot one leaf per data point Then sort leaves in rank order The student should construct a stem & leaf plot here using the first two digits as the stem and the last digit as the leaf. The shape of the stem & leaf plot should look similar to the bar graph shown on an upcoming slide. Chapter 1 11/22/2018 11/22/2018 15 Chapter 1

Each Stemplot will differ! Key: select the correct axis multiplier before breaking values in stem & leaf values Think of the stemplot as a histogram with between 4 – 12 stem “bins” 11/22/2018

Second Example (n = 8) Data (average coliform count): 1.47, 2.06, 2.36, 3.43, 3.74, 3.78, 3.94, 4.42 Stem = ones-place values Leaves = tenths-place Truncate extra digit (e.g., 1.47  1.4) The book rounds but we shall truncate Do not plot the decimal |1|4 |2|03 |3|4779 |4|4 (×1)Coliforms 11/22/2018

Third Example (n = 25) Data & stemplot Split stem values |1|4789 |2|223466789 |3|000123445678 (×1) Too squished! Split stem values First “1” on stem holds leaves between 0 to 4 Second “1” on stem holds leaves between 5 to 9 Etc. |1|4 |1|789 |2|2234 |2|66789 |3|00012344 |3|5678 (×1) Notice shape (negative skew) 11/22/2018

Interpreting Stemplots HS 67 Lectue Notes Thursday, November 22, 2018 Interpreting Stemplots Shape: Symmetrical? Mound(s)? Tails? Central location (the book uses the midpoint) Spread (for now use range; better methods next week) Outliers: fall outside regular pattern 11/22/2018 11/22/2018 Chapter 1 19 Chapter 1

Interpreting Stemplots 10|0166 11|009 12|0034578 13|00359 14|08 15|00257 16|555 17|000255 18|000055567 19|245 20|3 21|025 22|0 23| 24| 25| 26|0 (×10) Shape: tail toward larger numbers  positive skew Center: n = 53  use the 26.5th ranked value  between 157 & 165 Spread: 100 to 260 Outlier: “260” seems out there 11/22/2018 11/22/2018 Chapter 1 20

Back-to-Back Stemplots Women | | Men ------------------- |0| |0|9 |1| 8|1| |2| |3| minutes (×100) Data = Exercise 1.38 on page 35 I’ve plotted the first value for women (180 minutes) and the first value for men (90 minutes) 11/22/2018 11/22/2018 Chapter 1 21

Interpreting Histograms Example: 7th Grader Vocabulary Score Shape: symmetrical Center: around 7 Spread: from 2 to 12 Outlier: 12(?) 11/22/2018 11/22/2018 Chapter 1 22