Chapter 1 Picturing Distributions with Graphs BPS - 3rd Ed. Chapter 1 1 Statistics Statistics is a science that involves the extraction of information from numerical data obtained during an experiment or from a sample. It involves the design of the experiment or sampling procedure, the collection and analysis of the data, and making inferences (statements) about the population based upon information in a sample. BPS - 3rd Ed. Chapter 1 2 Individuals and Variables Individuals the objects described by a set of data may be people, animals, or things Variable any characteristic of an individual can take different values for different individuals BPS - 3rd Ed. Chapter 1 3 Chapter 1 1
Variables Categorical Places an individual into one of several groups or categories Quantitative (Numerical) Takes numerical values for which arithmetic operations such as adding and averaging make sense BPS - 3rd Ed. Chapter 1 4 The Effect of Hypnosis on the Immune System reported in Science News, Sept. 4, 1993, p. 153 BPS - 3rd Ed. Chapter 1 5 The Effect of Hypnosis on the Immune System Objective: To determine if hypnosis strengthens the disease-fighting capacity of immune cells. BPS - 3rd Ed. Chapter 1 6 Chapter 1 2
65 college students. 33 easily hypnotized 32 not easily hypnotized white blood cell counts measured all students viewed a brief video about the immune system. BPS - 3rd Ed. Chapter 1 7 Students randomly assigned to one of three conditions subjects hypnotized, given mental exercise subjects relaxed in sensory deprivation tank control group (no treatment) BPS - 3rd Ed. Chapter 1 8 white blood cell counts re-measured after one week the two white blood cell counts are compared for each group results hypnotized group showed larger jump in white blood cells easily hypnotized group showed largest immune enhancement BPS - 3rd Ed. Chapter 1 9 Chapter 1 3
Variables measured categorical quantitative Easy or difficult to achieve hypnotic trance Group assignment Pre-study white blood cell count Post-study white blood cell count BPS - 3rd Ed. Chapter 1 10 Weight Gain Spells Heart Risk for Women Weight, weight change, and coronary heart disease in women. W.C. Willett, et. al., vol. 273(6), Journal of the American Medical Association, Feb. 8, 1995. (Reported in Science News, Feb. 4, 1995, p. 108) BPS - 3rd Ed. Chapter 1 11 Weight Gain Spells Heart Risk for Women Objective: To recommend a range of body mass index (a function of weight and height) in terms of coronary heart disease (CHD) risk in women. BPS - 3rd Ed. Chapter 1 12 Chapter 1 4
Study started in 1976 with 115,818 women aged 30 to 55 years and without a history of previous CHD. Each woman s weight (body mass) was determined Each woman was asked her weight at age 18. BPS - 3rd Ed. Chapter 1 13 The cohort of women were followed for 14 years. The number of CHD (fatal and nonfatal) cases were counted (1292 cases). BPS - 3rd Ed. Chapter 1 14 Variables measured quantitative categorical Age (in 1976) Weight in 1976 Weight at age 18 Incidence of coronary heart disease Smoker or nonsmoker Family history of heart disease BPS - 3rd Ed. Chapter 1 15 Chapter 1 5
Distribution Tells what values a variable takes and how often it takes these values Can be a table, graph, or function BPS - 3rd Ed. Chapter 1 16 Displaying Distributions Categorical variables Pie charts Bar graphs Quantitative variables Histograms Stemplots (stem-and-leaf plots) BPS - 3rd Ed. Chapter 1 17 Class Make-up on First Day Data Table Year Count Percent Freshman 18 41.9% Sophomore 10 23.3% Junior 6 14.0% Senior 9 20.9% Total 43 100.1% BPS - 3rd Ed. Chapter 1 18 Chapter 1 6
Class Make-up on First Day Senior 20.9% Pie Chart Junior 14.0% Freshman 41.9% Sophomore 23.3% BPS - 3rd Ed. Chapter 1 19 Percent Class Make-up on First Day Bar Graph 45.0% 41.9% 40.0% 35.0% 30.0% 25.0% 23.3% 20.9% 20.0% 15.0% 14.0% 10.0% 5.0% 0.0% Freshman Sophomore Junior Senior Year in School BPS - 3rd Ed. Chapter 1 20 Example: U.S. Solid Waste (2000) Data Table Material Weight (million tons) Percent of total 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 5.5 % Yard trimmings 27.7 11.9 % Other 7.5 3.2 % Total 231.9 100.0 % BPS - 3rd Ed. Chapter 1 21 Chapter 1 7
Example: U.S. Solid Waste (2000) Pie Chart BPS - 3rd Ed. Chapter 1 22 Example: U.S. Solid Waste (2000) Bar Graph BPS - 3rd Ed. Chapter 1 23 Examining the Distribution of Quantitative Data Overall pattern of graph Deviations from overall pattern Shape of the data Center of the data Spread of the data (Variation) Outliers BPS - 3rd Ed. Chapter 1 24 Chapter 1 8
Shape of the Data Symmetric bell shaped other symmetric shapes Asymmetric right skewed left skewed Unimodal, bimodal BPS - 3rd Ed. Chapter 1 25 Symmetric Bell-Shaped BPS - 3rd Ed. Chapter 1 26 Symmetric Mound-Shaped BPS - 3rd Ed. Chapter 1 27 Chapter 1 9
Symmetric Uniform BPS - 3rd Ed. Chapter 1 28 Asymmetric Skewed to the Left BPS - 3rd Ed. Chapter 1 29 Asymmetric Skewed to the Right BPS - 3rd Ed. Chapter 1 30 Chapter 1 10
Outliers Extreme values that fall outside the overall pattern May occur naturally May occur due to error in recording May occur due to error in measuring Observational unit may be fundamentally different BPS - 3rd Ed. Chapter 1 31 Histograms For quantitative variables that take many values Divide the possible values into class intervals (we will only consider equal widths) Count how many observations fall in each interval (may change to percents) Draw picture representing distribution BPS - 3rd Ed. Chapter 1 32 Histograms: Class Intervals How many intervals? One rule is to calculate the square root of the sample size, and round up. Size of intervals? Divide range of data (max min) by number of intervals desired, and round to convenient number Pick intervals so each observation can only fall in exactly one interval (no overlap) BPS - 3rd Ed. Chapter 1 33 Chapter 1 11
Weight Data Introductory Statistics class Spring, 1997 Virginia Commonwealth University BPS - 3rd Ed. Chapter 1 34 Weight Data 192 110 195 180 170 215 152 120 170 130 130 125 135 185 120 155 101 194 110 165 185 220 180 128 212 175 140 187 180 119 203 157 148 260 165 185 150 106 170 210 123 172 180 165 186 139 175 127 150 100 106 133 124 BPS - 3rd Ed. Chapter 1 35 Weight Data: Frequency Table Weight Group Count 100 - <120 7 120 - <140 12 140 - <160 7 160 - <180 8 180 - <200 12 200 - <220 4 220 - <240 1 240 - <260 0 260 - <280 1 sqrt(53) = 7.2, or 8 intervals; range (260 100=160) / 8 = 20 = class width BPS - 3rd Ed. Chapter 1 36 Chapter 1 12
14 Weight Data: Histogram Number of students 12 10 8 6 4 2 0 100 120 140 160 180 200 220 240 260 280 Weight * Left endpoint is included in the group, right endpoint is not. Frequency BPS - 3rd Ed. Chapter 1 37 Stemplots (Stem-and-Leaf Plots) For quantitative variables Separate each observation into a stem (first part of the number) and a leaf (the remaining part of the number) Write the stems in a vertical column; draw a vertical line to the right of the stems Write each leaf in the row to the right of its stem; order leaves if desired BPS - 3rd Ed. Chapter 1 38 Weight Data 192 110 195 180 170 215 152 120 170 130 130 125 135 185 120 155 101 194 110 165 185 220 180 128 212 175 140 187 180 119 203 157 148 260 165 185 150 106 170 210 123 172 180 165 186 139 175 127 150 100 106 133 124 BPS - 3rd Ed. Chapter 1 39 Chapter 1 13
Weight Data: Stemplot (Stem & Leaf Plot) Key 20 3 means 203 pounds Stems = 10 s Leaves = 1 s 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 BPS - 3rd Ed. Chapter 1 40 Weight Data: Stemplot (Stem & Leaf Plot) Key 20 3 means 203 pounds Stems = 10 s Leaves = 1 s 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 BPS - 3rd Ed. Chapter 1 41 Extended Stem-and-Leaf Plots If there are very few stems (when the data cover only a very small range of values), then we may want to create more stems by splitting the original stems. BPS - 3rd Ed. Chapter 1 42 Chapter 1 14
Extended Stem-and-Leaf Plots Example: if all of the data values were between 150 and 179, then we may choose to use the following stems: 15 15 16 16 17 17 Leaves 0-4 would go on each upper stem (first 15 ), and leaves 5-9 would go on each lower stem (second 15 ). BPS - 3rd Ed. Chapter 1 43 Time Plots A time plot shows behavior over time. Time is always on the horizontal axis, and the variable being measured is on the vertical axis. Look for an overall pattern (trend), and deviations from this trend. Connecting the data points by lines may emphasize this trend. Look for patterns that repeat at known regular intervals (seasonal variations). BPS - 3rd Ed. Chapter 1 44 Class Make-up on First Day (Fall Semesters: 1985-1993) 70% Class Make-up On First Day Percent of Class That Are Freshman 60% 50% 40% 30% 20% 10% 0% 1985 1986 1987 1988 1989 1990 1991 1992 1993 Year of Fall Semester BPS - 3rd Ed. Chapter 1 45 Chapter 1 15
Average Tuition (Public vs. Private) BPS - 3rd Ed. Chapter 1 46 Chapter 1 16