# BPS - 5th Ed. Chapter 11 Picturing Distributions with Graphs.

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BPS - 5th Ed. Chapter 11 Picturing Distributions with Graphs

BPS - 5th Ed. Chapter 12 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 - 5th Ed. Chapter 13 Individuals and Variables u Individuals –the objects described by a set of data –may be people, animals, or things u Variable –any characteristic of an individual –can take different values for different individuals

BPS - 5th Ed. Chapter 14 Variables u Categorical –Places an individual into one of several groups or categories u Quantitative (Numerical) –Takes numerical values for which arithmetic operations such as adding and averaging make sense

BPS - 5th Ed. Chapter 15 Case Study The Effect of Hypnosis on the Immune System reported in Science News, Sept. 4, 1993, p. 153

BPS - 5th Ed. Chapter 16 Case Study The Effect of Hypnosis on the Immune System Objective: To determine if hypnosis strengthens the disease-fighting capacity of immune cells.

BPS - 5th Ed. Chapter 17 Case Study u 65 college students. –33 easily hypnotized –32 not easily hypnotized u white blood cell counts measured u all students viewed a brief video about the immune system.

BPS - 5th Ed. Chapter 18 Case Study u Students randomly assigned to one of three conditions –subjects hypnotized, given mental exercise –subjects relaxed in sensory deprivation tank –control group (no treatment)

BPS - 5th Ed. Chapter 19 Case Study u white blood cell counts re-measured after one week u the two white blood cell counts are compared for each group u results –hypnotized group showed larger jump in white blood cells –“easily hypnotized” group showed largest immune enhancement

BPS - 5th Ed. Chapter 110 Case Study Variables measured u Easy or difficult to achieve hypnotic trance u Group assignment u Pre-study white blood cell count u Post-study white blood cell count categorical quantitative

BPS - 5th Ed. Chapter 111 Case Study 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 - 5th Ed. Chapter 112 Case Study 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 - 5th Ed. Chapter 113 Case Study u Study started in 1976 with 115,818 women aged 30 to 55 years and without a history of previous CHD. u Each woman’s weight (body mass) was determined. u Each woman was asked her weight at age 18.

BPS - 5th Ed. Chapter 114 Case Study u The cohort of women were followed for 14 years. u The number of CHD (fatal and nonfatal) cases were counted (1292 cases).

BPS - 5th Ed. Chapter 115 Case Study u Age (in 1976) u Weight in 1976 u Weight at age 18 u Incidence of coronary heart disease u Smoker or nonsmoker u Family history of heart disease quantitative categorical Variables measured

BPS - 5th Ed. Chapter 116 Distribution u Tells what values a variable takes and how often it takes these values u Can be a table, graph, or function

BPS - 5th Ed. Chapter 117 Displaying Distributions u Categorical variables –Pie charts –Bar graphs u Quantitative variables –Histograms –Stemplots (stem-and-leaf plots)

BPS - 5th Ed. Chapter 118 YearCountPercent Freshman1841.9% Sophomore1023.3% Junior614.0% Senior920.9% Total43100.1% Data Table Class Make-up on First Day

BPS - 5th Ed. Chapter 119 Pie Chart Class Make-up on First Day

BPS - 5th Ed. Chapter 120 Class Make-up on First Day Bar Graph

BPS - 5th Ed. Chapter 121 Example: U.S. Solid Waste (2000) Data Table MaterialWeight (million tons)Percent of total Food scraps25.911.2 % Glass12.85.5 % Metals18.07.8 % Paper, paperboard86.737.4 % Plastics24.710.7 % Rubber, leather, textiles15.86.8 % Wood12.75.5 % Yard trimmings27.711.9 % Other7.53.2 % Total231.9100.0 %

BPS - 5th Ed. Chapter 122 Example: U.S. Solid Waste (2000) Pie Chart

BPS - 5th Ed. Chapter 123 Example: U.S. Solid Waste (2000) Bar Graph

BPS - 5th Ed. Chapter 124 Examining the Distribution of Quantitative Data u Overall pattern of graph u Deviations from overall pattern u Shape of the data u Center of the data u Spread of the data (Variation) u Outliers

BPS - 5th Ed. Chapter 125 Shape of the Data u Symmetric –bell shaped –other symmetric shapes u Asymmetric –right skewed –left skewed u Unimodal, bimodal

BPS - 5th Ed. Chapter 126 Symmetric Bell-Shaped

BPS - 5th Ed. Chapter 127 Symmetric Mound-Shaped

BPS - 5th Ed. Chapter 128 Symmetric Uniform

BPS - 5th Ed. Chapter 129 Asymmetric Skewed to the Left

BPS - 5th Ed. Chapter 130 Asymmetric Skewed to the Right

BPS - 5th Ed. Chapter 131 Outliers u 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 - 5th Ed. Chapter 132 Histograms u For quantitative variables that take many values u Divide the possible values into class intervals (we will only consider equal widths) u Count how many observations fall in each interval (may change to percents) u Draw picture representing distribution

BPS - 5th Ed. Chapter 133 Histograms: Class Intervals u How many intervals? –One rule is to calculate the square root of the sample size, and round up. u Size of intervals? –Divide range of data (max  min) by number of intervals desired, and round to convenient number u Pick intervals so each observation can only fall in exactly one interval (no overlap)

BPS - 5th Ed. Chapter 134 Case Study Weight Data Introductory Statistics class Spring, 1997 Virginia Commonwealth University

BPS - 5th Ed. Chapter 135 Weight Data

BPS - 5th Ed. Chapter 136 Weight Data: Frequency Table sqrt(53) = 7.2, or 8 intervals; range (260  100=160) / 8 = 20 = class width

BPS - 5th Ed. Chapter 137 Weight Data: Histogram 100120140160180200220240260280 Weight * Left endpoint is included in the group, right endpoint is not. Number of students

BPS - 5th Ed. Chapter 138 Stemplots (Stem-and-Leaf Plots) u For quantitative variables u Separate each observation into a stem (first part of the number) and a leaf (the remaining part of the number) u Write the stems in a vertical column; draw a vertical line to the right of the stems u Write each leaf in the row to the right of its stem; order leaves if desired

BPS - 5th Ed. Chapter 139 Weight Data 1212

BPS - 5th Ed. Chapter 140 Weight Data: Stemplot (Stem & Leaf Plot) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s 192 2 152 2 5 135

BPS - 5th Ed. Chapter 141 Weight Data: Stemplot (Stem & Leaf Plot) 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 Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s

BPS - 5th Ed. Chapter 142 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 - 5th Ed. Chapter 143 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 - 5th Ed. Chapter 144 Time Plots u A time plot shows behavior over time. u Time is always on the horizontal axis, and the variable being measured is on the vertical axis. u Look for an overall pattern (trend), and deviations from this trend. Connecting the data points by lines may emphasize this trend. u Look for patterns that repeat at known regular intervals (seasonal variations).

BPS - 5th Ed. Chapter 145 Class Make-up on First Day (Fall Semesters: 1985-1993)

BPS - 5th Ed. Chapter 146 Average Tuition (Public vs. Private)

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