Picturing Distributions with Graphs BPS - 5th Ed. 1 Chapter 1

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BPS - 5th Ed.Chapter 1 3 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.

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 - 5th Ed. 4 Chapter 1

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 - 5th Ed. 5 Chapter 1

BPS - 5th Ed.Chapter 1 6 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, (Reported in Science News, Feb. 4, 1995, p. 108)

BPS - 5th Ed.Chapter 1 7 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.

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 - 5th Ed. 8 Chapter 1

The cohort of women were followed for 14 years. The number of CHD (fatal and nonfatal) cases were counted (1292 cases). BPS - 5th Ed. 9 Chapter 1

BPS - 5th Ed.Chapter 1 10 Age (in 1976) Weight in 1976 Weight at age 18 Incidence of coronary heart disease Smoker or nonsmoker Family history of heart disease quantitative categorical Variables measured

Tells what values a variable takes and how often it takes these values Can be a table, graph, or function BPS - 5th Ed. 11 Chapter 1

Categorical variables Pie charts Bar graphs Quantitative variables Histograms Stemplots (stem-and-leaf plots) BPS - 5th Ed. 12 Chapter 1

BPS - 5th Ed. 13 Chapter 1 YearCountPercent Freshman1841.9% Sophomore1023.3% Junior614.0% Senior920.9% Total % Data Table

BPS - 5th Ed. 14 Chapter 1 Pie Chart

BPS - 5th Ed. 15 Chapter 1 Bar Graph

MaterialWeight (million tons)Percent of total Food scraps % Glass % Metals % Paper, paperboard % Plastics % Rubber, leather, textiles % Wood % Yard trimmings % Other % Total % BPS - 5th Ed.Chapter 1 16 Data Table

BPS - 5th Ed. 17 Chapter 1 Pie Chart

BPS - 5th Ed. 18 Chapter 1 Bar Graph

Overall pattern of graph Deviations from overall pattern Shape of the data Center of the data Spread of the data (Variation) Outliers BPS - 5th Ed. 19 Chapter 1

Symmetric bell shaped other symmetric shapes Asymmetric right skewed left skewed Unimodal, bimodal BPS - 5th Ed. 20 Chapter 1

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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. 26 Chapter 1

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 - 5th Ed. 27 Chapter 1

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 - 5th Ed. 28 Chapter 1

BPS - 5th Ed.Chapter 1 29 Weight Data Introductory Statistics class Spring, 1997 Virginia Commonwealth University

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BPS - 5th Ed.Chapter 1 31 sqrt(53) = 7.2, or 8 intervals; range (260  100=160) / 8 = 20 = class width

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

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 - 5th Ed. 33 Chapter 1

BPS - 5th Ed.Chapter

BPS - 5th Ed.Chapter Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s

BPS - 5th Ed.Chapter Key 20 | 3 means 203 pounds Stems = 10’s Leaves = 1’s

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. 37 Chapter 1

Example: if all of the data values were between 150 and 179, then we may choose to use the following stems: BPS - 5th Ed. 38 Chapter Leaves 0-4 would go on each upper stem (first “15”), and leaves 5-9 would go on each lower stem (second “15”).

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 - 5th Ed. 39 Chapter 1

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BPS - 5th Ed.Chapter 1 41 Average Tuition (Public vs. Private)