 # Essential Statistics Chapter 11 Picturing Distributions with Graphs.

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Essential Statistics Chapter 11 Picturing Distributions with Graphs

Essential Statistics Chapter 13 Statistics -Statistics is a science that extract information from numerical data collected during an experiment or from a sample. -It involves the design of the experiment or sampling procedure, the collection of the data. -After analysis of the data, and making inferences (statements) about the population based on information in a sample.

Essential Statistics Chapter 14 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

Essential Statistics Chapter 15 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

Essential Statistics Chapter 16 Distribution u Tells what values a variable takes and how far from average and how often it takes these values u Can be a table, graph, or function

Essential Statistics Chapter 17 Displaying Distributions u Categorical variables –Pie charts –Bar graphs u Quantitative variables –Histograms –Stemplots (stem-and-leaf plots)

Essential Statistics Chapter 18 YearCountPercent Freshman1841.9% Sophomore1023.3% Junior614.0% Senior920.9% Total43100.1% Data Table Class Make-up on First Day

Essential Statistics Chapter 19 Pie Chart Class Make-up on First Day

Essential Statistics Chapter 110 Class Make-up on First Day Bar Graph

Essential Statistics Chapter 111 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 %

Essential Statistics Chapter 112 Example: U.S. Solid Waste (2000) Pie Chart

Essential Statistics Chapter 113 Example: U.S. Solid Waste (2000) Bar Graph

Essential Statistics Chapter 114 Shape of the Distributiona u Symmetric –bell shaped –other symmetric shapes u Asymmetric –right skewed –left skewed u Unimodal, bimodal

Essential Statistics Chapter 115 Symmetric Bell-Shaped

Essential Statistics Chapter 116 Symmetric Mound-Shaped

Essential Statistics Chapter 117 Symmetric Uniform

Essential Statistics Chapter 118 Asymmetric Skewed to the Left

Essential Statistics Chapter 119 Asymmetric Skewed to the Right

Essential Statistics Chapter 120 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

Essential Statistics Chapter 121 Histograms u For quantitative variables that take many values u Divide the possible values into class intervals (equal widths) u Count how many observations fall in each interval (frequency) u Draw picture representing distribution

Histograms u A histogram is a graph that displays frequency of data, or say display the frequency distribution u The data is grouped in equal-sized intervals that do not overlap u The height of each bar shows the frequency of the data in a given intervals Essential Statistics Chapter 122

Frequency Histogram u Represents a frequency distribution of a data set u The horizontal scale represents interval of data values u The vertical scale represents frequencies u The height of each column correspond to the frequency or relative frequency, of the interval Essential Statistics Chapter 123

Essential Statistics Chapter 124 Histograms: Define 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)

Essential Statistics Chapter 125 Case Study Weight Data Introductory Statistics class Spring, 1997 Virginia Commonwealth University

Essential Statistics Chapter 126 Weight Data

Essential Statistics Chapter 127 Weight Data: Frequency Table sqrt(53) = 7.2, or 8 intervals; range (260  100=160) / 8 = 20 = class width

Essential Statistics Chapter 128 Weight Data: Histogram 100120140160180200220240260280 Weight * Left endpoint is included in the group, right endpoint is not. Number of students

Essential Statistics Chapter 129 Interpreting Histograms u Center of the data u Shape of the data u Spread of the data (Variation) u Outliers u The overall pattern u Deviations from overall pattern

Essential Statistics Chapter 130 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

Essential Statistics Chapter 131 Weight Data 1212

Essential Statistics Chapter 132 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

Essential Statistics Chapter 133 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

Essential Statistics Chapter 134 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”).

Essential Statistics Chapter 135 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).

Essential Statistics Chapter 136 Class Make-up on First Day (Fall Semesters: 1985-1993)

Essential Statistics Chapter 137 Average Tuition (Public vs. Private)