1. Essential Statistics Picturing Distributions with Graphs
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Picturing Distributions with Graphs
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2. 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.
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3. Individuals and Variables
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
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4. Variables Categorical Quantitative (Numerical)
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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
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5. Case Study The Effect of Hypnosis on the Immune System
reported in Science News, Sept. 4, 1993, p. 153
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6. Case Study The Effect of Hypnosis on the Immune System Objective:
To determine if hypnosis strengthens the
disease-fighting capacity of immune cells.
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7. Case Study 65 college students. white blood cell counts measured
33 easily hypnotized
32 not easily hypnotized
white blood cell counts measured
all students viewed a brief video about the immune system.
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8. Case Study Students randomly assigned to one of three conditions
subjects hypnotized, given mental exercise
subjects relaxed in sensory deprivation tank
control group (no treatment)
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9. Case Study 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
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10. Case Study Variables measured
Easy or difficult to achieve hypnotic trance
Group assignment
Pre-study white blood cell count
Post-study white blood cell count
categorical
quantitative
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11. 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)
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12. 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.
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13. Case Study
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.
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14. Case Study The cohort of women were followed for 14 years.
The number of CHD (fatal and nonfatal) cases were counted (1292 cases).
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15. Case Study Variables measured quantitative Age (in 1976)
Weight in 1976
Weight at age 18
Incidence of coronary heart disease
Smoker or nonsmoker
Family history of heart disease
categorical
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16. Distribution
Tells what values a variable takes and how often it takes these values
Can be a table, graph, or function
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17. Displaying Distributions
Categorical variables
Pie charts
Bar graphs
Quantitative variables
Histograms
Stemplots (stem-and-leaf plots)
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18. 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%
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19. Class Make-up on First Day
Pie Chart
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20. Class Make-up on First Day
Bar Graph
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21. 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
Yard trimmings
27.7
11.9 %
Other
7.5
3.2 %
Total
231.9
100.0 %
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22. Example: U.S. Solid Waste (2000)
Pie Chart
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23. Example: U.S. Solid Waste (2000)
Bar Graph
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24. Examining the Distribution of Quantitative Data
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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
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25. Shape of the Data Symmetric Asymmetric Unimodal, bimodal bell shaped
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Shape of the Data
Symmetric
bell shaped
other symmetric shapes
Asymmetric
right skewed
left skewed
Unimodal, bimodal
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26. Symmetric Bell-Shaped
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Symmetric Bell-Shaped
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27. Symmetric Mound-Shaped
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Symmetric Mound-Shaped
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28. Symmetric Uniform Chapter 1 Essential Statistics Essential Statistics

29. Asymmetric Skewed to the Left
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Asymmetric Skewed to the Left
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30. Asymmetric Skewed to the Right
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Asymmetric Skewed to the Right
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31. Outliers Extreme values that fall outside the overall pattern
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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
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32. 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
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33. 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)
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34. Case Study Weight Data Introductory Statistics class Spring, 1997
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Case Study
Weight Data
Introductory Statistics class Spring, 1997
Virginia Commonwealth University
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35. Weight Data
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36. Weight Data: Frequency Table
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Weight Data: Frequency Table
sqrt(53) = 7.2, or 8 intervals; range (260100=160) / 8 = 20 = class width
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37. Weight Data: Histogram
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Weight Data: Histogram
Number of students
100
120
140
160
180
200
220
240
260
280
Weight
* Left endpoint is included in the group, right endpoint is not.
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38. 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
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39. Weight Data 1 2
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40. Weight Data: Stemplot (Stem & Leaf Plot)
Essential Statistics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Weight Data: Stemplot (Stem & Leaf Plot)
192 5
152 2
135
Key
20|3 means 203 pounds
Stems = 10’s Leaves = 1’s 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.
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41. Weight Data: Stemplot (Stem & Leaf Plot)
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11 009
14 08
16 555
19 245
20 3
21 025
22 0 23 24 25
26 0
Weight Data: Stemplot (Stem & Leaf Plot)
Key
20|3 means 203 pounds
Stems = 10’s Leaves = 1’s
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.
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42. 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.
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43. 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:
Leaves 0-4 would go on each upper stem (first “15”), and leaves 5-9 would go on each lower stem (second “15”).
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44. 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).
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45. Class Make-up on First Day (Fall Semesters: 1985-1993)
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46. Average Tuition (Public vs. Private)
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Average Tuition (Public vs. Private)
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