1. The rise of statistics
Statistics is the science of collecting, organizing and interpreting data. The goal of statistics is to gain understanding from data.
Historically, the ideas and methods of statistics developed gradually as society became interested in collecting and using data for a variety of applications.
The discipline of statistics took shape in the twentieth century when methods for producing and understanding data grew in number and sophistication.
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2. Elements of Statistics - Introduction
Data are numerical facts with context and we need to understand
the context if we are to make sense of the numbers.
A set of data contains some information about a group of
individuals.
Individuals are the objects upon which we collect data.
Individuals can be people, animals, plots of land and many other
things.
A population is a set of individuals that we are interested in studying.
A variable is any characteristic of an individual.
A sample is a subset of the individuals of a population.
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3. Questions to ask when planning a statistical study
Why? What purpose do the data have? Do we hope to answer some specific questions? Do we want to draw conclusions about individuals other than the ones we actually have data for?
Who? What individuals do the data describe? How many individuals appear in the data?
What? How many variables do the data contain? Exact definitions of these variables. What are the units of measurements in which each variable is recorded? Weights for example, might be recorded in pounds, or in kg.
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4. Collecting Data
Generally, data can be obtained in four different ways.
Published source.
Designed experiment.
Survey.
Observational study.
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5. Types of Variables
A categorical variable places an individual into one of several groups or categories, e.g. gender, college major.
A quantitative variable takes numerical values for which arithmetic operations are defined, e.g. height, weight.
The distribution of a variable tells us what values it takes and how often it takes these values.
Example 1.4 on page 12.
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6. Displaying Distributions with Graphs
Statistical tools and ideas help us examine data in order to describe their main features. This examination is called exploratory data analysis.
Two basic strategies for exploration of data set:
Begin by examining each variable by itself. Then move on to study the relationships among the variables.
Begin with graphs. Then add numerical summaries of specified aspects of the data.
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7. Graphs for categorical variables
A class is one of the values of a categorical variable. It is
usually the label for the categories such as “male” and “female”.
The class frequency (count) is the number of observations in
the data set falling in a particular class.
The class relative frequency is the class frequency divided by
the total number of observation in the data set.
The distribution of a categorical variable lists the categories and gives either the count or the percent of individuals who fall in each category.
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8. Example : The of marital status for all Americans age 18+.
Count (millions)
Percent
Never married
Married
Widowed
Divorced
43.9
116.7
13.4
17.6
22.9
60.9
7.0
9.2
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10. Measurement - example
We want to compare the “size” of several statistics books.
Describe three possible numerical variables that
describes the “size” of a book.
In what units would you measure each variable?
What measuring instrument does each require?
Describe a variable that is appropriate for estimating how
long it would take to read the book?
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11. Describing Quantitative Data
The pattern of variation of a variable is called its distribution.
The distribution of a variable is best displayed graphically.
There are three main graphical methods for describing
summarizing and detecting patterns in quantitative data:
Dot plot
Stem-and-leaf plot
Histogram
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12. Stem-and-Leaf Plots To make a stem-and-leaf plot:
Separate each observation into a stem consisting of all but the final (rightmost) digit and a leaf, the final digit.
Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column.
Write each leaf in the row to the right of its stem, in increasing order out from the stem.
Examples 2.2 on page
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13. Example Here are the scores of a basketball player (say player A)
Make a stem-and-leaf plot of these data. Describe the main
features of the distribution.
Solution : Min. = 22, Max = 60 …
MINITAB command: Graph > Stem-and-Leaf
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14. Exercise Write down the min, max, and the median of the data sets
summarized by the following MINITAB stemplot.
Stem-and-leaf of Fuel Use N = 15
Leaf Unit = 1.0
(4)
1 1
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15. Exercise Write down the min., max, and the median of the data sets
summarized by the following MINITAB stemplot.
Stem-and-leaf of weight N = 12
Leaf Unit = 0.010
(4)
5 109
5 110
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16. Examining a distribution
In any graph of data, look for the overall pattern and for
striking deviations from that pattern.
Overall pattern of a distribution can be described by its shape, centre, and spread.
An important kind of deviation is an outlier, an individual value that falls outside the overall pattern.
Some other things to look for in describing shape are:
Does the distribution have one or several major peaks,
called modes? A distribution with one major peak is
called unimodal.
Is it approximately symmetric or skewed in one direction.
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17. Exercise Describe the shape of the distributions summarized
by the following stemplot.
Stem-and-leaf of sta220 marks N = 42
Leaf Unit = 1.0
(11)
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18. Exercise Describe the shape of the distributions summarized by the
following stemplots.
Stem-and-leaf of C1 N= 50
Leaf Unit = 0.10
(17)
2 2
1 3 .
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19. Exam Question
Forty students wrote a Statistics examination having a maximum of 50 marks.
The mark distribution is given in the following stem-and-leaf plot:
Stem Leaf
0 28
1 2245
5 000
State whether the following statements are true or false.
 (a) The distribution is right skewed.
 (b) The median of the distribution is 31.
 (c) The median of the distribution is 28.
(d) The mode of the distribution is 48.
  (e) More than 18% of the students scored 45 or more on the examination.
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20. Histograms A histogram breaks the range of values of a variable into
intervals and displays only the count or percent of the
observations that fall into each interval.
We can choose a convenient number of intervals.
Histograms do not display the actual values observed. (only counts in each interval).
Example:
Here is some data on the number of days lost due to illness
of a group of employees:
47, 1, 55, 30, 1, 3, 7, 14, 7, 66, 34, 6, 10, 5, 12, 5, 3, 9, 18,
45, 5, 8, 44, 42, 46, 6, 4, 24, 24, 34, 11, 2, 3, 13, 5, 5, 3, 4,
4, 1
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21. The main steps in constructing a histogram
Determine the Range of the data (largest and smallest values)
In our example the data ranges from a min.
of 1 day to a max. of 66 days.
Decide on the number of intervals (or classes) , and the
width of each class (usually equal).
Count the number of observations in each class. These counts are called class frequencies.
4. Draw the histogram.
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22. Class
No. of employees
(Frequency)
Cumulative. Frequency
Relative frequency
0-10
10-20
20-30
30-40
40-50
50-60
60-70 23 5 3 2 1 28 31 33 38 39 40
0.575
0.125
0.075
0.050
0.025
Total
1.000
A table with the first two columns above is called frequency table or frequency distribution.
A table with the first column and the third column is called relative frequency distribution.
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23. MINITAB command: Graph > Histogram
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24. Comments
The above histogram suggests that the distribution is skewed to the right. No gaps or outliers.
Since this data set is not very large (40 observations) we can also use a dot plot or a stem-and-leaf plot to represent the data.
MINITAB commands for dotplot:
Graph > Character Graphs > Dotplot
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25. Stem-and-leaf of days lost N = 40 Leaf Unit = 1.0   
(22)
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26. Question Which type of display
Uses the actual numbers as building blocks for the display?
Gives the most flexibility for setting the class width and number of classes?
Is most convenient for very large data sets?
Is quickest to construct?
Keeps the most detail re the actual data?
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27. Review Questions The purpose of a frequency distribution is to ____.
present scores and their frequency of occurrence
present data in a more meaningful way than single scores
provide more information than a graph
all of the above
a and b
Which of the following indicates the proportion of the total number of scores which occurred in each interval?
Relative frequency distribution
Cumulative frequency distribution
Cumulative percentage distribution
None of the above
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28. Relative frequency distribution Cumulative frequency distribution
3. Which of the following indicates the number of scores which fell below the upper limit of each interval?
Relative frequency distribution
Cumulative frequency distribution
Cumulative percentage distribution
None of the above
4. Which of the following is not a symmetrical distribution?
A bell-shaped distribution
A J-shaped distribution
A U-shaped distribution
An inverted U-shaped distribution
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