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![Page 1: Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 Overview and Descriptive Statistics.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551c2712550346a84f8b5e72/html5/thumbnails/1.jpg)
Chapter 1
Overview and
Descriptive Statistics
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1.1
Populations, Samples, and Processes
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Populations and Samples
A population is a well-defined collection of objects.
When information is available for the entire population we have a census. A subset of the population is a sample.
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Data and Observations
Univariate data consists of observations on a single variable (multivariate – more than two variables).
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Branches of Statistics
Descriptive Statistics – summary and description of collected data.
Inferential Statistics – generalizing from a sample to a population.
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Relationship Between Probability and Inferential Statistics
Population Sample
Probability
Inferential
Statistics
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1.2
Pictorial and Tabular Methods
in Descriptive Statistics
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Stem-and- Leaf Displays
1. Select one or more leading digits for the stem values. The trailing digits become the leaves.
2. List stem values in a vertical column.
3. Record the leaf for every observation.
4. Indicate the units for the stem and leaf on the disply.
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Stem-and-Leaf Example
9, 10, 15, 22, 9, 15, 16, 24,11 Observed values:
0 9 9
1 1 0 5 5 6
2 2 4
Stem: tens digit Leaf: units digit
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Stem-and- Leaf Displays• Identify typical value
• Extent of spread about a value
• Presence of gaps
• Extent of symmetry
• Number and location of peaks
• Presence of outlying values
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Dotplots
9, 10, 15, 22, 9, 15, 16, 24,11
Observed values:
5 10 15 20 25
Represent data with dots.
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Types of Variables
A variable is discrete if its set of possible values constitute a finite set or an infinite sequence. A variable is continuous if its set of possible values consists of an entire interval on a number line.
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Histograms Discrete Data
Determine the frequency and relative frequency for each value of x. Then mark possible x values on a horizontal scale. Above each value, draw a rectangle whose height is the relative frequency of that value.
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Ex. Students from a small college were asked how many charge cards that they carry. x is the variable representing the number of cards and the results are below.
x #people
0 12
1 42
2 57
3 24
4 9
5 4
6 2
Rel. Freq
0.08
0.28
0.38
0.16
0.06
0.03
0.01
Frequency Distribution
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Histograms
x Rel. Freq.
0 0.08
1 0.28
2 0.38
3 0.16
4 0.06
5 0.03
6 0.01
Credit card results:
Relative Frequency
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6
Number of Cards
ix
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Histograms Continuous Data:
Equal Class Widths
Determine the frequency and relative frequency for each class. Then mark the class boundaries on a horizontal measurement axis. Above each class interval, draw a rectangle whose height is the relative frequency.
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Histograms Continuous Data: Unequal Widths
After determining frequencies and relative frequencies, calculate the height of each rectangle using:
The resulting heights are called densities and the vertical scale is the density scale.
relative frequency of the classrectangle height =
class width
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Histogram Shapes
symmetric unimodal bimodal
positively skewed negatively skewed
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1.3
Measures of Location
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The Mean
The average (mean) of the n numbers
1 2, ,..., nx x x is wherex
1 2 ... nx x xx
n
1
n
ii
x
n
Population mean:
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The sample median, is the middle value in a set of data that is arranged in ascending order. For an even number of data points the median is the average of the middle two.
Median
,x
Population median:
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Three Different Shapes for a Population Distribution
symmetric
positive skewnegative skew
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1.4
Measures of
Variability
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22
1 1
i xxx x S
sn n
Sample VarianceVariance is a measure of the spread of the data.
The sample variance of the sample x1, x2, …xn of n values of X is given by
We refer to s2 as being based on n – 1 degrees of freedom.
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2s s
The sample standard deviation is the square root of the sample variance:
Standard Deviation
Standard deviation is a measure of the spread of the data using the same units as the data.
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Formula for s2
22 2 i
xx i i
xS x x x
n
An alternative expression for the numerator of s2 is
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Properties of s2
Let x1, x2,…,xn be any sample and c be any nonzero constant.
2 21 11. If ,..., , then n n y xy x c y x c s s
2 2 21 12. If ,..., , then ,n n y xy cx y cx s c s
where is the sample variance of the x’s and is the sample variance of the y’s.
2xs
2ys
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Upper and Lower Fourths
After the n observations in a data set are ordered from smallest to largest, the lower (upper) fourth is the median of the smallest (largest) half of the data, where the median is included in both halves if n is odd. A measure of the spread that is resistant to outliers is the fourth spread fs = upper fourth – lower fourth.
x
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Outliers
Any observation farther than 1.5fs from the closest fourth is an outlier. An outlier is extreme if it is more than 3fs from the nearest fourth, and it is mild otherwise.
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Boxplots
medianextreme outliers
upper fourthlower fourth
mild outliers