Download - Descriptive statistics
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Descriptive Statistics
2-1 Overview
2-2 Summarizing Data with Frequency Tables
2-3 Pictures of Data
2-4 Measures of Center
2-5 Measures of Variation
2-6 Measures of Position
2-7 Exploratory Data Analysis (EDA)
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Descriptive Statistics
summarize or describe the important characteristics of a known set of population data
Inferential Statistics
use sample data to make inferences (or generalizations) about a population
2 -1 Overview
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1. Center: A representative or average value that indicates where the middle of the data set is located
2. Variation: A measure of the amount that the values vary among themselves
3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed)
4. Outliers: Sample values that lie very far away from the vast majority of other sample values
5. Time: Changing characteristics of the data over time
Important Characteristics of Data
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Frequency Table
lists classes (or categories) of values,
along with frequencies (or counts) of
the number of values that fall into each
class
2-2 Summarizing Data With Frequency Tables
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Qwerty Keyboard Word RatingsTable 2-1
2 2 5 1 2 6 3 3 4 2
4 0 5 7 7 5 6 6 8 10
7 2 2 10 5 8 2 5 4 2
6 2 6 1 7 2 7 2 3 8
1 5 2 5 2 14 2 2 6 3
1 7
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Frequency Table of Qwerty Word RatingsTable 2-3
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
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Frequency Table
Definitions
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Lower Class Limits
Lower ClassLimits
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
are the smallest numbers that can actually belong to different classes
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Upper Class Limits
Upper ClassLimits
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
are the largest numbers that can actually belong to different classes
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are the numbers used to separate classes, but without the gaps created by class limits
Class Boundaries
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number separating classes
Class Boundaries
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency- 0.5
2.5
5.5
8.5
11.5
14.5
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Class Boundaries
ClassBoundaries
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency- 0.5
2.5
5.5
8.5
11.5
14.5
number separating classes
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midpoints of the classes
Class Midpoints
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midpoints of the classes
Class Midpoints
ClassMidpoints
0 - 1 2 20
3 - 4 5 14
6 - 7 8 15
9 - 10 11 2
12 - 13 14 1
Rating Frequency
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is the difference between two consecutive lower class limits or two consecutive class boundaries
Class Width
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Class Width
Class Width
3 0 - 2 20
3 3 - 5 14
3 6 - 8 15
3 9 - 11 2
3 12 - 14 1
Rating Frequency
is the difference between two consecutive lower class limits or two consecutive class boundaries
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1. Be sure that the classes are mutually exclusive.
2. Include all classes, even if the frequency is zero.
3. Try to use the same width for all classes.
4. Select convenient numbers for class limits.
5. Use between 5 and 20 classes.
6. The sum of the class frequencies must equal the number of original data values.
Guidelines For Frequency Tables
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3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score.
4. Add the class width to the starting point to get the second lower class limit, add the width to the second lower limit to get the
third, and so on.
5. List the lower class limits in a vertical column and enter the upper class limits.
6. Represent each score by a tally mark in the appropriate class.
Total tally marks to find the total frequency for each class.
Constructing A Frequency Table1. Decide on the number of classes .
2. Determine the class width by dividing the range by the number of classes (range = highest score - lowest score) and round
up.class width round up of
range
number of classes
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Relative Frequency Table
relative frequency =class frequency
sum of all frequencies
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Relative Frequency Table
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
0 - 2 38.5%
3 - 5 26.9%
6 - 8 28.8%
9 - 11 3.8%
12 - 14 1.9%
RatingRelative
Frequency
20/52 = 38.5%
14/52 = 26.9%
etc.
Total frequency = 52
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Cumulative Frequency Table
CumulativeFrequencies
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
Less than 3 20
Less than 6 34
Less than 9 49
Less than 12 51
Less than 15 52
RatingCumulativeFrequency
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Frequency Tables
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
0 - 2 38.5%
3 - 5 26.9%
6 - 8 28.8%
9 - 11 3.8%
12 - 14 1.9%
RatingRelative
Frequency
Less than 3 20
Less than 6 34
Less than 9 49
Less than 12 51
Less than 15 52
RatingCumulative Frequency
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a value at the
center or middle of a data set
Measures of Center
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Mean
(Arithmetic Mean)
AVERAGE
the number obtained by adding the values and dividing the total by the number of values
Definitions
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Notation
denotes the addition of a set of values
x is the variable usually used to represent the individual data values
n represents the number of data values in a sample
N represents the number of data values in a population
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Notationis pronounced ‘x-bar’ and denotes the mean of a set of sample values
x =n
xx
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Notation
µ is pronounced ‘mu’ and denotes the mean of all values in a population
is pronounced ‘x-bar’ and denotes the mean of a set of sample values
Calculators can calculate the mean of data
x =n
xx
Nµ =
x
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
often denoted by x (pronounced ‘x-tilde’)~
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
often denoted by x (pronounced ‘x-tilde’)
is not affected by an extreme value
~
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6.72 3.46 3.60 6.44
3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers
3.60 + 6.44
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(even number of values)
MEDIAN is 5.02
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6.72 3.46 3.60 6.44 26.70
3.46 3.60 6.44 6.72 26.70
(in order - odd number of values)
exact middle MEDIAN is 6.44
6.72 3.46 3.60 6.44
3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers
3.60 + 6.44
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(even number of values)
MEDIAN is 5.02
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Definitions Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
denoted by M
the only measure of central tendency that can be used with nominal data
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a. 5 5 5 3 1 5 1 4 3 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
c. 1 2 3 6 7 8 9 10
Examples
Mode is 5
Bimodal - 2 and 6
No Mode
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Midrange
the value midway between the highest and lowest values in the original data set
Definitions
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Midrange
the value midway between the highest and lowest values in the original data set
Definitions
Midrange =highest score + lowest score
2
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SymmetricData is symmetric if the left half of its histogram is roughly a mirror of its right half.
SkewedData is skewed if it is not symmetric and if it extends more to one side
than the other.
Definitions
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Skewness
Mode = Mean = Median
SYMMETRIC
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
SKEWED RIGHT(positively)
Mean Mode Median
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Waiting Times of Bank Customers at Different Banks
in minutes
Jefferson Valley Bank
Bank of Providence
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0
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Jefferson Valley Bank
Bank of Providence
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0
Jefferson Valley Bank
7.15
7.20
7.7
7.10
Bank of Providence
7.15
7.20
7.7
7.10
Mean
Median
Mode
Midrange
Waiting Times of Bank Customers at Different Banks
in minutes
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Dotplots of Waiting Times
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Measures of Variation
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Measures of Variation
Range
valuehighest lowest
value
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a measure of variation of the scores about the mean
(average deviation from the mean)
Measures of Variation
Standard Deviation
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Sample Standard Deviation Formula
calculators can compute the sample standard deviation of data
(x - x)2
n - 1S =
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Sample Standard Deviation Shortcut Formula
n (n - 1)
s =n (x2) - (x)2
calculators can compute the sample standard deviation of data
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x - x
Mean Absolute Deviation Formula
n
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Population Standard Deviation
calculators can compute the population standard deviation
of data
2 (x - µ)
N =
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Measures of Variation
Variance
standard deviation squared
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Measures of Variation
Variance
standard deviation squared
s
2
2
}
use square keyon calculatorNotation
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SampleVariance
PopulationVariance
Variance
(x - x )2
n - 1s2 =
(x - µ)2
N 2 =
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Estimation of Standard DeviationRange Rule of Thumb
x - 2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
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Estimation of Standard DeviationRange Rule of Thumb
x - 2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
Range
4s =
highest value - lowest value
4
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x
The Empirical Rule(applies to bell-shaped distributions)FIGURE 2-15
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x - s x x + s
68% within1 standard deviation
34% 34%
The Empirical Rule(applies to bell-shaped distributions)FIGURE 2-15
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x - 2s x - s x x + 2sx + s
68% within1 standard deviation
34% 34%
95% within 2 standard deviations
The Empirical Rule(applies to bell-shaped distributions)
13.5% 13.5%
FIGURE 2-15
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x - 3s x - 2s x - s x x + 2s x + 3sx + s
68% within1 standard deviation
34% 34%
95% within 2 standard deviations
99.7% of data are within 3 standard deviations of the mean
The Empirical Rule(applies to bell-shaped distributions)
0.1% 0.1%
2.4% 2.4%
13.5% 13.5%
FIGURE 2-15
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Chebyshev’s Theorem applies to distributions of any shape.
the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K2 , where K is any positive number greater than 1.
at least 3/4 (75%) of all values lie within 2 standard deviations of the mean.
at least 8/9 (89%) of all values lie within 3 standard deviations of the mean.
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Measures of Variation Summary
For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations.
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z Score (or standard score)
the number of standard deviations that a given value x is above or
below the mean
Measures of Position
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Sample
z = x - xs
Population
z = x - µ
Measures of Positionz score
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- 3 - 2 - 1 0 1 2 3
Z
Unusual Values
Unusual Values
OrdinaryValues
Interpreting Z Scores
FIGURE 2-16
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Measures of Position
Quartiles, Deciles,
Percentiles
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Q1, Q2, Q3 divides ranked scores into four equal parts
Quartiles
25% 25% 25% 25%
Q3Q2Q1(minimum) (maximum)
(median)
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D1, D2, D3, D4, D5, D6, D7, D8, D9
divides ranked data into ten equal parts
Deciles
10% 10% 10% 10% 10% 10% 10% 10% 10% 10%
D1 D2 D3 D4 D5 D6 D7 D8 D9
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99 Percentiles
Percentiles
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Quartiles, Deciles, Percentiles
Fractiles
(Quantiles)partitions data into approximately equal parts
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Exploratory Data Analysis
the process of using statistical tools (such as graphs, measures of center, and measures of variation) to investigate the data sets in order to understand their important characteristics
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Outliers
a value located very far away from almost all of the other values
an extreme value
can have a dramatic effect on the mean, standard deviation, and on the scale of the histogram so that the true nature of the distribution is totally obscured
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Boxplots(Box-and-Whisker Diagram)
Reveals the: center of the data spread of the data distribution of the data presence of outliers
Excellent for comparing two or more data sets
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Boxplots
5 - number summary
Minimum
first quartile Q1
Median (Q2)
third quartile Q3
Maximum
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Boxplots
Boxplot of Qwerty Word Ratings
2 4 6 8 10 12 14
02 4 6
14
0
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Bell-Shaped Skewed
Boxplots
Uniform
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Exploring Measures of center: mean, median, and mode
Measures of variation: Standard deviation and range
Measures of spread and relative location: minimum values, maximum value, and quartiles
Unusual values: outliers
Distribution: histograms, stem-leaf plots, and boxplots