section 3.1 measures of center. mean (average) sample mean
TRANSCRIPT
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Section 3.1
Measures of Center
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Mean (Average)
The mean of a data set is the sum of the observations divided by the number of observations.
Symbolically, = , Where is the mean of the set of values,
is the sum of all the values, and
is the number of values.
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Sample Mean
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Example
The mean or average of 10, 20, 5, 25, 20 and 10 is calculated as: Mean
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Median
The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.
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Median of a Data Set
Arrange the data in increasing order.
• If the number of observations is odd, then the median is
the observation exactly in the middle of the ordered list.
• If the number of observations is even, then the median is
the mean of the two middle observations in the ordered list.
In both cases, if we let n denote the number of observations,
then the median is at position (n + 1) / 2 in the ordered list.
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Example• The marks of seven students in a test that had a
maximum possible mark of 50 are given below: 40 35 37 30 38 36 35
Find the median of this set of data values.Solution:• Arrange the data values in order from the lowest
value to the highest value: 30 35 35 36 37 38 40• The fourth data value, 36, is the middle value in
this arrangement.Therefore, the median is 36.
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Mode
The mode of a set of data values is the value(s) that occurs most often.
The mode has applications in manufacturing.For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others.
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Mode of a Data Set
• Find the frequency of each value in the data set.
• If no value occurs more than once, then the data set has no mode.
• Otherwise, any value that occurs with the greatest frequency is a mode of the data set.
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ExampleData Set IIData Set I
Means, medians, and modes of salaries in Data Set I and Data Set II
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Figure 3.1
Relative positions of the mean and median for (a) right-skewed, (b) symmetric, and (c) left-skewed
distributions
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Section 3.2
Measures of Variation
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Figure 3.2
Five starting players on two basketball teams
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Figure 3.3Shortest and tallest starting players on the teams
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Range of a Data Set
The range of a data set is given by the formula
Range = Max – Min,
where Max and Min denote the maximum and minimum observations, respectively.
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Data sets that have different variation
Means and standard deviations of the data sets in Table above.
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Section 3.3
• The Five-Number Summary• Boxplots
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QuartilesArrange the data in increasing order and determine the
median.
• The first quartile is the median of the part of the entire
data set that lies at or below the median of the entire data
set.
• The second quartile is the median of the entire data set.
• The third quartile is the median of the part of the entire
data set that lies at or above the median of the entire data
set.
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Interquartile Range
The interquartile range, or IQR, is the difference between the first and third quartiles;
that is, IQR = Q3 – Q1.
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Five-Number Summary
The five-number summary of a data set is
Min, Q1, Q2, Q3, Max.
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Procedure 3.1
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Figure 3.4
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Section 3.4
• Descriptive Measures for Populations• Use of Samples
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Population and sample for bolt diameters
Parameter and Statistic
Parameter: A descriptive measure for a population.
Statistic: A descriptive measure for a sample.
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z-Score
For an observed value of a variable x, the corresponding value of the standardized variable z is called the z-score of the observation. The term standard score is often used instead of z-score.