SLIDES PREPARED

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STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

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Chapter 2

• Data Description Numerical Measures of Central
  Tendency for Ungrouped Univariate Data

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Outline

• Do I Need to Read This Chapter?
• 2-1 The Mean
• 2-2 The Median
• 2-3 The Mode
• 2-4 Shapes (Skewness)
• Its a Wrap

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Objectives

• Introduction of some basic statistical
  measurements of central tendency.
• Introduction of some graphical displays to
  explain these measures of central tendency and to
  explain certain shapes.

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Introduction

• A measure of central tendency for a collection of
  data values is a number that is meant to convey
  the idea of centralness for the data set.
• The most commonly used measures of central
  tendency for sample data are the mean, median,
  and mode.

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2-1 The Mean

• Explanation of the term mean The mean of a set
  of numerical (data) values is the (arithmetic)
  average for the set of values.
• NOTE When computing the value of the mean, the
  data values can be population values or sample
  values.
• Hence we can compute either the population mean
  or the sample mean

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2-1 The Mean

• Explanation of the term population mean If the
  numerical values are from an entire population,
  then the mean of these values is called the
  population mean.
• NOTATION The population mean is usually denoted
  by the Greek letter µ (read as mu).

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2-1 The Mean

• Explanation of the term sample mean If the
  numerical values are from a sample, then the mean
  of these values is called the sample mean.
• NOTATION The sample mean is usually denoted by
  (read as x-bar).

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The Mean -- Example

• Example What is the mean of the following 11
  sample values?
• 3 8 6 14 0 -4
• 0 12 -7 0 -10

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The Mean -- Example (Continued)

• Solution

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The Mean Why do we use the MEAN as a measure of
the center of a set of values?
Dot Plot with the data values and where the mean
is located.
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The Mean Why do we use the MEAN as a measure
of the center of a set of values?

• Next we will compute the deviations from the
  sample mean (of 2 for this example).
• These values are shown on the next slide.

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The Mean Why do we use the MEAN as a measure of
the center of a set of values?
NOTE
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The Mean Why do we use the MEAN as a measure of
the center of a set of values?

Display of the deviations from the sample
mean.
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The Mean Why do we use the MEAN as a measure of
the center of a set of values?
Sum of negative Deviations 33
Sum of positive Deviations 33
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The Mean Why do we use the MEAN as a measure of
the center of a set of values?

• NOTE
• When the deviations on the left and on the right
  of the sample mean are added, disregarding the
  the sign of the deviations, we see that when the
  balancing point is the sample mean, then these
  sums are equal in absolute value.

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The Mean Why do we use the MEAN as a measure of
the center of a set of values?

• NOTE
• Thus, the mean is that central point where the
  sum of the negative deviations (absolute value)
  from the mean and the sum of the positive
  deviations from the mean are equal.

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The Mean Why do we use the MEAN as a measure of
the center of a set of values?
NOTE This is why the mean is considered a
measure of central tendency.
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Quick Tip

• When a data set has a large number of values, we
  sometimes summarize it as a frequency table. The
  frequencies represent the number of times each
  value occurs.

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• Example The at-rest pulse rate for 16 athletes
  at a meet were 57, 57, 56, 57, 58, 56, 54, 64,
  53, 54, 54, 55, 57, 55, 60, and 58. Summarize
  the information with an ungrouped frequency
  distribution.

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Quantitative Frequency Distributions Ungrouped
-- Example Continued
Note The (ungrouped) classes are the observed
values themselves.
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Quantitative Frequency Distributions Ungrouped
-- Example Continued
Find the mean for frequency table.
Solution
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2-2 The Median

• Explanation of the term median The median of a
  set of numerical (data) values is that numerical
  value in the middle when the data set is arranged
  in order.
• NOTE When computing the value of the median,
  the data values can be population values or
  sample values.
• Hence we can compute either the population median
  or the sample median.

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Quick Tip

• When the number of values in the data set is odd,
  the median will be the middle value in the
  ordered array.
• When the number of values in the data set is
  even, the median will be the average of the two
  middle values in the ordered array.

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The Median -- Example

• Example What is the median for the following
  sample values?
• 3 8 6 14 0 -4
• 2 12 -7 -1 -10

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The Median -- Example (Continued)

• Solution First of all, we need to arrange the
  data set in order. The ordered set is
• -10 -7 -4 -1 0 2 3 6 8 12 14

6th value
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The Median -- Example (Continued)

• Solution (Continued) Since the number of values
  is odd, the median will be found in the 6th
  position in the ordered set.
• Thus, the value of the median is 2.

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The Median Why do we use the MEDIAN as a
measure of central tendency?
Dot Plot with the data values and where the
median is located.
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The Median Why do we use the MEDIAN as a
measure of central tendency?
A list of the values that are above the median
and below the median is given below.
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The Median Why do we use the MEDIAN as a
measure of central tendency?
When the values above and below the median are
counted, we see that if the balancing point is
the sample median,
Median 2
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The Median Why do we use the MEDIAN as a
measure of central tendency?
Observe that there are the same number of values
above the median as there are below the
median. This is why the median is considered as
a measure of central tendency.
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The Median -- Example

• Example Find the median age for the following
  eight college students.
• 23 19 32 25 26 22 24 20

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The Median Example (continued)

• Example First we have to order the values as
  shown below.
• 19 20 22 23 24 25 26 32

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The Median Example (continued)

• Example Since there is an even number of ages,
  the median will be the average of the two middle
  values.
• Thus, median (23 24)/2 23.5.

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2-3 The Mode

• Explanation of the term mode The mode of a set
  of numerical (data) values is the most frequently
  occurring value in the data set.

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Quick Tip

• If all the elements in the data set have the same
  frequency of occurrence, then the data set is
  said to have no mode.
• Example of data set with no mode.

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Quick Tip

• If the data set has one value that occurs more
  frequently than the rest of the values, then the
  data set is said to be unimodal.

Example of A Unimodal Data set.
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Quick Tip

• If two data values in the set are tied for the
  highest frequency of occurrence, then the data
  set is said to be bimodal.

Example of a bimodal set of data.
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2-4 Shapes (Skewness)

• Positively skewed Distribution
• In a positively skewed distribution, most of the
  data values fall to the left of the mean, and the
  tail of the distribution is to the right.
• The mean is to the right of the median and the
  mode is to the left of the median.

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Positively skewed Distribution
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Negatively Skewed Distribution

• In a negatively skewed distribution, most of the
  data values fall to the right of the mean, and
  the tail of the distribution is to the left.
• The mean is to the left of the median and the
  mode is to the right of the median.

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Negatively skewed Distribution
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Symmetrical Distribution

• In a symmetrical distribution, the data values
  are evenly distributed on both sides of the mean.
• When the distribution is unimodal, the mean, the
  median, and the mode are all equal to one another
  and are located at the center of the
  distribution.

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Symmetrical Distribution