Describing distributions with numbers

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Chapter 2
Describing distributions with numbers
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Chapter Outline

• 1. Measuring center the mean
• 2. Measuring center the median
• 3. Comparing the mean and the median
• 4. Measuring spread the quartiles
• 5. The five-number summary and boxplots
• 6. Measuring spread the standard deviation
• 7. Choosing measures of center and spread

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Measuring center the mean

• Notation
• It is simply the ordinary arithmetic average.
• Suppose that we have n observations (data size,
  number of individuals).
• Observations are denoted as x1, x2, x3, xn.

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Measuring center the mean

• How to get ?
• Example 2.1 (P.33)

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Measuring center the median

• Notation M
• Median M is the midpoint of a distribution? half
  the observations are smaller than M and the
  other half are larger than M.

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Measuring center the median

• How to find M?
• 1. Sort all observations in increasing order
  (This step is important!!!)
• 2. If n is odd, observation is M. if n is
  even, average of two center values is M.
• Note that is the location of the median
  in the ordered list, not the median value.

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Measuring center the median

• Examples
• Case 1. 11, 21, 13, 24, 15, 26, 17
• Case 2. 11, 21, 13, 24, 15, 26
• Example 2.2, 2.3 (P.35)

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Mean vs. Median

• Median is more resistant than the mean.
• The mean and median of a symmetric distribution
  are close together. If the distribution is
  exactly symmetric, the mean and median are
  exactly the same. In a skewed distribution, the
  mean is farther out in the long tail than is the
  median.
• Example
• 1, 2, 3, 4, 5, 6, 10000

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Inference

• Strongly skewed distributions are reported with
  median than the mean.

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Measuring Spread The Quartiles

• The quartiles mark out the middle half of the
  distribution.

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Calculating the Quartiles

• Step1.
• Arrange the observations in increasing order
  and locate the median M in the ordered list of
  observations.
• Step2.
• The first quartile Q1 is the median of the
  observations whose position in the ordered list
  is to the left of the location of the overall
  median.
• Step3.
• The third quartile Q3 is the median of the
  observations whose position in the ordered list
  is to the right of the location of the overall
  median.

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Measuring spread the quartiles

• Example 2.4 (P. 37)
• Example 2.5 (P. 38)
• Note
• (1) It is important to sort data first before
• we try to find quartiles!
• (2) Quartiles are resistant.

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The five-number summary and boxplots

• The five-number summary
• Minimum, Q1, M, Q3, Maximum.
• Boxplot is a graph of five number
• summary.
• Boxplots are most useful for side-by-side
  comparison of several distributions.

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Boxplot

• 1. A boxplot is a graph of the five-number
  summary
• 2. A central box spans the quartiles
• 3. A line in the box marks the median
• 4. Lines extended from the box out to the minimum
  and maximum
• 5. Range maximum - minimum

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The five-number summary and boxplot

• Figure 2.2(P.39) side-by-side boxplots comparing
  the distributions of earning for two levels of
  education.

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The five-number summary and boxplots
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Inference

• Boxplot also gives an indication of the symmetry
  or skewness of a distribution.
• -- In a symmetric distribution Q1 and Q3
• are equally distant from the median,
• but in case of right skewed one the
• third quartile would be further above the
• median than the first quartile bellow it.

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Measuring spread the standard deviation

• It says how far the observations are from their
  mean.
• The variance s2 of a set of observations is an
  average of the squares of the deviations of the
  observations from their mean.
• Notation s2 for variance and s for standard
  deviation

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Why (n-1) ?

• As the sum of the deviations
• always equals 0, so the knowledge of (n-1) of
  them determines the last one.
• --- Only (n-1) of the squared deviations are
  variable but not the last one, so we average by
  dividing the total by (n-1).
• The number (n-1) is called the degrees of
  freedom of the variance or standard deviation

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Measuring spread the standard deviation

• To find the variance and the standard deviation
• 1. Find the mean of the data set
• 2. Subtract the mean from each number (we call
  that deviation)
• 3. Square each result
• 4. Sum all the square
• 5. Divide the sum of square by n-1, where n is
  the number of all observations. Now you get
  variance
• 6. Standard deviation is just the positive square
  root of the variance.

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Measuring spread the standard deviation

• Example 2.6 (P.42)

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Properties of s2 and s

• s measures spread about the mean and should be
  used only when the mean is chosen as the measure
  of center.
• s 0 and s0 only when each of the observation
  values does not differ from each other.
• S is not resistant.

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Choosing measures of center and spread

• With a skewed distribution or with a distribution
  with extreme outliers, five-number summary is
  better.
• With a symmetric distribution (without outliers),
  mean and standard deviation are better.