PPA 415 Research Methods in Public Administration

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PPA 415 Research Methods in Public
Administration

• Lecture 3 Measures of Central Tendency

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Introduction

• The benefit of frequency distributions, graphs,
  and charts is their ability to summarize the
  overall shape of a distribution.

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Introduction

• To completely summarize a distribution, however,
  you need two additional pieces of information
  some idea of the typical or average case in the
  distribution and some idea about how much variety
  or heterogeneity there is in the distribution.
• The typical case involves measures of central
  tendency.

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Introduction

• The three most common measures of central
  tendency are the mode, median, and the mean.
• The mode is the most common score.
• The median is the middle score.
• The mean is the typical score.
• If the distribution has a single peak and is
  perfectly symmetrical, all three are the same.

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Mode

• The value that occurs most frequently.
• Best used when dealing with nominal level
  variables, although it can be used for higher
  levels of measurement.
• Limitations some distributions have no mode or
  too many modes.
• For ordinal and interval-ratio data, the mode may
  not be central to the distribution.

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Median

• Always represents the exact center of a
  distribution of scores.
• The median is the score of the case where half of
  the cases are higher and half of the cases are
  lower. If the median family income is 30,000,
  half of the families make less than 30,000 and
  half make more.

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Median

• Before finding the median, the scores must be
  arranged in order from lowest to highest or
  highest to lowest.
• When the number of cases is odd, the central case
  is the median (N1)/2 case.

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Median

• When the number of cases is even, the median is
  the arithmetic average of the two central cases
  the mean of case N/2 and case (N/21).
• The median can be calculated for ordinal and
  interval-ratio data.

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Percentiles

• The median is a subset of a larger group of
  positional measures called percentiles.
• The median is the 50th percentile (50 of the
  scores are lower.
• The 25th percentile would mean that 25 of the
  scores are lower (and 75 higher).

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Percentiles

• Deciles divide distribution into ten equal
  segments. The score at the first decile has 10
  of the scores lower, the second decile had 20 of
  the scores lower, etc.
• Quartiles divide the distribution into quarters.
• The second quartile, the fifth decile and the
  median are all the same value.

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Mean

• The calculation of the mean is straightforward
  add the scores and divide by the number of
  scores.
• Mathematical formula

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Characteristics of the Mean

• The mean is the point around which all of the
  scores (Xi) cancel out.
• The sum of the squared differences from the mean
  is smaller than the difference for any other
  point.

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Characteristics of the Mean

• Every score in the distribution affects it.
• Advantage the mean utilizes all the available
  information.
• Disadvantage a few extreme cases can make the
  mean misleading.
• Relative to the median, the mean is always pulled
  in the direction of extreme scores.
• Positive skew mean higher than the median.
• Negative skew mean lower than the median.

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Rules for the Selection of Measures of Central
Tendency

• Use the mode when
• Variables are measured at the nominal level.
• You want a quick and easy measure for ordinal or
  interval measures.
• You want to report the most common score.
• Use the median when
• Variables are measured at the ordinal level.
• Variables measured at the interval-ratio level
  have highly skewed distributions.
• You want to report the central score.

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Rules for the Selection of Measures of Central
Tendency

• Use the mean when
• Variables are measured at the interval-ratio
  level (except for highly skewed distributions).
• You want to report the most typical score. The
  mean is the fulcrum that exactly balances all
  scores.
• You anticipate additional statistical analyses.

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Grouped Median

• If you do not have raw data, but have only the
  grouped frequency distribution, assume all cases
  are evenly distributed across each interval and
  estimate the median with the following formula.

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Grouped Median
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Grouped Mean

• If you do not have raw data, but only have the
  grouped frequency distribution, assign the
  midpoint to each interval, multiply the midpoint
  by the number of cases in each interval, sum, and
  divide by the number of cases to get an estimate
  of the mean.

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Example Mode
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Example Median
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Example Mean
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Example Mean
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Example Data for Grouped Median
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Example Group Mean NES 1948-2000
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Example Group Median NES 1948-2000