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PPA 415 – Research Methods in Public Administration. Lecture 3 – Measures of Central Tendency. Introduction. The benefit of frequency distributions, graphs, and charts is their ability to summarize the overall shape of a distribution. Introduction.
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PPA 415 – Research Methods in Public Administration Lecture 3 – Measures of Central Tendency
Introduction • The benefit of frequency distributions, graphs, and charts is their ability to summarize the overall shape of a distribution.
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.
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.
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.
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.
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 [(N+1)/2 case].
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/2+1)]. • The median can be calculated for ordinal and interval-ratio data.
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).
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.
Mean • The calculation of the mean is straightforward: add the scores and divide by the number of scores. • Mathematical formula:
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.
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.
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.
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.
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.
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.
