Measures of Central Tendency

• What is the "average" of a set of data.
• Mean or arithmetic mean, (xbar)
  • sometimes μ
  • Σ₁¹⁰i = 1 + 2 + 3 + ... + 10
  • (xbar) = Σx/n, where n is the number of data items.
  • Data: 1 2 3 6 7 8 10, xbar = (1+2+3+6+7+8+10)/7 = 37/7 = 5.3
• Median or middle number
  • Arrange the data low to high.
  • If there is an odd number of data, find the middle one.
  • If there is an even number, take the arithmetic mean of the two middle.
  • 1 2 3 7 9 , the median is 3
  • 1 2 3 4 7 9, the median is 3+4/2 = 3.5
• Mode or the number that occurs most frequently.
  • Order the data
  • Look for the most frequently occurring.
  • If multiple data occur most frequently, they are the mode
  • If no data occurs most frequently there is no mode.
  • This definition may be different in different books.
  • 1 2 2 3 4 5, 2 is the mode
  • 1 2 2 3 3 4 5, 2 and 3 are the mode
  • 1 2 3 4 5, there is no mode.
• Midrange the middle of the range.
  • (Low + High) / 2
  • 1 2 3 5 7 9, midrange = (1+9/2) = 5
• When to use
  • Median is most common.
  • It assumes all data is of equal weight.
  • Median is used when there are a few outliers
  • Median income ignores the riches 10%
  • Mode is important when selling things
  • Midrange is sometimes used when the data is constantly changing.
  • High and low temp for example.
• Measures of position.
  • Used for large data sets.
  • For example standardized tests.
  • Used to describe where data is in the data set.
  • Percentile:
    • The data is partitioned into 100 bins.
    • The percentile score for a given piece of data is the corresponding bin.
    • If you score in the 75 percentile
      • You did better than 75% of the people taking the test.
      • You did worse than (100-75) or 25% of the people taking the test.
  • Quartile
    • Q₁ Q₂ Q₃ correspond to the 25^th, 50^th and 75^th percentile.
    • Q₂ is the median.
    • Q₁ is the median of the lower half of the data.
    • Q₂ is the median of the upper half of the data.

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