Measures of Central Tendency

• At least four words mean average.
  • Mean - the arithmetic mean.
  • Median - the middle number
  • Mode - the most frequent number.
  • Midrange - The average of the extremes.
• If a sample has n items, x₁ through x_n
  • The mean (x with a bar over it) = Σx_i/n
  • (μ for populations, xbar for samples)
  • This is the most common use of the word average.
  • Use data from #47. Find Mean.
  • The mean can be thrown off by a single large (or small, outlier)
  • Consider 1, 2, 3, 4, 10,000,000
• The median is the middle value (or the average of the middle values) of the data.
  • First arrange the data in order.
  • If there are an odd number of data, pick the middle one n/2 rounded up.
  • do 1, 4, 6, 8, 9
  • If there is an even number of data, average the two middle values.
  • do 1, 4, 5, 6, 8, 9
  • With the mode, 1/2 the population is above, and 1/2 is below.
  • The median is not thrown off by outliers
• Closely related to the median is the concept of measure of position.
  • Percentile describes the position of a piece of data.
  • If you score in the 72^nd percentile, you did better than 72% of the people who took the test and well well than 28% of the people who took the test.
  • Percentile divides the data into 100 bins.
  • Quartile divides the population into 4 bins.
    • The second quartile, or Q₂ is the median.
    • The first quartile, or Q₁ is the median of the data less than Q₂
    • The third quartile is the median of the data greater than Q₂
  • To find the quartiles:
    • Order the data.
    • Find the median, this is Q₂.
    • Find the median of the lower half, this is Q₁
• The mode is the number which occurs most frequently.
  • Data can have several modes, or none at all.
• Midrange = (highest value + lowest value) /2

Measures of Dispersion

• How much does the data differ?
• range = high - low
• standard deviation (s for samples, σ for populations)
• s = sqrt(Σ(x-xbar)²/(n-1))