Measures of Central Tendency

• What is the average of the data?
• Four measures, mean, median, mode, midrange.
• Mean:
  • xbar (but I can't do that in html easily) for sample
  • μ for population.
  • xbar = Σx/n
  • This is what is commonly referred to as average.
  • For the data set 7 8 8 10 12 12 12 23 25
  • xbar = 127/9 = 14.1
• Median
  • This is the middle number.
  • Arrange the numbers in order
  • If there are an odd number, the median is the middle number.
  • If there are an even number, the median is the mean of the two middle numbers.
  • For the data set 7 8 8 10 12 12 12 23 25
  • median is 12
  • For the data set 7 8 8 10 12 12 23 25
  • The median is (10+12)/2 = 11
• Mode
  • The mode is the number which occurs most frequently
  • For the data set 7 8 8 10 12 12 12 23 25
  • 12 is the mode.
  • A data set may have no mode.
  • A dataset may be multimodal.
• Midrange
  • midrange is (low+high)/2
  • For the data set 7 8 8 10 12 12 12 23 25
  • The midrange is (7+25)/2 = 16
• Discussion of which is best:
  • The mean considers all of the data
  • But can be changed by a single outlier
  • The median is less effected by outliers
  • But it only considers part of the data.
  • Mode is a good measure for marketing (think shirt size)
• Example: Problem 19 page 867
• Problem 22
• Problem 30
• Problem 36

Measures of Position

• Frequently test results are given as percentile scores.
• You may score a 57%, but be in the 80^th percentile.
• Break the population into 100 parts, based on test scores.
• percentile tells us what percent of the population is below and above a given score.
• We will deal with quartiles or 1/4^th
• Find the median, call this Q₂
• Q₁ is the median of the data less than Q₂
• Q₃ is the median of the data greater than Q₂
• Problem 48

Measures of Dispersion

• There are two measures of dispersion
• Range:
  • high - low
  • For the data set 7 8 8 10 12 12 12 23 25
  • The range is 25-7 = 18
• Standard Deviation
  • σ for populations, s for samples.
  • s = sqrt(Σ(x-xbar)²/(n-1))