Measures of Central Tendency

• What is the "average" data?
  • We will use four terms
  • They each measure different things.
  • They each can be a useful measurement of the average.
• The arithmetic mean or mean
  • written $\bar{x}$ for a sample
  • Written μ for a population.
  • $\bar{x} = \frac{Σ x}{n}$
  • This is the most common use of the word "average".
  • It uses all of the data in the computation.
  • Every piece of data has an impact on the mean.
• The median is the middle piece of data.
  • If there are an odd number of data points
  • It is the average of the two middle data points if they are even.
  • To find the median
    • Arrange the data in order
    • if n is odd, take the middle number
    • If n is even the median is $\frac{x_{n/2-1} + x_{n/2+1}}{2}$
    • All numbers sort of have an impact on the median.
    • It ignores the values of extreme numbers (or outliers)
    • Frequently used for income.
• The mode is the number which occurs the most frequently.
  • In some books there can be only one mode.
  • In others there can be multiple modes.
  • But only if a number repeats.
• The midrange is a quick estimation of the average
  • $midrange = \frac{lowest value + highest value}{2}$
• Measurements of position
  • Percentiles and quartiles tell us the position of a piece of data.
  • $Q_2$ is the median
  • $Q_1$ is the median of the data less than $Q_2$
  • $Q_3$ is the median of the data greater than $Q_2$
• Do some problems Page 753 11-21, 38-46, 47-48