Measures of Central Tendency

• The average is a number that is representative of a group of data.
• But there are many ways to measure the average.
• What is the average age?
• What is the average shoe size?
• What is the average opinion?
• We must be careful about what average we are talking about.
• The mean is the sum of the data divided by the number of pieces of data.
  • x̄ is used for samples
  • μ is used for populations.
  • This is the arithmetic mean.
  • This is the most common meaning of the word average.

  • 
        Σ x
    x̄ = ---
         n
    
    where Σx represents the sum of the data 
    and n is the number of data
        

  • The salaries for ten employees are given, find the mean.

    25,000   26,000    27,000   27,000    28,000
    29,000   30,000    32,000   36,000    65,000
    
    The sum is 325,000, so the mean is 32,500
    	

  • The mean is not a good measure for average in salary.
  • Extreme cases can throw it off, For example, the mean of the first ten numbers is 28,889
• The median is the value in the middle of a set of ranked data
  • Arrange the data in order.
  • If there is an odd number of data it is the middle number.
  • If there is an even number of data it is the average of the two middle numbers.
  • For the above case, (28,000 + 29,000)/2 = 28,500
  • The median is used for cases where there is a small extreme
• In the mean all values contribute directly to the computation.
• In the median, all values influence the computation.
• The mode is the value which occurs most frequently.
  • The definition is fine if there is a single mode.
  • Sometimes there are two values that occur most frequently, then the data is bimodal
  • Or there may be no mode.
  • In the above data set 27,000 is the mode as it occurs twice.
  • The mode is good for retail situations.
• The midrange is the mean of the low and high values.
  • midrange = (low data value + high data value)/2
  • midrange = (25,000+65,000)/2 = 45,000
  • The midrandge is good when data is fluctuating.

Measures of Position

• Measures of Position are used to compare scores across a population.
• They are very useful when the population is large.
• This includes the Quartile and Percentile rank.
• Percentile
  • This measures the position of a piece of data in one of 100 locations.
  • If a piece of data is in the 50^th percentile then it is in the middle of the data. 50% is lower and 50% is higher.
  • On a test, you might score a 75%, but be in the 50^th percentile.
  • The percent correct and the percentile are related, but not directly.
  • Frequently, you will be given your percentile scores on standardized tests.
• Quartiles
  • The population is divided into four equally sized sets.
  • The lowest 25% of values are in the first quartile.
  • The next 25% of the values are in the second quartile
  • and so on.
  • Three positions are calculated Q₁, Q₂ and Q₃.
    • Q₂ is the median of the entire data set.
    • Q₁ is the median of the lower half of the data set (not counting Q₂).
    • Q₃ is the median of the lower half of the data set (not counting Q₂).
  • For our data 27,000, 28,500, and 32,000 are Q₁, Q₂ and Q₃