Measures of Central Tendency

Discussion

• There are at least four different ways statisticians calculate an average
  • The arithmetic mean, or just mean
    • Represented by x, read x bar
    • Calculated by summing all of the data and dividing by the number of data points.
    • Σ x means to sum over all of the values
    • x = Σ x / n
    • It may not be part of the data.
    • Example 1, find the mean of 3, 7, 12, 13

      Σx = 3 + 7 + 12 + 14 = 35
      x = Σx/n = 36/4 = 9 
      	     

    • This is the most common use of the word average
  • The median is the middle data value.
    • Arrange the data in order.
    • If there is an odd number od data, it is the middle data
    • If there is an even number of data, it is the average of the two middle values.
    • Example 2: Find the median of 3, 7 , 12, 13

       
      Since there is an even number of data items 
      it is the average of the two middle values
      
      median = 7+12/2 = 19/2 = 9.5
      	     

    • Example 3, Find the median of 2 5 9 12 14

      There are an odd number of numbers so
      the median is the middle number or 9
      	     

    • Remember you need to put the data in order first
    • Example 4, find the median of 22 2 5 3 8

      First Order the data 2 3 5 8 22
      There is an odd number of data, so
      the median is the middle number or 5
      	     

  • The mode of the data, is the data item, or items that occur most often.
    • Example 5, find the mode of 2 2 4 5 6 8 9,

      the mode is 2 since it occurs most often.
      	   

    • The mode might not unique
    • Example 6, what is the mode of 3 4 5 5 6 7 8 8 9
    • This data is bimodal
    • Example 7, what is the mode of 2 4 5 6 7 8 8 9 9 9 10 11 12
  • The midrange is the average of the maximum and minimum values
    • Find the max, and min
    • Sum them and divide by two.
    • What is the midrange of 4 5 6 7 9 10 12

      4 + 12 = 16
      16/2 = 8
      	     

  • Each measure has problems. Please read the text on 781
• Measures of Position.
  • Percentile
    • Dividing the data into 100 equal parts. 1^st percentile to 99^th percentile
    • If something is in the second percentile, it is better than 2%, and worse than 98%.
    • If something is in the 3^rd percentile, it is better than 3% of the data, and not as good as 97%
    • If something is in the 78^th percentile, it is better than 77% of the data, and not as good as 21% of the data.
  • Quartile
    • Divide the data into four sections.
    • Example:

      Find the quartiles for the following data:
      
      2 3 5 6 6 7 9 12 14 16 17 19 20 21 25 28 29 31 33 35
      
      The median value is  16 + 17 /2 = 16.5
      The end of Q1 is  6+7/2  = 6.5
      The end of Q3 is 25+28/2 = 26.5
      
      Q1 is 6.5
      Q2 is 16.5
      Q3 is 26.5
      
         
      	    

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