Measure For Center

31 Jul 2022 in Study / Statistics on Measure for center

mean, median, mode

• Mean
• Median
• Mode

Mean

• Population mean : calculation of mean in population
  
  

Example
  Population A
  10, 15, 28, 58, 59, 67, 89, 104, 14, 57, 65, 105

  μ = (10 + 15 + 28 + 58 + 59 + 67 + 89 + 104 + 14 + 57 + 65 + 105) / 12 = 55.91667

• Sample mean : calculation of mean in a specific sample
  
  

Example
  Sample (from population A) : n = 5
  58, 67, 89, 14, 57

  Sample mean (X-bar) = (58 + 67 + 89 + 14 + 57) / 5 = 285

• Trimmed mean (절사 평균)
  • Data should be sorted
  • α% trimmed mean : After removing (n x 0.0α) data from each side, calculate the mean of data
  • If (n x 0.0α) is not integer (m.xxx) : remove m data from each side → calculate the mean, but each end data should be multiplied by 0.xxx
    

Example
  Population A
  10, 15, 28, 58, 59, 67, 89, 104, 14, 57, 65, 105

  10% trimmed mean
  → sorting : 10 14 15 28 57 58 59 65 67 89 104 105
  → (14 x 0.2) + 15 + 28 + 57 + 58 + 59 + 65 + 67 + 89 + (104 x 0.2) / 10 = 461.6

Properties Of Mean

• Sensitive to outlier
  → can use trimmed mean instead of general mean

Median

• Data should be sorted
• Middle position of data
• if n is odd : (n + 1)/2
• if n is even : average for n/2 and n/2 + 1
• not sensitive to outlier
  

Example 1
  Population A
  10, 15, 28, 58, 59, 67, 89, 104, 14, 57, 65, 105

  → sorting : 10 14 15 28 57 58 59 65 67 89 104 105
  Median = (58 + 59)/2 = 58.5

Example 2
  Sample (from population A) : n = 5
  58, 67, 89, 14, 57

  → sorting : 14 57 58 67 89
  Median = 58/2 = 29

Mode

• Data that have the highest frequency
  

Example
  Population B
  8, 9, 20, 9, 8, 2, 35, 35, 9, 50, 9, 11, 9

  Mode = 9

Estimation For Distribution

• Mean is sensitive to outlier

• Symmetric
  
  

• Skewed to the right
  
  

• Skewed to the left