Understanding Central Tendency Properties (Mean, Median and Mode) in Statistics

11.02.2011 | Author: albertfrank | Posted in Reference and Education
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In Statistics, Measures of Central Tendency are numerical values that locate, in some sense, the centre of a set of data. The term average is often associated with all measures of central tendency.

Mean

  1. Measure of central tendency
  2. Most common measure
  3. Acts as ‘balance point’
  4. Affected by extreme values (‘outliers’)
  5. Formula (sample mean)

Raw Data:       10.3    4.9       8.9       11.7    6.3       7.7

X = ∑ x/n = x1 + x2 + x3 …..xn /n

Here in this case 10.3+ 4.9+ 8.9 + 11.7 +6.3 + 7.7/6 = 8.3

The mean is 8.3

Median

  1. Measure of central tendency
  2. Middle value in ordered sequence
  • If n is odd, middle value of sequence
  • If n is even, average of 2 middle values
  1. Position of median in sequence

Positioning Point = n+1/2

  1. Not affected by extreme values

Calculating Median from an Odd-sized example

  • Raw Data:       24.1    22.6    21.5    23.7    22.6
  • Ordered:         21.5    22.6    22.6 23.7    24.1
  • Position:         1          2          3 4          5

Positioning Point = n+1/2 = 5+1/2 = 3

Median = 22.6

Median Example from an Even-Sized Sample

  • Raw Data:       10.3    4.9       8.9       11.7    6.3       7.7
  • Ordered:         4.9       6.3       7.7 8.9 10.3    11.7
  • Position:         1          2          3 4 5          6

Positioning Point = n+1/2 = 6+1/2 = 3.5

Median = 7.7 + 8.9/2 = 8.3

Mode

  1. Measure of central tendency
  2. Value that occurs most often
  3. Not affected by extreme values
  4. May be no mode or several modes
  5. May be used for quantitative or qualitative data
  • No Mode
    Raw Data:       10.3    4.9       8.9       11.7    6.3       7.7

    • One Mode
      Raw Data:       6.3       4.9 8.9       6.3 4.9      4.9
    • More Than 1 Mode
      Raw Data:       21 28        28 41        43 43

You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues:  17, 16, 21, 18, 13, 16, 12, 11.

Describe the stock prices
in terms of central tendency.

Central Tendency Solution

Mean

X = ∑x/n = 17 + 16 + 21 + 18 + 13 + 16 + 12 + 11/8 = 15.5

Median

  • Raw Data:       17        16        21        18        13        16        12        11
  • Ordered:         11        12        13        16        16        17        18        21
  • Position:         1          2          3          4          5          6          7          8

Positioning Point = n+1/2 = 8+1/2 = 4.5

Median = 16+16/2 = 16

Mode

Raw Data:  17, 16, 21, 18, 13, 16, 12, 11
Mode = 16

Central Tendency Measures
Measure Formula Description
Mean ∑x/n Balance Point
Median n+1/2 Position Middle Value when ordered
Mode None Most frequent

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