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Introduction to Statistics

By John A. Knox and Alexandra Estrella

I - Definitions

  1. mean: the average of a set of numbers
    median: The number found in the middle when looking at the set of numbers from smallest to largest
    mode: most commonly occuring value in a set of numbers

  2. Example for mean, median, and mode:
    Sample data set:   5°     10°     10°     7°     3°    
    mean = 5 + 10 + 10 + 7 + 3 = 35, 35/5 = 7
    median = 3° 5° 7° 10° 10°
    mode = 10°

I - Advanced Definitions

  1. Variance: a measure of how data points differ from the mean

  2. Data Set 1: 3, 5, 7, 10, 10
    Data Set 2: 7, 7, 7, 7, 7

    - Data Set 1: mean = 7, median = 7
    - Data Set 2: mean = 7, median = 7

    But we know that the two data sets are not identical! The variance shows how they are different.
  3. Formula for variance:
    S2 = (1/(N-1) × (the sum of (each data point - mean)2)

    Formula applied to data set 1:
    S2data set 1 = ( 1/(5-1) ) × ( (3-7)2 + (5-7)2 + (7-7)2 + (10-7)2 + (10-7)2 )
    note: N = number of data points

    S2data set 1 = 1/4 × { (-4)2 + (-2)2 + (0)2 + (3)2 + (3)2 }

    S2data set 1 = 1/4 × ( 16 + 4 + 0 + 9 + 9 )

    S2 = 1/4 × 38

    S2 = 38/4

    S2 = 9.5 for data set 1

    Formula applied to data set 2:
    S2data set 2 = (1/4) × (0 + 0 + 0 + 0 + 0)

    S2 = 1/4(0)

    S2 = 0 for data set 2

  4. Standard Deviation is "S," the square root of the variance:

    S = Square root of [(1/(N-1)) × (sum of (each data point - mean)2)]
    • measure of the difference from the mean. Large S means the data is spread widely around the mean.
    • units are the same as the data itself
  5. S for data set 1 above is 3.08. S for data set 2 above is 0.

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