My interest in statistics is definitely pedestrian: I don't sit around doing statistical analysis with SPSS or SAS.

Assume a set of numbers, X:

X = {x1, x2, ..., xN}

Large naturally occurring sets of random numbers tend to have a normal distribution (aka Gaussian distribution; the famous bell curve).

The mode is the most frequently occurring value in the set of numbers, X. EG:

If X = {5, 6, 8, 9,  9}, then 9 is the mode.

The median is the nth of set of 2n values most frequently occurring value in the set of random variables, X. EG:

If X = {5, 6, 8, 9,  9}, then 8 is the mode.

The mean (aka average), x̄, is the sum of the set divided by the count of the set:

x̄ = (x1 + x2 + ... + xN) / N

If X = {5, 6, 8, 9,  9}, the mean is:
x̄ = (5 + 6 + 8 + 9 +9) / 5
x̄ = 37 / 5
x̄ = 7.4

The standard deviation, σ, is the "average" of the differences of each number in the set from the mean. Standard deviation is the root mean square deviation from the mean.

σ = sqrt( summation( square( xi - x̄ ) ) / N )

If X = {5, 6, 8, 9, 9}, then the standard deviation can be calculated:

  1. Find the differences from the mean for each number in the set:
    {5-7.4, 6-7.4, 8-7.4, 9-7.4, 9-7.4} =
    {-2.4, -1.4, -0.6, 1.6, 1.6}
  2. Square each number in the set:
    {-2.4*-2.4, -1.4*-1.4, -0.6*-0.6, 1.6*1.6, 1.6*1.6} =
    {5.76, 1.96, 0.36, 2.56, 2.56}
  3. Sum the set:
    5.76 + 1.96 + 0.36 + 2.56 + 2.56 =
    13.2
  4. Find the square root of that:
    sqrt(13.2) =
    3.63318

     

(The variance, σ2, is simply the square of the standard deviation.)

By math magic (which I won't show here), for a normal distribution, the numbers in a set will be distributed as follows:

See also this nice chart from Statistics [W]:

The normal distribution

A standard error uses the standard deviation in relation to the sample size. That is the greater the sample, the smaller the standard error.

standard error = σ / sqrt(samples)

If X = {5, 6, 8, 9, 9}, then the standard error can be found:
3.63318 / sqrt(5) =
3.63318 / 2.236067977 =
1.624807681

Page Modified: (Hand noted: 2007-10-23 23:39:34Z) (Auto noted: 2007-11-17 06:23:22Z)