**HP 71B: Population Deviation Derivation and Program**

**Derivation**

The standard formula for the population deviation for a set of statistical data is given by:

σx = √( Σ(x - μ)^2 / n)

where μ is the mean, where μ = Σx / n

Σx = the sum of all the data points

n = the number of data points

If this function needs to be programmed, using the definition as will probably require the use of two FOR structures: one for the mean, and one for the deviation. Is there a way to use a formula where a FOR structure can be saved?

Turns out, the answer is yes.

σx = √( Σ(x - μ)^2 / n)

= √( Σ(x^2 - 2*x*μ + μ^2) / n)

Note that:

Σ( f(x) + g(x) ) = Σ f(x) + Σ g(x)

For a constant, a: Σ( a * f(x) ) = a * Σ f(x)

And: Σ a = a * n (sum from 1 to n)

Then:

σx = √( Σ(x - μ)^2 / n)

= √( Σ(x^2 - 2*x*μ + μ^2) / n)

= √( (Σ(x^2) - 2*μ*Σx + Σ( μ^2 )) / n)

= √( (Σ(x^2) - 2*μ*Σx + n*μ^2) / n)

= √( (Σ(x^2) - 2*Σx*Σx/n + n*(Σx)^2/n^2 ) / n)

= √( (Σ(x^2) - 2*(Σx)^2/n + (Σx)^2/n) / n)

= √( (Σ(x^2) - (Σx)^2/n ) / n)

The above formula allows to use a sum and sum of square of data points in calculating population deviation, eliminating the need for an additional FOR structure.

Standard deviation follows a similar formula:

sx = √( (Σ(x^2) - (Σx)^2/n ) / (n - 1))

**HP 71B Program: Population Deviation**

File: PDEV

20 N=0

22 A=0

24 B=0

30 INPUT "X? ";X

40 N=N+1

50 A=A+X

60 B=B+X^2

70 INPUT "DONE? NO=0"; C

80 IF C=0 THEN 30

90 M=A/N

95 S=SQR((B-A^2/N)/N)

100 DISP "MEAN= ";M

105 PAUSE

110 DISP "PDEV= ";S

120 END

**Example**

Data Set: 150, 178, 293, 426, 508, 793, 1024

MEAN = 481.7142857

PDEV = 300.6320553

Eddie

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