Standard deviation question+answer

Question

The question:

 

The ages, x years, of 150 cars are summarised by

 

 

∑x = 645 and

∑x² = 8287.5

 

Find ∑(x-y)² where y denotes the mean of x.

 

 

The answer:

 

y= 4.3

 

standard deviation = √ (8287.5/150 − 4.32²) = √36.76 = 6.063....

 

∑(x − y)² = 150 × 6.063²

 

= 5514 (5510)

 

 

 

What I need to know is how to get this answer?

I only know that you have to find the mean but I don't know what to do next.

Why do you have to use the standard deviation formula?

And why do you then multiply 150 by 6.0632?

Answer 1

You can approach this as playing with the equations you've been given until things work out. I'll approach it from a slightly different perspective. I'll use n to represent the number of cars, 150. First, by the definition of the summation operator (and that y is the same, not indexed).

 

Σ(x-y)² = (x₁ - y)² + (x₂ - y)² + ... + (x_n - y)²

 

Each of those terms on the right-hand side can be expanded.

 

Σ(x-y)² = x₁² - 2x₁y + y² + x₂² - 2x₂y + y² + ... + x_n² - 2x_ny + y²

 

This can be rewritten as:

 

Σ(x-y)² = (Σx²) - 2y(Σx) + ny²

 

The mean is simply the total of the ages divided by the number of cars.

 

y = 645 / 150 = 4.3

 

So,

 

Σ(x-y)² = 8287.5 - 2(4.3)(645) + (150)(4.3)² = 5514

 

Which rounds to 5510.

 

Answer 2

Hang on - it's a bit of a ride.

 

The mean (μ) is:

 

     μ = (Σx)/n = 645/150 = 4.3

 

Standard deviation is:

 

     σ² = (1/n)Σ(x-μ)²

 

Now (x-μ)² = x² - 2μx + μ², so:

 

     σ² = (1/n)Σ(x-μ)² = (1/n)(∑x²/n - 2μ∑x/n + Σμ²/n)

         = 8287.5/150 - (2)(4.3)(645)/150 + (150)(4.3)²/150

 

Now 645/150 = μ = 4.3, so:

 

      σ² = 8287.5/150 - (2)(4.3)² + (4.3)² = 8287.5/150 - (4.3)² = 36.76

Finally:

 

     σ² = (1/n)Σ(x-μ)²

     n* σ² = Σ(x-μ)²

     (150)(36.76) = Σ(x-μ)² = 5514

 

Questions?