William W. answered • 05/23/19
Math and science made easy - learn from a retired engineer
The equation for mean (μ) is:
μ = (Σx)/n
The equation for standard deviation (σ) is:
σ = √(∑(x - μ)/n)
In this case μ must be 4 since there are only 3 numbers and to get a mean of 4 the middle number must be equal to 4 with the other two numbers being ± the same thing.
Let the three numbers be x1, x2, and x3. We know that x2 = 4. We know n = 3, μ = 4 and σ = 4.
Writing out the equations in terms of x1, x2, and x3, we get:
Mean:
4 = (x1 + x2 + x3)/3
12 = x1 + 4 + x3
x1 + x3 = 8
x1 = 8 - x3
Standard Deviation:
4 = √[(x1 - 4)2+ (x2 - 4)2 + (x3 - 4)2]/3
Squaring both sides gives:
16 = [(x1 - 4)2+ (4 - 4)2 + (x3 - 4)2]/3
48 = (x1 - 4)2+ (0)2 + (x3 - 4)2
Multiplying out gives:
x12 - 8x1 + 16 + x32 - 8x3 +16 - 48 = 0
x12 - 8x1 + x32 - 8x3 - 16 = 0
Plug in the equation from the mean (x1 = 8 - x3) which gives:
(8 - x3)2 - 8(8 - x3) + x32 - 8x3 - 16 = 0
Multiplying out and simplifying gives:
2x32 - 16x3 -16 = 0
Dividing by 2 gives:
x32 - 8x3 - 8 = 0
Solve by using the quadratic formula:
x3 = 4 ± 2√6
So the numbers are:
x1 = 4 - 2√6
x2 = 4
x3 = 4 + 2√6
