Question

Calculate the the averages of numbers with standard deviations?

Like say I have you 3 values:

89 +\- 11; 73.4 +\- 18; 81.6 +\- 7.53

Would you be able to calculate the mean and the resultant standard deviation?

And show me please how you calculate them

Answer 1

I'll assume the numbers you gave are: 89, 73.4 and 81.6

First, the mean is calculated by the formula represented by ($\mu$): adding all the numbers dividing by the total numbers:

In this case, the sum of numbers is 244. And dividing the sum with 3 (total numbers)

Mean ($\mu$) is 244/3 = 81.33 (approx)

The Standard Deviation is a measure of how spread out numbers are. It is represented by $\sigma$ (the Greek letter sigma).

This is the formula for Standard Deviation

$\sigma = \sqrt{1/N * \sum_{i=1}^{N}(x_{i} - \mu^{2})}$

To calculate the standard deviation of those numbers:

1. Work out the Mean (the simple average of the numbers)

2. Then for each number: subtract the Mean and square the result

3. Then work out the mean of those squared differences.

4. Take the square root of that and we are done!

Step 1: Calculate Mean i.e., 81.33 (from above)

Step 2: Then for each number: subtract the Mean and square the result

i.e., the part of the formula:  $(x_{i} - \mu^{2})$. So what is x_i ? They are the individual x values 89, 73.4 and 81.6. In other words x₁ = 89, x₂ = 73.4, x₃ = 81.6, etc.

So, do it like this

$(89-81.33)^{2}$ = $(7.67)^{2}$ = 58.83

$(73.4-81.33)^{2}$ = $(-7.93)^{2}$ = 62.88

$(81.6-81.33)^{2}$ = $(0.27)^{2}$ = 0.0729

Step 3: Then work out the mean of those squared differences

To work out the mean, add up all the values then divide by how many.

First add up all the values from the previous step.

i.e., $\sum_{i=1}^{N}(x_{i} - \mu^{2})$

Which means: Sum all values from (x₁-81.33)² to (x_N-81.33)²

We already calculated (x₁-81.33)² = 58.83 etc. in the previous step, so just sum them up:

58.83 + 62.88 + 0.0729 = 121.7829

But that isn't the mean yet, we need to divide by how many, which is done by multiplying by 1/N (the same as dividing by N):

$1/N * \sum_{i=1}^{N}(x_{i} - \mu ^{2})$

Mean of squared differences = (1/3) * 121.7829 = 40.5943 (this value is called the variance)

Step 4: Take the square root of that:

$\sigma = \sqrt{1/N * \sum_{i=1}^{N}(x_{i} - \mu^{2})}$

$\sigma = \sqrt{40.5943}$ = 6.37

DONE!