# Thread: Estimate mean and std dev?

1. ## Estimate mean and std dev?

A random sample of 13 observations from a population yielded $\displaystyle \sum$ x=488.8 and $\displaystyle \sum$ $\displaystyle x^2$ = 18950.2. Estimate $\displaystyle \mu$ and $\displaystyle \sigma.$

I can't find this in my notes, I think $\displaystyle \sum$ x = 488.8 would be the sum of the sample, not sure where to go from there.

Thank you

2. Originally Posted by rba_mandy
A random sample of 13 observations from a population yielded $\displaystyle \sum$ x=488.8 and $\displaystyle \sum$ $\displaystyle x^2$ = 18950.2. Estimate $\displaystyle \mu$ and $\displaystyle \sigma.$

I can't find this in my notes, I think $\displaystyle \sum$ x = 488.8 would be the sum of the sample, not sure where to go from there.

Thank you
$\displaystyle \overline{x} = E(X) = \frac{\sum_{i=1}^n x_i}{n}$

$\displaystyle s_X^2 = Var(X) = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n} = \frac{\left(\sum_{i=1}^n x_i^2\right) - 2 \overline{x} \left( \sum_{i=1}^n x_i \right) + n \overline{x}^2}{n}$.

$\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - 2 \overline{x} \frac{\sum_{i=1}^n x_i}{n} + \overline{x}^2$

$\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - \overline{x}^2$.

Substitute your data and do the calculations.

3. Originally Posted by mr fantastic
$\displaystyle \overline{x} = E(X) = \frac{\sum_{i=1}^n x_i}{n}$

$\displaystyle s_X^2 = Var(X) = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n} = \frac{\left(\sum_{i=1}^n x_i^2\right) - 2 \overline{x} \left( \sum_{i=1}^n x_i \right) + n \overline{x}^2}{n}$.

$\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - 2 \overline{x} \frac{\sum_{i=1}^n x_i}{n} + \overline{x}^2$

$\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - \overline{x}^2$.

Substitute your data and do the calculations.

Ok, but I don't understand these formulas. What is $\displaystyle \frac{\sum_{i=1}^n x_i}{n}$ the n over i?

Is the first formula for $\displaystyle \mu$ and the second for $\displaystyle \sigma$?

Could you possibly give me a head start or an example maybe?

4. Originally Posted by rba_mandy
Ok, but I don't understand these formulas. What is $\displaystyle \frac{\sum_{i=1}^n x_i}{n}$ the n over i?

Is the first formula for $\displaystyle \mu$ and the second for $\displaystyle \sigma$?

Could you possibly give me a head start or an example maybe?

$\displaystyle \sum_{i=1}^n x_i$ is the sum of all the data.
$\displaystyle \sum_{i=1}^n x_i^2$ is the sum of the squares of all the data.
$\displaystyle n$ is the number of data points.