# Estimate mean and std dev?

• April 8th 2009, 11:50 AM
rba_mandy
Estimate mean and std dev?
A random sample of 13 observations from a population yielded $\sum$ x=488.8 and $\sum$ $x^2$ = 18950.2. Estimate $\mu$ and $\sigma.$

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

Thank you
• April 8th 2009, 02:37 PM
mr fantastic
Quote:

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

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

Thank you

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

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

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

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

Substitute your data and do the calculations.
• April 8th 2009, 03:47 PM
rba_mandy
Quote:

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

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

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

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

Substitute your data and do the calculations.

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

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

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

• April 9th 2009, 02:43 AM
mr fantastic
Quote:

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

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

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

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