Lecture 5 - "Standard Error on the mean"

Goal of “Best Estimate of Parent Mean”

to find the best estimate of the mean of a parent distribution

Probability for each xi(data point) is proportional

to the exponential expression involving μ and the standard deviation σ

Maximum likelihood method

Means we calculate the value of μ that maximizes the probability of observing the given data

maximize the overall probability of observing all the data points (multiplying individual probabilites)

• need to solve for μ

• leads to conclusion that the best etimate for the mean is the sample mean (μ’), which is simply the average of all the data points

Using maximum likelihood approach shows the sample mean is the best estimate for:

the true mean of the parent distribution

Sample Mean

best estimate of the true mean of a parent distribution is the sample mean

Error in the mean (error in estimating the mean D_N)

difference between the the sample mean xˉ and the true mean μ

Deriving the Standard Error (compute variance of the error in the mean D_N)

take square of the error and sum it over all measurements… shows that error decreases as the number of measurements N increases

Standard Error on the mean

• standard deviation of the distribution of means

• tells you how precise your sample mean is, based on number of measurements

  • more measurements take, the smaller the error

What two factors do the precision of experimental results depend on?

• the precision of individual measurements (std, )

• and number of measurements N

Standard Error quantifies:

how much the sample mean is likely to vary from the true population mean

Larger sample sizes reduce:

the standard error, improving the precision of sample mean as an estimate of the true mean