RM - Unit 102 - Standard error and t-distribution
Book: Analysing Data Using Linear Models
Chapter 2.4, 2.5, 2.6, 2.7, 2.8
Chapter 2.4: The standard error
The standard error of the mean is the standard deviation of the sample
means and serves as a measure of the uncertainty about the population mean.
The larger the sample size, the smaller the standard error, the closer a sample
mean is expected to be around the population mean, the more certain we can be about the population
mean.
The standard error of the variance - the standard deviation of the sampling distribution of the sample
variance. Says something about how spread out the values of the sample variances are. It can be used to
quantify the uncertainty about the population variance when we only have the variance of the sample
values to go on.
Chapter 2.5: Confidence intervals
Confidence interval - an interval centered around the sample mean.
If it’s based on 95% of the sampling distribution (centered around the population mean) it is
called a 95% confidence interval. It basically represents 95% of the sample means had the population
been equal to the sample mean.
Chapter 2.6: The t-statistic
T-values - the standardized sample mean
In summary: if you know the standard error (because you know the population variance), the
standardized sample means will show a normal distribution. If you don’t know the standard error, you
have to estimate it based on the sample variance. If sample size is really large, you can estimate the
population variance pretty well, and the sample variances will be very similar to each other. In that case,
the sampling distribution will look very much like a normal distribution. But if sample size is relatively
small, each sample will show a different sample variance, resulting in different standard error estimates.
If you standardise each sample mean with a different standard error, the sampling distribution will not
look normal. This distribution is called a t-distribution.
Heavy-tailed - relatively more observations in the tails than around the mean.
With this t-distribution, 95% of the observations lie between -3.18 and +3.18.
Chapter 2.7: Interpreting confidence intervals
A confidence interval is constructed as if you know the population mean and variance, which you don’t.
We assume that the population is a certain value, say µ = m0, we assume that the standard error of the
mean is equal to σy¯ = q s 2 n , and we know that if we would look at many many samples and compute
standardised sample means, their distribution would be a t-distribution. Based on that t-distribution, we
Book: Analysing Data Using Linear Models
Chapter 2.4, 2.5, 2.6, 2.7, 2.8
Chapter 2.4: The standard error
The standard error of the mean is the standard deviation of the sample
means and serves as a measure of the uncertainty about the population mean.
The larger the sample size, the smaller the standard error, the closer a sample
mean is expected to be around the population mean, the more certain we can be about the population
mean.
The standard error of the variance - the standard deviation of the sampling distribution of the sample
variance. Says something about how spread out the values of the sample variances are. It can be used to
quantify the uncertainty about the population variance when we only have the variance of the sample
values to go on.
Chapter 2.5: Confidence intervals
Confidence interval - an interval centered around the sample mean.
If it’s based on 95% of the sampling distribution (centered around the population mean) it is
called a 95% confidence interval. It basically represents 95% of the sample means had the population
been equal to the sample mean.
Chapter 2.6: The t-statistic
T-values - the standardized sample mean
In summary: if you know the standard error (because you know the population variance), the
standardized sample means will show a normal distribution. If you don’t know the standard error, you
have to estimate it based on the sample variance. If sample size is really large, you can estimate the
population variance pretty well, and the sample variances will be very similar to each other. In that case,
the sampling distribution will look very much like a normal distribution. But if sample size is relatively
small, each sample will show a different sample variance, resulting in different standard error estimates.
If you standardise each sample mean with a different standard error, the sampling distribution will not
look normal. This distribution is called a t-distribution.
Heavy-tailed - relatively more observations in the tails than around the mean.
With this t-distribution, 95% of the observations lie between -3.18 and +3.18.
Chapter 2.7: Interpreting confidence intervals
A confidence interval is constructed as if you know the population mean and variance, which you don’t.
We assume that the population is a certain value, say µ = m0, we assume that the standard error of the
mean is equal to σy¯ = q s 2 n , and we know that if we would look at many many samples and compute
standardised sample means, their distribution would be a t-distribution. Based on that t-distribution, we
