Estimating the mean response
In some situations, we are interested in estimating the mean response at some x-value,
μy = β0 + β1x
The least squares estimate,
= b0 + b1 x
becomes increasingly accurate as the sample size increases (since b0 and b1 become more accurate estimates of β0 and β1).
Predicting a single item's response
To predict the response for a single new individual with a known x-value, the same prediction would be used,
= b0 + b1 x
However no matter how accurately we estimate the mean response for such individuals, a single new individual's response will have a distribution with standard deviation σ around this mean and we have no information to help us predict how far it will be from its mean. The prediction error cannot have a standard deviation that is less than σ.
The error in predicting an individual's response is usually greater than the error in estimating the mean response.
Simulation
The diagram below contrasts estimation of the mean response and prediction of a new individual's response at x = 5.5. Least squares lines have been fitted to several simulated data sets, one of which is shown on the left. The two kinds of errors from the simulations are shown on the right, showing that the prediction errors are usually greater.