Worked solution
Estimate mean and variance from seven raw values
Given the data , find unbiased estimates of the mean and variance of the population from which the data are drawn.
CJC · Lecture · Example 2
Method 1: raw-data summaries
These are seven raw observations, not a frequency table. and . Therefore and The denominator is .
Method 2: calculator and notation
Press STAT → 1:Edit. Highlight L1 and press CLEAR to remove old entries, then enter 52, 50, 65, 59, 70, 63, 67 in L1. Choose STAT → CALC → 1:1-Var Stats, set List L1 with no frequency list, and Calculate → ENTER. The screen gives , , , , , , minimum 50 and first quartile 52. Retrieve VARS → 5:Statistics → 3:Sx and square it with x² → ENTER: . Use unrounded stored values.
Common mistakes
The teacher’s larger-value rule distinguishes the n−1 and n denominators for the same nonconstant sample. Sx² is unbiased for population variance; Sx is not generally unbiased for population standard deviation, despite the source’s wording. If all observations are equal both displayed SDs are zero.
Solution notes
Teacher page 10 flags raw data; page 11 explicitly contrasts Sx and σx. The source’s “unbiased standard deviation” wording is qualified rather than silently repeated. No printed marks.