Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — variance and standard deviation
Find the variance and standard deviation of 4,7,9,12,18.
- 1xˉ=550=10The mean is needed for either formula.
- 2∑x2=16+49+81+144+324=614Using σ2=n∑x2−xˉ2, the faster form.
- 3σ2=5614−100=122.8−100
- 4=22.8Check by the definition: 536+9+1+4+64=22.8 ✓
- 5σ=22.8=4.77 (3 s.f.)In the same units as the data, unlike the variance.
Example 2 — quartiles and outliers
For 3,5,7,8,10,12,15,18, find the interquartile range and test for outliers.
- 1The data is ordered with 8 values, so the median is 28+10=9.
- 2Lower half 3,5,7,8: Q1=25+7=6
- 3Upper half 10,12,15,18: Q3=212+15=13.5
- 4IQR =13.5−6=7.5
- 5Boundaries: 6−1.5(7.5)=−5.25 and 13.5+1.5(7.5)=24.75The standard 1.5 × IQR rule.
- 6All values lie inside, so there are no outliers.
Example 3 — the effect of coding
Data x is coded as y=x−100, giving yˉ=3.2 and σy=1.5. Find xˉ and σx.
- 1The coding subtracts a constant; it does not scale.b=1 in y=bx−a.
- 2xˉ=100+yˉ=100+3.2Reverse the shift.
- 3xˉ=103.2
- 4σx=σy=1.5Adding a constant moves every value together, so the spread is untouched.
- 5Had the coding been y=10x−100, the standard deviation would have been 10×1.5=15.Scaling affects the spread; shifting does not.