- Mean x̄
- 7
- Median
- 7
- Mode
- 6, 9
- Sample s²
- 3.3333
- Sample s
- 1.8257
- Range
- 5
- Pop. σ²
- 2.8571
- Pop. σ
- 1.6903
- IQR
- 3
- Mean x̄
- 11.4545
- Median
- 10
- Mode
- none
- Sample s²
- 50.2727
- Sample s
- 7.0903
- Range
- 26
- Pop. σ²
- 45.7025
- Pop. σ
- 6.7604
- IQR
- 7
Side-by-side comparison
Sample measures divide by n−1; population measures divide by n.
| Measure | Set A | Set B | |
|---|---|---|---|
| n | 7 | 11 | |
| Σxᵢ | 49 | 126 | |
| Σxᵢ² | 363 | 1946 | |
| Mean x̄ | 7 | 11.4545 | |
| Median | 7 | 10 | |
| Mode | 6, 9 | none | |
| Modality | bimodal | no mode | |
| Min | 4 | 4 | |
| Max | 9 | 30 | |
| Range | 5 | 26 | |
| Q1 | 6 | 7 | |
| Q3 | 9 | 14 | |
| IQR | 3 | 7 | |
| Σ(xᵢ−x̄)² | 20 | 502.7273 | |
| Sample variance s² (÷ n−1) | 3.3333 | 50.2727 | |
| Sample sd s | 1.8257 | 7.0903 | |
| Population variance σ² (÷ n) | 2.8571 | 45.7025 | |
| Population sd σ | 1.6903 | 6.7604 | |
| 1.5·IQR fences | 1.5 … 13.5 | -3.5 … 24.5 | |
| Whisker ends | 4 … 9 | 4 … 15 | |
| Potential outliers | none | 30 | |
| Five-number summary | 4, 6, 7, 9, 9 | 4, 7, 10, 14, 30 |
Graphs
Box = Q1 to Q3, thick line = median, dot = mean, circles = values beyond the 1.5·IQR fences.
Step-by-step working
The deviation table, both variance formulas and the n vs n−1 check.
| xᵢ | 4 | 6 | 6 | 7 | 8 | 9 | 9 | 49 |
|---|---|---|---|---|---|---|---|---|
| xᵢ − x̄ | -3 | -1 | -1 | 0 | 1 | 2 | 2 | 0 |
| (xᵢ − x̄)² | 9 | 1 | 1 | 0 | 1 | 4 | 4 | 20 |
Sample variance
s² = Σ(xᵢ−x̄)² / (n−1) = 20 / 6 = 3.3333
Shortcut check
s² = (Σxᵢ² − n·x̄²)/(n−1) = (363 − 343) / 6 = 3.3333
Sample sd
s = √s² = 1.8257
Population variance
σ² = Σ(xᵢ−x̄)² / n = 20 / 7 = 2.8571 (σ = 1.6903)
Signed deviations sum to 0 — they always cancel, which is why we square them. Dividing the squared total by n instead of n−1 gives the population value, not the sample value.
Chebyshev & the empirical rule
Guaranteed minimums for any shape versus approximate percentages for normal data.
Chebyshev — any shape
Interval μ ± kσ = [70, 90]
At least 1 − 1/4 = 75% of the data lies inside.
So Pr(|X − μ| ≥ kσ) ≤ 25%.
These are minimum guarantees, true for every distribution.
Empirical rule — approximately normal
About 95.45% of the data lies within 2σ.
Outside: about 4.55%; above μ + kσ: about 2.28%.
z-score of x = (90 − 80) / 5 = 2
Approximations — valid only if the shape is roughly normal and symmetric with no strong outliers.
Reference values
k = 1: Chebyshev ≥ —, normal ≈ 68.27%
k = 1.5: Chebyshev ≥ 55.6%, normal ≈ 86.64%
k = 2: Chebyshev ≥ 75%, normal ≈ 95.45%
k = 2.5: Chebyshev ≥ 84%, normal ≈ 98.76%
k = 3: Chebyshev ≥ 88.9%, normal ≈ 99.73%
k = 5: Chebyshev ≥ 96%, normal ≈ 100%
For a guarantee of at least p, use k = 1/√(1−p). For 96%: k = 5.