The expanded form of the Empirical Rule:
Below μ - 3σ lie about 0.15% of observations.
Between μ - 3σ and μ - 2σ lie about 2.35% of observations.
Between μ - 2σ and μ - σ lie about 13.5% of observations.
Between μ - σ and μ lie about 34% of observations.
Between μ and μ + σ lie about 34% of observations.
Between μ + σ and μ + 2σ lie about 13.5% of observations.
Between μ + 2σ and μ + 3σ lie about 2.35% of observations.
Above μ + 3σ lie about 0.15% of observations.
Between μ - 3σ and μ + 3σ lie about 99.7% of observations.
Between μ - 2σ and μ + 2σ lie about 95% of observations.
Between μ - σ and μ + σ lie about 68% of observations.
(A picture would be very convenient here, but I got an error when I tried to submit a picture.)
For this problem the mean μ = 50 and the standard deviation σ = 5.
| μ - 3σ | μ - 2σ | μ - σ | μ | μ + σ | μ + 2σ | μ + 3σ |
| 35 | 40 | 45 | 50 | 55 | 60 | 65 |
(a) 99.7% of the widget weights lie between μ - 3σ and μ + 3σ, so between 35 oz and 65 oz.
(b) Between 40 oz (μ - 2σ) and 65 oz (μ + 3σ) lie about 13.5% + 34% + 34% + 13.5% + 2.35% = 97.35% of the widget weights.
(c) Below 55 oz (μ + σ) lie about 0.15% + 2.35% +13.5% + 34% + 34% = 84% of the widget weights.
(Alternate calculations: 50% (below μ = 50) + 34% = 84%, or 100% - (13.5% + 2.35% + 0.15%) = 84%)
