1. **Problem Statement:** Calculate the average and standard deviation of the number of accidents for the years 2078 and 2079. 2. **Data:** - Year 2078 accidents: $[15, 18, 20, 22, 16, 12, 10, 14, 15, 18, 16, 20]$ - Year 2079 accidents: $[18, 20, 22, 24, 18, 10, 8, 15, 13, 15, 16, 22]$ 3. **Formulas:** - Average (mean) $\mu = \frac{1}{n} \sum_{i=1}^n x_i$ - Standard deviation $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}$ 4. **Calculations for 2078:** - Sum: $15+18+20+22+16+12+10+14+15+18+16+20 = 196$ - Number of months $n=12$ - Average $\mu_{2078} = \frac{196}{12} = 16.33$ - Calculate squared differences: $$(15-16.33)^2=1.77, (18-16.33)^2=2.78, (20-16.33)^2=13.44, (22-16.33)^2=32.11,$$ $$(16-16.33)^2=0.11, (12-16.33)^2=18.78, (10-16.33)^2=40.11, (14-16.33)^2=5.44,$$ $$(15-16.33)^2=1.77, (18-16.33)^2=2.78, (16-16.33)^2=0.11, (20-16.33)^2=13.44$$ - Sum of squared differences $= 132.64$ - Standard deviation $\sigma_{2078} = \sqrt{\frac{132.64}{12}} = \sqrt{11.05} = 3.32$ 5. **Calculations for 2079:** - Sum: $18+20+22+24+18+10+8+15+13+15+16+22 = 201$ - Average $\mu_{2079} = \frac{201}{12} = 16.75$ - Calculate squared differences: $$(18-16.75)^2=1.56, (20-16.75)^2=10.56, (22-16.75)^2=27.56, (24-16.75)^2=52.56,$$ $$(18-16.75)^2=1.56, (10-16.75)^2=45.56, (8-16.75)^2=76.56, (15-16.75)^2=3.06,$$ $$(13-16.75)^2=14.06, (15-16.75)^2=3.06, (16-16.75)^2=0.56, (22-16.75)^2=27.56$$ - Sum of squared differences $= 264.6$ - Standard deviation $\sigma_{2079} = \sqrt{\frac{264.6}{12}} = \sqrt{22.05} = 4.70$ 6. **Comparison of accident occurrences:** - Average accidents in 2078: $16.33$ - Average accidents in 2079: $16.75$ - Year 2079 has a slightly higher average number of accidents. 7. **Uniformity of accidents:** - Standard deviation measures spread; smaller $\sigma$ means more uniform data. - $\sigma_{2078} = 3.32$ and $\sigma_{2079} = 4.70$ - Year 2078 has more uniform number of accidents as it has a smaller standard deviation. **Final answers:** - Average accidents 2078: $16.33$, standard deviation: $3.32$ - Average accidents 2079: $16.75$, standard deviation: $4.70$ - Year with greater accidents: 2079 - Year with more uniform accidents: 2078