Answer :
85.7% of the data falls within 1 population standard deviation of the mean.
- First, we need to calculate the mean (μ):
Sum of all numbers: 72 + 69 + 82 + 74 + 69 + 78 + 68 + 72 + 69 + 82 + 74 + 69 + 78 + 68 = 1024
Number of data points: 14
Mean (μ) = 1024 / 14 = 73.14285714
- Subtract the mean from each number, square the result, and sum:
(72 - 72.43)² = 0.43² = 0.1849
(69 - 72.43)² = -3.43² = 11.7649
(82 - 72.43)² = 9.57² = 91.5849
(74 - 72.43)² = 1.57² = 2.4649
(69 - 72.43)² = -3.43² = 11.7649
(78 - 72.43)² = 5.57² = 31.0849
(68 - 72.43)² = -4.43² = 19.6449
(72 - 72.43)² = -0.43² = 0.1849
(69 - 72.43)² = -3.43² = 11.7649
(82 - 72.43)² = 9.57² = 91.5849
(74 - 72.43)² = 1.57² = 2.4649
(69 - 72.43)² = -3.43² = 11.7649
(78 - 72.43)² = 5.57² = 31.0849
(68 - 72.43)² = -4.43² = 19.6449
Summing these squared differences:
0.1849 + 11.7649 + 91.5849 + 2.4649 + 11.7649 + 31.0849 + 19.6449 + 0.1849 + 11.7649 + 91.5849 + 2.4649 + 11.7649 + 31.0849 + 19.6449 = 336.98
- Divide by n:
336.98 / 14 = 24.07
- Now we can calculate the population standard deviation (σ):
σ = √24.07 = 4.90
- The range within one standard deviation of the mean is:
Lower bound: 73.14285714 - 4.90 = 68.24285714
Upper bound: 73.14285714 + 4.90 = 78.04285714
- Count how many data points fall within this range:
72, 69, 74, 69, 78, 68, 72, 69, 74, 69, 78, 68
12 out of 14 data points are within this range
- Calculate the percentage:
(12 / 14) × 100 = 85.71428571%
- Rounding to the nearest 0.1%:
= 85.7%
