Answer:
The sample standard deviation is 14.458
The population standard deviation is 13.967
Step-by-step explanation:
* Lets revise the population standard deviation (σ)
1. Work out the Mean (μ) (average of the numbers)
2. Then for each number subtract the Mean and square
the result (xi - μ)²
3. Then work out the mean of those squared differences
[∑(xi - μ)²/N]
4. Take the square root of that (√[∑(xi - μ)²/N]) and to find σ
* Lets revise the sample standard deviation (s)
1. Work out the Mean (x) (average of the numbers)
2. Then for each number subtract the Mean and square
the result (xi - x)²
3. Then work out the mean of those squared differences [∑(xi - x)²/N - 1]
4. Take the square root of that (√[∑(xi - x)²/N - 1]) and to find s
* Now lets solve the problem
- The data are 15 numbers
17 , 37 , 56 , 16 , 12 , 16 , 19 , 45 , 14 , 37 , 21 , 26 , 43 , 46 , 42
∵ x = μ = sum/number
∴ x = μ = (17+37+56+16+12+16+19+45+14+37+21+26+43+46+42)÷15=29.8
- Subtract the mean from each number and square the result
∵ (17 - 29.8)² = 163.84
∵ (37 - 29.8)² = 51.84
∵ (56 - 29.8)² = 686.44
∵ (16 - 29.8)² = 190.44
∵ (12 - 29.8)² = 316.84
∵ (16 - 29.8)² = 190.44
∵ (19 - 29.8)² = 116.64
∵ (45 - 29.8)² = 231.04
∵ (14 - 29.8)² = 249.64
∵ (37 - 29.8)² = 51.84
∵ (21 - 29.8)² = 77.44
∵ (26 - 29.8)² = 14.44
∵ (43 - 29.8)² = 174.24
∵ (46 - 29.8)² = 262.44
∵ (42 - 29.8)² = 148.84
∴ ∑(xi - μ)² = ∑(xi - x)² = 2926.36
∵ N = 15
∴ The sample standard deviation = √[∑(xi - x)²/(N - 1)]
∴ s = √[2926.36/(15 - 1)] = 14.458
∴ The population standard deviation = √[∑(xi - μ)²/N]
∴ σ = √[2926.36/15] = 13.967
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