Question

Find the sample Standard deviation and the population standard deviation of the data set? 17,37,56,16,12,16,19,45,14,37,21,26,43,46,42

Answer 1

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