Question

The numbers of rooms for 15 homes recently sold were: 8, 8, 8, 5, 9, 8, 7, 6, 6, 7, 7, 7, 7, 9, 9. what is the sample standard deviation?

Answer 1

Hence, the sample standard deviation is:

1.1832

Explanation:

The data is given by:

8, 8, 8, 5, 9, 8, 7, 6, 6, 7, 7, 7, 7, 9, 9.

The mean of these data points is given by:

`Mean(x')=(8+8+8+5+9+8+7+6+6+7+7+7+7+9+9)/(15)\\\\i.e.\\\\Mean(x')=(111)/(15)\\\\i.e.\\\\Mean(x')=7.4`

Now,

x x-x' (x-x')²

8 8-7.4=0.6 0.36

8 8-7.4=0.6 0.36

8 8-7.4=0.6 0.36

5 5-7.4= -2.4 5.76

9 9-7.4=1.6 2.56

8 8-7.4=0.6 0.36

7 7-7.4= -0.4 0.16

6 6-7.4= -1.4 1.96

6 6-7.4= -1.4 1.96

7 7-7.4= -0.4 0.16

7 7-7.4= -0.4 0.16

7 7-7.4= -0.4 0.16

7 7-7.4= -0.4 0.16

9 9-7.4=1.6 2.56

9 9-7.4=1.6 2.56

∑(x-x')²=19.69

Now the variance of the sample population is given by:

`Variance=(\sum (x-x')^2)/(n-1)`

where n is the number of data points.

Here n= 15

Hence, n-1=14

Hence, we get:

`Variance=(19.6)/(14)\\\\i.e.\\\\Variance=1.4`

We know that the standard deviation is the square root of the variance.

i.e.

`Standard\ deviation=√(1.4)\\\\i.e.\\\\Standard\ deviation=1.1832`

Answer 2

From the data set given:
8, 8, 8, 5, 9, 8, 7, 6, 6, 7, 7, 7, 7, 9, 9
to get the standard deviation we use the formula for variance given by:
σ²=[Σ(x-μ)²]/(n-1)
σ² is the variance
μ is the mean
The mean of the data will be:
μ=(8+8+8+5+9+8+7+6+6+7+7+7+7+9+9)/15=7.4
The variance will be found as follows:
σ²=[(8-7.4)²×4+(5-7.4)²+3×(9-7.4)²+5×(7-7.4)²+2×(6-7.4)²]/(15-1)
σ²=1.4
thus
σ=√1.4=1.1183