Question

Write a table to show your steps for calculating the standard deviation and then use your table to calculate the standard deviation of the values listed:

10, 11, 12, 10.5, 11.25, 12.25, 10

Answer 1

The standard deviation for the given data set is 1.65.

Explanation:

The Standard Deviation is defined as

`\sigma =\sqrt{(\Sigma (x_(i) -\mu)^(2) )/(N-1) }`

Where
`\sigma` is the standard deviation,
`\mu` is the mean and
`N` is the total number of data.

So, first, we need to find the mean of the data set.

`\mu = (10+11+12+10.5+11.25+12.25+10)/(7) =11`

Now, we have to subtract each data with the mean to then elevate the differnece to the square power.

`10-11 = -1 \implies (-1)^(2)=1\\ 11-11 = 0 \\12-11 = 1 \implies 1^(2)=1\\ 10.5-11 = -0.5 \implies (-0.5)^(2) =0.25\\11.25 - 11 = 0.25 \imples (1.25)^(2)=0.0625\\12.25 - 11 = 1.25 \implies (1.25)^(2)=1.5625\\10-11 = -1 \implies (-1)^(2)=1`

Then, we sum all these results.

`\Sigma (x_(i)- \mu )^(2)=1+0+1+0.25+0.0625+1.5625+1=16.278`

Next, we divide the sum by
`N-1=7-1=6`

`(\Sigma (x_(i)-\mu )^(2) )/(N-1)=(16.278)/(6) \approx 2.713`

Finally, we apply the square root.

`\sigma = √(2.713) \approx 1.65`

Therefore, the standard deviation for the given data set is 1.65.