Respuesta :
Answer:
The standard deviation for the given data set is 1.65.
Step-by-step explanation:
The Standard Deviation is defined as
[tex]\sigma =\sqrt{\frac{\Sigma (x_{i} -\mu)^{2} }{N-1} }[/tex]
Where [tex]\sigma[/tex] is the standard deviation, [tex]\mu[/tex] is the mean and [tex]N[/tex] is the total number of data.
So, first, we need to find the mean of the data set.
[tex]\mu = \frac{10+11+12+10.5+11.25+12.25+10}{7} =11[/tex]
Now, we have to subtract each data with the mean to then elevate the differnece to the square power.
[tex]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[/tex]
Then, we sum all these results.
[tex]\Sigma (x_{i}- \mu )^{2}=1+0+1+0.25+0.0625+1.5625+1=16.278[/tex]
Next, we divide the sum by [tex]N-1=7-1=6[/tex]
[tex]\frac{\Sigma (x_{i}-\mu )^{2} }{N-1}=\frac{16.278}{6} \approx 2.713[/tex]
Finally, we apply the square root.
[tex]\sigma = \sqrt{2.713} \approx 1.65[/tex]
Therefore, the standard deviation for the given data set is 1.65.
