Question

the standard deviation of the data set {5,6,9,10,10,11,13,14} is about ______. 2.905 9.641 3.105

Explanation:

Step1: Calculate the mean ($\mu$)

The data set is $\{5, 6, 9, 10, 10, 11, 13, 14\}$. The number of data points $n = 8$.
The mean $\mu=\frac{5 + 6 + 9 + 10 + 10 + 11 + 13 + 14}{8}=\frac{78}{8} = 9.75$.

Step2: Calculate the squared differences from the mean

For each data point $x_i$:

• $(5 - 9.75)^2=(-4.75)^2 = 22.5625$
• $(6 - 9.75)^2=(-3.75)^2 = 14.0625$
• $(9 - 9.75)^2=(-0.75)^2 = 0.5625$
• $(10 - 9.75)^2=(0.25)^2 = 0.0625$
• $(10 - 9.75)^2=(0.25)^2 = 0.0625$
• $(11 - 9.75)^2=(1.25)^2 = 1.5625$
• $(13 - 9.75)^2=(3.25)^2 = 10.5625$
• $(14 - 9.75)^2=(4.25)^2 = 18.0625$

Step3: Calculate the variance ($\sigma^2$)

Variance is the average of the squared differences.
Sum of squared differences $=22.5625+14.0625 + 0.5625+0.0625+0.0625+1.5625+10.5625+18.0625 = 67.5$
Variance $\sigma^2=\frac{67.5}{8}=8.4375$

Step4: Calculate the standard deviation ($\sigma$)

Standard deviation is the square root of the variance.
$\sigma=\sqrt{8.4375}\approx2.905$

Answer:

2.905