Question

calculate the mean and sample standard deviation of the data shown. round to two decimal places - anything less will be marked wrong.
x 3.6 8.3 15.4 15.9 16.1 19.9 22.4 25.1
mean:
sample standard deviation:

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Explanation:

Step1: Calculate the sum of data

$3.6 + 8.3+15.4 + 15.9+16.1+19.9+22.4+25.1=126.7$

Step2: Calculate the mean

There are $n = 8$ data - points. The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{126.7}{8}=15.84$

Step3: Calculate the squared - differences from the mean

$(3.6 - 15.84)^2=(- 12.24)^2 = 149.82$
$(8.3 - 15.84)^2=(-7.54)^2 = 56.85$
$(15.4 - 15.84)^2=(-0.44)^2 = 0.19$
$(15.9 - 15.84)^2=(0.06)^2 = 0.0036$
$(16.1 - 15.84)^2=(0.26)^2 = 0.07$
$(19.9 - 15.84)^2=(4.06)^2 = 16.48$
$(22.4 - 15.84)^2=(6.56)^2 = 43.03$
$(25.1 - 15.84)^2=(9.26)^2 = 85.75$
The sum of squared - differences $\sum_{i = 1}^{n}(x_{i}-\bar{x})^2=149.82+56.85 + 0.19+0.0036+0.07+16.48+43.03+85.75=352.2$

Step4: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}}$. Here $n = 8$, so $s=\sqrt{\frac{352.2}{8 - 1}}=\sqrt{\frac{352.2}{7}}\approx7.08$

Answer:

mean: $15.84$
Sample standard deviation: $7.08$