Question

calculate the mean and sample standard deviation of the data shown. round to two decimal places.
x 2.4 4.3 5.7 8.1 9.9 24.8
mean:
sample standard deviation:
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Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 6$, and $\sum_{i=1}^{6}x_i=2.4 + 4.3+5.7+8.1+9.9+24.8=55.2$. So $\bar{x}=\frac{55.2}{6}=9.20$.

Step2: Calculate the squared - differences

First, find the differences between each data - point and the mean:
$d_1=2.4 - 9.2=-6.8$, $d_2=4.3 - 9.2=-4.9$, $d_3=5.7 - 9.2=-3.5$, $d_4=8.1 - 9.2=-1.1$, $d_5=9.9 - 9.2 = 0.7$, $d_6=24.8 - 9.2 = 15.6$.
Then square them: $d_1^{2}=(-6.8)^{2}=46.24$, $d_2^{2}=(-4.9)^{2}=24.01$, $d_3^{2}=(-3.5)^{2}=12.25$, $d_4^{2}=(-1.1)^{2}=1.21$, $d_5^{2}=(0.7)^{2}=0.49$, $d_6^{2}=(15.6)^{2}=243.36$.

Step3: Calculate the sum of squared - differences

$\sum_{i = 1}^{6}d_i^{2}=46.24+24.01+12.25+1.21+0.49+243.36=327.56$.

Step4: Calculate the sample variance

The formula for the sample variance $s^{2}$ is $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^{2}}{n - 1}$. Here $n = 6$, so $s^{2}=\frac{327.56}{6 - 1}=\frac{327.56}{5}=65.512$.

Step5: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^{2}}$. So $s=\sqrt{65.512}\approx8.09$.

Answer:

mean: $9.20$
Sample standard deviation: $8.09$