Question

calculate the mean and sample standard deviation of the data shown. round to two decimal places - anything less will be marked wrong.
x 12.5 15.4 18 21.8 22 23.6
mean:
sample standard deviation:

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Explanation:

Step1: Calculate the sum of data

$12.5 + 15.4+18 + 21.8+22+23.6=113.3$

Step2: Calculate the mean

There are $n = 6$ data - points. The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{113.3}{6}\approx18.88$

Step3: Calculate the deviations from the mean

$(12.5 - 18.88)^2=(-6.38)^2 = 40.7044$, $(15.4 - 18.88)^2=(-3.48)^2 = 12.1104$, $(18 - 18.88)^2=(-0.88)^2 = 0.7744$, $(21.8 - 18.88)^2=(2.92)^2 = 8.5264$, $(22 - 18.88)^2=(3.12)^2 = 9.7344$, $(23.6 - 18.88)^2=(4.72)^2 = 22.2784$

Step4: Calculate the sum of squared deviations

$40.7044+12.1104 + 0.7744+8.5264+9.7344+22.2784=94.1284$

Step5: Calculate the sample - variance

The sample - variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{94.1284}{6 - 1}=\frac{94.1284}{5}=18.82568$

Step6: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^{2}}=\sqrt{18.82568}\approx4.34$

Answer:

mean: $18.88$
Sample standard deviation: $4.34$