Question

question 6
the numbers of false fire alarms were counted each month at all sites. the results are given in the following table.
label the following numerical measures according to the context:
size:?
mean:?
variance:?
standard deviation:?

Explanation:

Step1: Count data - points

Count the number of values in the data - set. There are 24 values. So, the size of the data - set $n = 24$.

Step2: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
$\sum_{i=1}^{24}x_{i}=12 + 14+6 + 11+5+11+9+12+9+6+15+12+5+7+6+6+7+7+11+8+8$
$=216$
$\bar{x}=\frac{216}{24}=9$

Step3: Calculate the variance

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$
$(x_1-\bar{x})^2=(12 - 9)^2 = 9$, $(x_2-\bar{x})^2=(14 - 9)^2 = 25$, $\cdots$
$\sum_{i = 1}^{24}(x_{i}-9)^{2}=228$
$s^{2}=\frac{228}{24 - 1}=\frac{228}{23}\approx9.913$

Step4: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}=\sqrt{\frac{228}{23}}\approx3.148$

Answer:

Size: 24
Mean: 9
Variance: $\frac{228}{23}\approx9.913$
Standard deviation: $\sqrt{\frac{228}{23}}\approx3.148$