Question

the bureau of tobacco and alcohol took a sample of cigarettes and measured the amount of tar (in milligram) in each. the data is shown below 16 18 8 6 5 18 12 9 17 15 13 15 18 2 11 19 find the mean, mode and median, range, standard deviation, variance and interquartile range for this data set. round to 3 decimal places and use the correct symbol.

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 16$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{16}x_{i}=16 + 18+8 + 6+5 + 18+12 + 9+17+15+13+15+18+2+11+19=190$
$\bar{x}=\frac{190}{16}=11.875$

Step2: Find the mode

The mode is the most frequently occurring value. In the data - set, 18 appears 3 times, more frequently than any other number. So the mode is 18.

Step3: Calculate the median

First, order the data: 2, 5, 6, 8, 9, 11, 12, 13, 15, 15, 16, 17, 18, 18, 18, 19.
Since $n = 16$ (an even number), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values.
The 8th value is 13 and the 9th value is 15. So the median $M=\frac{13 + 15}{2}=14$

Step4: Calculate the range

The range $R=x_{\max}-x_{\min}$, where $x_{\max}=19$ and $x_{\min}=2$. So $R=19 - 2=17$

Step5: Calculate the variance

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$
$(x_{1}-\bar{x})^{2}=(16 - 11.875)^{2}=4.125^{2}=17.015625$
$(x_{2}-\bar{x})^{2}=(18 - 11.875)^{2}=6.125^{2}=37.515625$
$\cdots$
$(x_{16}-\bar{x})^{2}=(19 - 11.875)^{2}=7.125^{2}=50.765625$
$\sum_{i = 1}^{16}(x_{i}-\bar{x})^{2}=383.875$
$s^{2}=\frac{383.875}{15}\approx25.592$

Step6: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}=\sqrt{25.592}\approx5.059$

Step7: Calculate the inter - quartile range

First, find the first quartile $Q_{1}$ and the third quartile $Q_{3}$.
The lower half of the data (first 8 values: 2, 5, 6, 8, 9, 11, 12, 13) has a median $Q_{1}=\frac{6 + 8}{2}=7$
The upper half of the data (last 8 values: 15, 15, 16, 17, 18, 18, 18, 19) has a median $Q_{3}=\frac{17+18}{2}=17.5$
The inter - quartile range $IQR = Q_{3}-Q_{1}=17.5 - 7 = 10.5$

Answer:

Mean: $\bar{x}=11.875$
Mode: 18
Median: $M = 14$
Range: $R = 17$
Variance: $s^{2}\approx25.592$
Standard deviation: $s\approx5.059$
Inter - quartile range: $IQR = 10.5$