Question

the highway mileage (mpg) for a sample of 8 different models of a car company can be found below. find the mean, median, mode, and standard deviation. round to one decimal place as needed. 19, 23, 26, 28, 29, 32, 35, 35 mean = median = mode = standard deviation =

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 8$ and $x_{i}$ are the data - points. $\sum_{i=1}^{8}x_{i}=19 + 23+26+28+29+32+35+35=227$. So, $\bar{x}=\frac{227}{8}=28.4$.

Step2: Calculate the median

First, arrange the data in ascending order: $19,23,26,28,29,32,35,35$. Since $n = 8$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values. $\frac{n}{2}=4$ and $\frac{n}{2}+1 = 5$. The 4 - th value is $28$ and the 5 - th value is $29$. So, the median $=\frac{28 + 29}{2}=28.5$.

Step3: Calculate the mode

The mode is the value that appears most frequently in the data - set. Here, $35$ appears twice and all other values appear once. So, the mode is $35$.

Step4: Calculate the standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(19-28.4)^{2}=(-9.4)^{2}=88.36$, $(23 - 28.4)^{2}=(-5.4)^{2}=29.16$, $(26-28.4)^{2}=(-2.4)^{2}=5.76$, $(28-28.4)^{2}=(-0.4)^{2}=0.16$, $(29-28.4)^{2}=(0.6)^{2}=0.36$, $(32-28.4)^{2}=(3.6)^{2}=12.96$, $(35-28.4)^{2}=(6.6)^{2}=43.56$, $(35-28.4)^{2}=(6.6)^{2}=43.56$.
$\sum_{i = 1}^{8}(x_{i}-28.4)^{2}=88.36+29.16+5.76+0.16+0.36+12.96+43.56+43.56=223.8$.
$s=\sqrt{\frac{223.8}{7}}\approx5.6$.

Answer:

Mean = $28.4$
Median = $28.5$
Mode = $35$
Standard Deviation = $5.6$