find the mean, standard deviation, and five - number summary for the data set. assume population data are…

find the mean, standard deviation, and five - number summary for the data set. assume population data are given. 8, 15, 18, 22, 22, 31, 35, 40, 46, 50\nthe mean is \n(round to one decimal place as needed.)

find the mean, standard deviation, and five - number summary for the data set. assume population data are given. 8, 15, 18, 22, 22, 31, 35, 40, 46, 50\nthe mean is \n(round to one decimal place as needed.)

Answer

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 10$, and $\sum_{i=1}^{10}x_i=8 + 15+18+22+22+31+35+40+46+50=287$. So, $\bar{x}=\frac{287}{10}=28.7$.

Step2: Calculate the variance

The variance $\sigma^{2}$ for a population is $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}$. $(8 - 28.7)^2=(-20.7)^2 = 428.49$ $(15 - 28.7)^2=(-13.7)^2 = 187.69$ $(18 - 28.7)^2=(-10.7)^2 = 114.49$ $(22 - 28.7)^2=(-6.7)^2 = 44.89$ $(22 - 28.7)^2=(-6.7)^2 = 44.89$ $(31 - 28.7)^2=(2.3)^2 = 5.29$ $(35 - 28.7)^2=(6.3)^2 = 39.69$ $(40 - 28.7)^2=(11.3)^2 = 127.69$ $(46 - 28.7)^2=(17.3)^2 = 299.29$ $(50 - 28.7)^2=(21.3)^2 = 453.69$

$\sum_{i = 1}^{10}(x_i - 28.7)^2=428.49+187.69+114.49+44.89+44.89+5.29+39.69+127.69+299.29+453.69 = 1741.1$

$\sigma^{2}=\frac{1741.1}{10}=174.11$

Step3: Calculate the standard deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. So, $\sigma=\sqrt{174.11}\approx13.2$.

Step4: Find the five - number summary

  1. Minimum: The minimum value in the data - set is $8$.
  2. First quartile ($Q_1$): The data - set has $n = 10$ values. The position of $Q_1$ is $\frac{n + 1}{4}=\frac{10+1}{4}=2.75$. So, $Q_1$ is the value $0.25$ of the way between the 2nd and 3rd ordered values. The 2nd value is $15$ and the 3rd value is $18$. $Q_1=15+(18 - 15)\times0.25=15 + 0.75=15.75$.
  3. Median ($Q_2$): The position of the median for $n = 10$ (an even - numbered data - set) is $\frac{n}{2}=5$ and $\frac{n}{2}+1 = 6$. The median is the average of the 5th and 6th ordered values. The 5th value is $22$ and the 6th value is $31$. So, $Q_2=\frac{22 + 31}{2}=26.5$.
  4. Third quartile ($Q_3$): The position of $Q_3$ is $\frac{3(n + 1)}{4}=\frac{3\times(10 + 1)}{4}=8.25$. So, $Q_3$ is the value $0.25$ of the way between the 8th and 9th ordered values. The 8th value is $40$ and the 9th value is $46$. $Q_3=40+(46 - 40)\times0.25=40 + 1.5=41.5$.
  5. Maximum: The maximum value in the data - set is $50$.

The five - number summary is: Minimum = $8$, $Q_1=15.75$, Median = $26.5$, $Q_3=41.5$, Maximum = $50$.

Answer:

Mean: $28.7$ Standard deviation: $13.2$ Five - number summary: Minimum: $8$, $Q_1:15.75$, Median: $26.5$, $Q_3:41.5$, Maximum: $50$