Question

pq3: measures of center & spread
score: 2/3 answered: 2/3
18mins × progress saved submit and end
question 3
0/1 pt 2 details
find the mode, median, mean, range, and standard deviation for the population of numbers:
17 15 8 8 18 15 21 24
you may round your answers to two decimals, if needed. use dne for any answers that do not exist.
mean =
median =
mode =
range =
standard deviation =

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a set of numbers $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 8$, and $\sum_{i=1}^{8}x_i=17 + 15+8+8+18+15+21+24=126$. So $\bar{x}=\frac{126}{8}=15.75$.

Step2: Calculate the median

First, order the data set: $8,8,15,15,17,18,21,24$. Since $n = 8$ (an even - numbered data set), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values. The $\frac{n}{2}=4$th value is $15$ and the $(\frac{n}{2}+1)=5$th value is $17$. So the median $M=\frac{15 + 17}{2}=16$.

Step3: Calculate the mode

The mode is the number that appears most frequently in the data - set. The numbers $8$ and $15$ both appear twice, while the other numbers appear only once. So the modes are $8$ and $15$.

Step4: Calculate the range

The range $R$ of a data set is the difference between the maximum and minimum values. Here, the maximum value $x_{max}=24$ and the minimum value $x_{min}=8$. So $R=24 - 8=16$.

Step5: Calculate the standard deviation

The formula for the population standard deviation $\sigma$ of a data set $x_1,x_2,\cdots,x_n$ with mean $\mu$ is $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\mu)^2}{n}}$.
$\sum_{i=1}^{8}(x_i - 15.75)^2=(17 - 15.75)^2+(15 - 15.75)^2+(8 - 15.75)^2+(8 - 15.75)^2+(18 - 15.75)^2+(15 - 15.75)^2+(21 - 15.75)^2+(24 - 15.75)^2$
$=(1.25)^2+(-0.75)^2+(-7.75)^2+(-7.75)^2+(2.25)^2+(-0.75)^2+(5.25)^2+(8.25)^2$
$=1.5625 + 0.5625+60.0625+60.0625+5.0625+0.5625+27.5625+68.0625$
$=223.5$.
$\sigma=\sqrt{\frac{223.5}{8}}\approx5.28$.

Answer:

Mean = 15.75
Median = 16
Mode = 8, 15
Range = 16
Standard Deviation = 5.28