Question

determine the mean, median, and mode of the unsorted data set given below, where the data is in terms of a variable $x$:
$a_1 = x + 2$
$a_2 = x + 5$
$a_3 = x - 5$
$a_4 = x + 4$
$a_5 = x - 2$
$a_6 = x - 4$
$a_7 = x - 1$
$a_8 = x - 2$
mean:
median:
mode:

Explanation:

Step1: Calculate the sum of the data - set

Sum $S=(x + 2)+(x + 5)+(x - 5)+(x + 4)+(x - 2)+(x - 4)+(x - 1)+(x - 2)=8x - 3$

Step2: Calculate the mean

The number of data - points $n = 8$. Mean $\bar{x}=\frac{S}{n}=\frac{8x - 3}{8}=x-\frac{3}{8}$

Step3: Sort the data - set

Sorting the data set: $x-5,x - 4,x - 2,x - 2,x - 1,x + 2,x + 4,x + 5$

Step4: Calculate the median

Since $n = 8$ (an even number), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered data - points. The 4th and 5th ordered data - points are $x - 2$ and $x - 1$. Median $M=\frac{(x - 2)+(x - 1)}{2}=x-\frac{3}{2}$

Step5: Calculate the mode

The mode is the data - point that appears most frequently. Here, $x - 2$ appears twice and other values appear once. So the mode is $x - 2$

Answer:

Mean: $x-\frac{3}{8}$
Median: $x-\frac{3}{2}$
Mode: $x - 2$