Question

lets practice: given the data set 2,4,6,8,11,14,12,16, find the following

1. calculate (round to nearest whole number):

a. mean:
b. median:
c. range:
d. standard deviation:

1. multiply to each value of the data set by 3

new data set:

1. calculate:

a. mean:
b. median:
c. range:
d. standard deviation:

Explanation:

Step1: Calculate the mean of the original data - set

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
The original data - set is $2,4,6,8,11,14,12,16$. Here $n = 8$ and $\sum_{i=1}^{8}x_i=2 + 4+6 + 8+11+14+12+16=73$.
So, $\bar{x}=\frac{73}{8}=9.125\approx9$.

Step2: Calculate the median of the original data - set

First, arrange the data in ascending order: $2,4,6,8,11,12,14,16$.
Since $n = 8$ (an even number), the median $M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}=\frac{8 + 11}{2}=9.5\approx10$.

Step3: Calculate the range of the original data - set

The range $R$ is the difference between the maximum and minimum values.
The maximum value $x_{max}=16$ and the minimum value $x_{min}=2$. So, $R=16 - 2=14$.

Step4: Calculate the standard deviation of the original data - set

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.
$\sum_{i = 1}^{8}(x_i - 9.125)^2=(2 - 9.125)^2+(4 - 9.125)^2+(6 - 9.125)^2+(8 - 9.125)^2+(11 - 9.125)^2+(14 - 9.125)^2+(12 - 9.125)^2+(16 - 9.125)^2$
$=(-7.125)^2+(-5.125)^2+(-3.125)^2+(-1.125)^2+(1.875)^2+(4.875)^2+(2.875)^2+(6.875)^2$
$=50.765625+26.265625+9.765625+1.265625+3.515625+23.765625+8.265625+47.265625 = 170.125$.
$s=\sqrt{\frac{170.125}{7}}\approx4.95\approx5$.

Step5: Multiply each value of the data - set by 3

The new data - set is $6,12,18,24,33,42,36,48$.

Step6: Calculate the mean of the new data - set

$\sum_{i = 1}^{8}y_i=6 + 12+18+24+33+42+36+48=219$.
The mean $\bar{y}=\frac{219}{8}=27.375\approx27$.

Step7: Calculate the median of the new data - set

Arrange the new data in ascending order: $6,12,18,24,33,36,42,48$.
Since $n = 8$ (an even number), the median $M=\frac{24 + 33}{2}=28.5\approx29$.

Step8: Calculate the range of the new data - set

The maximum value $y_{max}=48$ and the minimum value $y_{min}=6$. So, $R = 48-6 = 42$.

Step9: Calculate the standard deviation of the new data - set

First, find the mean $\bar{y}=27.375$.
$\sum_{i = 1}^{8}(y_i - 27.375)^2=(6 - 27.375)^2+(12 - 27.375)^2+(18 - 27.375)^2+(24 - 27.375)^2+(33 - 27.375)^2+(42 - 27.375)^2+(36 - 27.375)^2+(48 - 27.375)^2$
$=(-21.375)^2+(-15.375)^2+(-9.375)^2+(-3.375)^2+(5.625)^2+(14.625)^2+(8.625)^2+(20.625)^2$
$=457.890625+236.390625+87.890625+11.390625+31.640625+213.890625+74.390625+425.390625 = 1548.8125$.
$s=\sqrt{\frac{1548.8125}{7}}\approx14.85\approx15$.

Answer:

1.
a. Mean: 9
b. Median: 10
c. Range: 14
d. Standard Deviation: 5

1. New Data Set: 6,12,18,24,33,42,36,48

3.
a. Mean: 27
b. Median: 29
c. Range: 42
d. Standard Deviation: 15