Question

How can I solve this?

Answer 1

Mean: 160

Median: 150

Mode: none

Range: 105

Standard deviation: 35.3

Explanation:

Given:

• Mean = 320
• Median = 300
• Mode = none
• Range = 210
• Standard deviation = 70.6

If each value in the data set is multiplied by a constant value, the mean, median, mode, range, and standard deviation will all be scaled by the same amount.

Therefore, if each value in the data set is multiplied by 0.5:

• Mean: 320 × 0.5 = 160
• Median: 300 × 0.5 = 150
• Mode: none × 0.5 = none
• Range: 210 × 0.5 = 105
• Standard deviation: 70.6 × 0.5 = 35.3

Proof

Data set: {1, 2, 3, 4, 5}

`\sf Mean\:\mu =(\sum x_i)/(n)= (1+2+3+4+5)/(5) = 3`

`\sf Median = 3`

`\sf Mode = none`

`\sf Range = 5-1=4`

`\sf Standard\:deviation\:\sigma=\sqrt{(\sum (x_i-\mu)^2)/(n)}=\sqrt{(10)/(5)}=√(2)`

If we multiply each value by 0.5, the new data set is:

{0.5, 1, 1.5, 2, 2.5}

`\sf Mean\:\mu =(\sum x_i)/(n)= (0.5+1+1.5+2+2.5)/(5) = 1.5`

`\sf Median = 1.5`

`\sf Mode = none`

`\sf Range = 2.5-0.5=2`

`\sf Standard\:deviation\:\sigma=\sqrt{(\sum (x_i-\mu)^2)/(n)}=\sqrt{(2.5)/(5)}=(√(2))/(2)`

Therefore, if each value in the data set is multiplied by 0.5, the mean, median, mode, range, and standard deviation are all scaled by the same amount.