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The arithmetic mean of 10 numbers is 20. If 5 is added to each number, what will be the new mean?
20
25
30
15
25
When a constant value k is added to every observation in a data set, the mean of the data set simply increases by the same value k.
Number of observations (n) = 10, Initial arithmetic mean = 20, Constant value added to each number (k) = 5
MeannewтАЛ=MeanoldтАЛ+k
When a constant value k is added to every observation in a data set, the mean of the data set simply increases by the same value k.
Students often try to calculate the sum of the original 10 numbers (10├Ч20=200), add 50 to the sum (200+50=250), and then divide by 10, which is time-consuming and prone to arithmetic errors.
Identify given parameters
We are given the initial mean of 10 numbers as 20. We are adding 5 to each number.
MeanoldтАЛ=20,k=5
Apply the Property of Mean
The arithmetic mean follows the linear transformation property, which states that if xiтАЛ is replaced by xiтАЛ+k, the new mean becomes x+k.
MeannewтАЛ=20+5
Final Calculation
Adding the constant 5 to the initial mean of 20 gives the new mean.
MeannewтАЛ=25
B is correct because adding a constant value to every observation in a set increases the mean by that exact constant, resulting in 20 + 5 = 25.
This property also holds for subtraction, multiplication, and division: if each term is multiplied by k, the mean is multiplied by k.
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