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The mean of 10 observations is 15. If each observation is increased by 3, what is the new mean?
15
18
45
12
18
If each observation is increased by a constant k, the mean simply increases by the same constant k without needing to calculate the sum of all values.
Number of observations n=10, original mean x=15, increment applied to each observation k=3.
xnewтАЛ=xoldтАЛ+k
If each observation is increased by a constant k, the mean simply increases by the same constant k without needing to calculate the sum of all values.
Students often attempt to calculate the total sum 15imes10=150, add 3imes10=30 to it, and then divide by 10, which is time-consuming and error-prone.
Identify property of mean
When every value in a data set is increased by a constant k, the new mean is equal to the old mean plus that constant.
xnewтАЛ=x+k
Substitute given values
Substitute the original mean 15 and the increment 3 into the property formula.
xnewтАЛ=15+3
Calculate final result
Adding the values gives the new mean.
xnewтАЛ=18
B is correct because adding a constant to every observation in a set increases the mean by that same constant, resulting in 15+3=18.
This concept applies to all measures of central tendency under linear transformations; if you multiply all observations by c, the mean is also multiplied by c.
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