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What is the sum of the deviations of the values of the data set 5,6,7,9,10,12 and 14, from the mean of the values given in the same data set?
1
0
1
2
0
The sum of deviations of any set of numbers from their arithmetic mean is always equal to 0 by definition.
A data set of seven numbers: 5, 6, 7, 9, 10, 12, and 14.
тИСi=1nтАЛ(xiтАЛтИТx)=0
The sum of deviations of any set of numbers from their arithmetic mean is always equal to 0 by definition.
Students often spend time calculating the mean (9) and then manually calculating each deviation (5тИТ9,6тИТ9,...) and summing them, which is unnecessary and prone to calculation errors.
Calculate the Arithmetic Mean
First, sum all the values in the data set and divide by the total count of numbers (n=7). Sum: 5+6+7+9+10+12+14=63. Mean: x=763тАЛ=9.
x=9
Calculate individual deviations
The deviation of each value xiтАЛ from the mean x is given by (xiтАЛтИТx).
(5тИТ9),(6тИТ9),(7тИТ9),(9тИТ9),(10тИТ9),(12тИТ9),(14тИТ9)
Sum the deviations
Adding these deviations together: (тИТ4)+(тИТ3)+(тИТ2)+(0)+(1)+(3)+(5)=0.
тИТ4тИТ3тИТ2+0+1+3+5=0
B is correct because the sum of the deviations of any set of numbers from their arithmetic mean is mathematically defined to be 0.
This property is the foundation for understanding variance and standard deviation, where we square the deviations precisely because their linear sum is always zero.
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