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If the mean of 10, 15, 20, 25, and y is 20, then the value of y is:
20
25
30
35
30
Calculate the deviation of each known number from the mean (20): 10 is -10, 15 is -5, 20 is 0, 25 is +5. The sum of these deviations is -10. To balance the mean, y must offset this by +10 from the mean, so 20 + 10 = 30.
A set of five observations {10, 15, 20, 25, y} and their arithmetic mean, which is 20.
Mean=nтИСi=1nтАЛxiтАЛтАЛ
Calculate the deviation of each known number from the mean (20): 10 is -10, 15 is -5, 20 is 0, 25 is +5. The sum of these deviations is -10. To balance the mean, y must offset this by +10 from the mean, so 20 + 10 = 30.
Students often forget to include the variable y in the count of observations (n=5) and incorrectly divide the sum of known numbers by 4.
Set up the Mean equation
The mean of n observations is the sum of observations divided by n. Given n=5, we set up the equation.
510+15+20+25+yтАЛ=20
Calculate the sum of known values
Add the constants in the numerator: 10+15+20+25=70.
70+y=20├Ч5
Solve for y
Multiply the mean by the number of observations to get the total sum, then subtract 70.
y=100тИТ70=30
C is correct because the total sum of the five values must be 100, and subtracting the known sum of 70 leaves y equal to 30.
This concept of 'balancing deviations from the mean' is highly useful in Statistics for calculating weighted means and finding missing frequencies in a data set.