If the mean and the variance of $6,\ 4,\ a,\ 8,\ b,\ 12,\ 10,\ 13$ are $9$ and $9.25$ respectively, then $a + b + ab$ is equal to:

Step 1: Calculate the sum of observations using the mean

The mean of a set of observations is calculated by dividing the sum of all observations by the total number of observations. We are given the mean as $9$ and the number of observations $n=8$. We can use this to find the sum of the observations.

Step 2: Formulate the first equation for a and b

Substitute the given values into the mean formula. The sum of the known numbers is $6+4+8+12+10+13 = 53$. So, $53 + a + b = 72$. This gives us our first equation relating $a$ and $b$.

Step 3: Calculate the sum of squares using the variance

The variance is given as $9.25$. We can use the formula for variance to find the sum of the squares of the observations. We know the mean is $9$ and $n=8$.

Step 4: Formulate the second equation for a and b

Substitute the known values into the variance formula. The sum of the squares of the known numbers is $36+16+64+144+100+169 = 529$. So, $529 + a^2 + b^2 = 8 \times (9.25 + 81) = 8 \times 90.25 = 722$. This gives us our second equation.

Step 5: Solve the system of equations for a and b

From the first equation, $a+b = 72 - 53 = 19$. From the second equation, $a^2+b^2 = 722 - 529 = 193$. We know that $(a+b)^2 = a^2 + b^2 + 2ab$. Substitute the values to find $ab$.

Step 6: Calculate the value of ab

Using the identity $(a+b)^2 = a^2 + b^2 + 2ab$, we can solve for $2ab = (a+b)^2 - (a^2+b^2)$. Substituting the values, $2ab = (19)^2 - 193 = 361 - 193 = 168$. Therefore, $ab = 84$.

Step 7: Calculate a + b + ab

Now that we have the values for $a+b$ and $ab$, we can find the required expression $a+b+ab$.

More from this chapter

• Calculate the mean and variance of the first n natural numbers.
• For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
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• The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
• If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
• The variance of the numbers 8, 21, 34, 47,, 320 is:
• The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.

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