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.

Step 1: Set up equations for mean and variance

We are given the mean and variance of 5 observations. Let the two unknown observations be $x$ and $y$. We can set up two equations based on the definitions of mean and variance. The mean is the sum of all observations divided by the number of observations, and the variance is the mean of the squares of the observations minus the square of the mean.

Step 2: Simplify the mean equation

First, let's simplify the equation for the mean. Multiply both sides by 5 and sum the known observations. This gives us a linear equation relating $x$ and $y$.

Step 3: Simplify the variance equation

Next, simplify the equation for the variance. Calculate the squares of the known observations and the square of the mean. Rearrange the terms to get a quadratic equation involving $x^2$ and $y^2$.

Step 4: Solve the system of equations

Now we have a system of two equations: one linear and one quadratic. Substitute the expression for $x$ from equation (1) into equation (2). This will result in a quadratic equation in terms of $y$.

Step 5: Solve the quadratic equation for y

Solve the quadratic equation $y^2 - 13y + 36 = 0$ by factoring. We look for two numbers that multiply to 36 and add up to -13. These numbers are -4 and -9, so the factors are $(y-4)$ and $(y-9)$. This gives us two possible values for $y$.

Step 6: Find the corresponding values for x

Substitute the values of $y$ back into equation (1) ($x = 13 - y$) to find the corresponding values for $x$. This shows that the two unknown observations are 4 and 9, regardless of which variable we assign them to.

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
• The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observ
• Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n
• 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.
• Find the mean deviation about the median for the following data: x_i: 3, 6, 9, 12, 13, 15, 21, 22 f_i: 3, 4, 5, 2, 4, 5, 4, 3.

All Statistics solutions →