Q.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.

Concept: Effect of Scaling Variance – The variance formula uses the sum of squares of deviations from the mean, so we can set up equations for the sum and sum of squares of the unknown observations.

Let the two unknown observations be $x$ and $y$.

Step 1: Use the mean.

Mean $= 4.4$, so total sum $= 5 \times 4.4 = 22$.

Given $1 + 2 + 6 = 9$, therefore

Step 2: Use the variance.

Variance $= 8.24$, so

Thus $\sum x_i^2 = 5(8.24 + 19.36) = 5 \times 27.6 = 138$.

Sum of squares of known observations: $1^2 + 2^2 + 6^2 = 1 + 4 + 36 = 41$.

Hence

Step 3: Solve for $x$ and $y$.

We have $x + y = 13$ and $x^2 + y^2 = 97$.

Using $(x+y)^2 = x^2 + y^2 + 2xy$,

So $x$ and $y$ are roots of $t^2 - 13t + 36 = 0$, giving $t = 4$ or $t = 9$.

Effect of Scaling Variance — Why This Works

When you know the mean and variance of a dataset, you have two powerful constraints. The mean pins down the sum of all values. The variance pins down the sum of squares of the values (since variance = $\frac{\sum x_i^2}{n} - (\text{mean})^2$). With three observations already known, the two unknowns must satisfy both a linear equation (from the mean) and a quadratic equation (from the variance). That’s enough to find them uniquely — up to order.

Let the two unknown observations be $a$ and $b$.

1. Use the mean to get the sum of $a$ and $b$.

The mean of 5 observations is 4.4:

Multiply through:

So:

2. Use the variance to get the sum of squares of $a$ and $b$.

Variance is given as 8.24. The formula for variance of $n$ observations is:

Here $n = 5$, mean $= 4.4$, so:

Compute the known squares:

And $(4.4)^2 = 19.36$.

So:

Add 19.36 to both sides:

Multiply by 5:

Thus:

3. Solve the system of equations.

From (1): $b = 13 - a$.

Substitute into (2):

Expand:

Divide by 2:

Factor:

So $a = 4$ or $a = 9$.

If $a = 4$, then $b = 13 - 4 = 9$.

If $a = 9$, then $b = 13 - 9 = 4$.

4. Verify quickly.

Sum: $1 + 2 + 6 + 4 + 9 = 22$, mean $= 22/5 = 4.4$ ✓

Sum of squares: $1 + 4 + 36 + 16 + 81 = 138$

Variance: $138/5 - 19.36 = 27.6 - 19.36 = 8.24$ ✓