Animated Solution for Mathematics - Statistics: Let the mean and the variance of 6 observation a, b, 68, 44, 48, 60 be 55 and 194 , respectively if $a>b$, then $a+3b$ is

Understanding Mean and Variance

Welcome, future IITian! Today, we are not just solving a statistics problem; we are learning to see the hidden structure behind numbers. Statistics is the language of uncertainty, and mean and variance are its two most important pillars.

The mean, $\bar{x}=\frac{\sum x_i}{n}$, is the balance point—the center of gravity of your data. The variance, ${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}$, is the measure of chaos, telling us how far our data points stray from that center. Let's dive in.

The Anchor (The Mean)

We are given six observations: $a,b,68,44,48,60$. The mean is $55$. Our first step is to find the sum of these observations.

Using the formula $\bar{x}=\frac{\sum x_i}{n}$, we write:

Summing the known values, $68+44+48+60=220$. Thus, $\frac{a+b+220}{6}=55$.

Multiplying by $6$, we get $a+b+220=330$, which simplifies to $a+b=110$. This is our first anchor, Equation 1.

The Measure of Chaos (The Variance)

Now, for the variance. We know ${\sigma }^2=194$. The formula is ${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}$. Substituting our values:

Let's calculate the squared deviations for the knowns: $(68-55{)}^2=169$, $(44-55{)}^2=121$, $(48-55{)}^2=49$, and $(60-55{)}^2=25$. The sum of these is $169+121+49+25=364$.

Plugging this back, we get:

Multiplying by $6$ gives $(a-55{)}^2+(b-55{)}^2+364=1164$. Subtracting $364$ yields $(a-55{)}^2+(b-55{)}^2=800$. This is Equation 2.

The Elegant Substitution

Here is where the JEE magic happens. Instead of expanding $(a-55{)}^2$, which is a recipe for algebraic disaster, let's use substitution.

Let $x=a-55$ and $y=b-55$. From Equation 1, we know $a+b=110$. Subtracting $110$ from both sides, we get $(a-55)+(b-55)=0$, which means $x+y=0$, or $y=-x$.

Now, look at Equation 2: $x^2+y^2=800$. Substituting $y=-x$, we get $x^2+(-x{)}^2=800$, which simplifies to $2x^2=800$, so $x^2=400$. Thus, $x=\pm 20$.

The Final Verdict

If $x=20$, then $a-55=20\Rightarrow a=75$. Since $y=-x=-20$, then $b-55=-20\Rightarrow b=35$. If $x=-20$, then $a=35$ and $b=75$.

The problem explicitly states $a>b$. Therefore, we must choose $a=75$ and $b=35$.

Finally, we calculate the target expression:

And there it is! By respecting the symmetry of the equations and using a clever substitution, we turned a daunting problem into a simple, elegant solution. Keep practicing, and keep that curiosity alive! The final answer is 180.