Concept: Mean and Variance of Grouped Data (Natural Numbers).

We compute the mean $\bar{x}$ using $\frac{\sum f_i x_i}{\sum f_i}$, then variance using $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$.

Step 1 – Compute $\sum f_i$ and $\sum f_i x_i$

$\sum f_i = 3+2+3+2+6+3+3 = 22$

$\sum f_i x_i = 92\cdot3 + 93\cdot2 + 97\cdot3 + 98\cdot2 + 102\cdot6 + 104\cdot3 + 109\cdot3$

$= 276 + 186 + 291 + 196 + 612 + 312 + 327 = 2200$

Step 2 – Mean

$\bar{x} = \frac{2200}{22} = 100$

Step 3 – Variance

Compute deviations:

$(92-100)^2\cdot3 = 64\cdot3 = 192$

$(93-100)^2\cdot2 = 49\cdot2 = 98$

$(97-100)^2\cdot3 = 9\cdot3 = 27$

$(98-100)^2\cdot2 = 4\cdot2 = 8$ …

Mean and variance for frequency data

With frequencies, $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$ and $\sigma^2 = \dfrac{\sum f_i (x_i-\bar{x})^2}{\sum f_i}$.

Step-by-step solution

Mean. $\sum f_i = 3+2+3+2+6+3+3 = 22$ and

so $\bar{x} = \dfrac{2200}{22} = 100.$

Variance. Since $\bar{x}=100$, each deviation is $x_i-100$: