Q.The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is
(A) $\sqrt{\dfrac{52}{7}}$
(B) $\dfrac{52}{7}$
(C) $\sqrt{6}$
(D) 6

The standard deviation measures the spread of data about the mean.

Step 1: Find the mean

Step 2: Compute squared deviations from the mean

$(6-9)^2 = 9$, $(5-9)^2 = 16$, $(9-9)^2 = 0$, $(13-9)^2 = 16$, $(12-9)^2 = 9$, $(8-9)^2 = 1$, $(10-9)^2 = 1$

Step 3: Find the variance

$\sigma^2 = \frac{9 + 16 + 0 + 16 + 9 + 1 + 1}{7} = \frac{52}{7}$ …

Standard deviation quantifies how much individual data points deviate from the average. It's the square root of variance, which itself is the average of squared deviations. Squaring ensures all deviations contribute positively (no cancellation) and penalizes larger deviations more heavily.

The formula for standard deviation of $n$ observations $x_1, x_2, \ldots, x_n$ is:

where $\bar{x}$ is the mean.

Let me work through this dataset: $6, 5, 9, 13, 12, 8, 10$.

1. Find the mean

2. Calculate each deviation from the mean

3. Sum the squared deviations …