Given that the sum of squares of the deviation of seven observations about their arithmetic mean is 77, we need to find the variance (VAR) and the corrected estimate of the variance (VARA) of the observations.

The sum of squares of the deviation about the mean is given by:

$\sum_{i=1}^{n} (x_i - \bar{x})^2 = 77$

Here, $n$ is the number of observations, which is 7.

Variance (VAR)

The variance (VAR) is calculated by dividing the sum of squares of the deviation by the number of observations ($n$):

$\text{VAR} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

Substitute the given values:

$\text{VAR} = \frac{77}{7} = 11$

Corrected Estimate of the Variance (VARA)

The corrected estimate of the variance (VARA), also known as the sample variance, is calculated by dividing the sum of squares of the deviation by $n - 1$:

$\text{VARA} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$

Substitute the given values:

$\text{VARA} = \frac{77}{7 - 1} = \frac{77}{6} \approx 12.83$

So, the variance (VAR) of the observations is 11 and the corrected estimate of the variance (VARA) is approximately 12.83.

Would you like more details or have any questions?

Here are some more questions you can ask:

1. What is the standard deviation and how is it calculated?
2. How do you calculate the mean of a set of observations?
3. What is the difference between population variance and sample variance?
4. Can you explain the concept of standard error?
5. How do outliers affect the variance of a dataset?

Tip: The corrected estimate of variance (sample variance) is used when estimating the variance of a population from a sample. It accounts for the fact that a sample may not fully represent the population by dividing by $n - 1$ instead of $n$.