The table below summarizes the number of surface flaws found on the paintwork of new cars following their inspection after primer paint was applied by a new method:
Find the variance of the number of flaws per car.
I found the mean but I don't know how to find the variance. Also, what kind of distribution is this?
Many thanks!
Here are the formulas I would use. I will leave it up to you to match them with the notation in your textbook or notes, and to do the computation.
You have $k = 7$ values $v_1 = 0,\, v_2 = 1,\, \dots,\, v_7 = 6,$ and you have $k$ corresponding frequencies $f_1 = 3,\, f_2 = 7,\, \dots,\, f_7 = 2.$ The total number of observations is $\sum_{i=1}^k f_i = 40.$ You say you have found the sample mean $\bar X = \frac{1}{n} \sum_{i=1}^k f_iv_i = 2.45.$ (Your book might call the values $x_i$ instead of my $v_i.$)
Then the sample variance is $$ S_X^2 = \frac{1}{n-1} \sum_{i-1}^k f_i(v_i - \bar X)^2.$$
You might want to make a table with columns headed $i,\, f_i,\, v_i, v_i - \bar X,\, (v_i - \bar X)^2,$ and $f_i(v_i - \bar X)^2.$ (The body of the table will have seven rows.) Then find the total of the last column and divide that total by $n-1 = 39.$
Note: This is a sample from some unknown discrete probability distribution. My best guess is that the population distribution from which the data were randomly sampled might be a Poisson distribution with mean approximately 2.45. But that is only speculation. Samples from Poisson populations often have sample means and variances that are numerically not far apart. My (unchecked!!) value for the sample variance is a little above 2, so that encouraged me to mention the Poisson idea. Maybe later in your course you will do a formal test to see if the data are truly consistent with a Poisson population.
The sketch below shows your frequencies (bars) along with frequencies that would be 'expected' if data were sampled from a Poisson distribution (red dots). The fit does not look fantastic, but it is actually not bad for a sample as small as $n = 40.$
