A way to find out if my coin tosses are independent from each other or not Background:
Suppose I toss a coin 9 times. The outcome (i.e., $H ~or ~T$) of these $9$ tosses turn out to be:
$H ~T ~H ~H ~H~ T~ H~ T~ H$
Now, I have $10,000$ other people each toss a similar coin 9 times and I record the outcome of their tosses as well.
As a crude measure of correlation between my tosses, I define the following:
I consider the number of switches from  $H$ to $T$  or from $T$ to $H$.
Using this measure, I find 6 such switches in the outcome of my own tosses.
I apply this measure to the outcome of the tosses of the other $10,000$ people as well which results in the following frequency distribution (see below).
Question
My number of switches (i.e., 6) doesn't seem to be among the most frequently occurring number of switches by $10,000$ people, who just repeated my coin toss experiment.
What can I say about the independence of my coin tosses from each other based on this frequency distribution?

 A: If you toss the coin a large number of times you can test for independence verse correlation with just one sequence using the Wald-Wolfowitz runs test which counts the number of runs which is the number of times you get a sequence of runs.  In your example of 9 tosses, you have seven runs.  A small number of runs indicates independence.  Say the number of tosses is N.  Let N$_h$ be the number of runs of heads and N$_t$ be the number of runs of tails. Then N=N$_h$ + N$_t$ and it is known that under the null hypothesis that the distribution of runs is approximately normal with mean 2N$_h$ N$_t$/N +1 and variance 2 N$_h$ N$_t$(2 N$_h$ N$_t$-N)/(N$^2$(N-1)). Using this you can apply the test and reject the null hypothesis if the number of runs is much higher or lower than the mean.
In a way, this is similar to your counting of switches but it is known what is large and what is small. I do not think the distribution of your switches is established and there is no need to repeat the experiment many times with different coins.
You can read more about this by googling Wald-Wofowitz runs test.
