In chi square contingency table 2x2: why we sum up all four cells, but compare with chi square distribution with 1 df (only one square)? I have read this and this and I understand where squared standard normal distribution comes from. I also understand why df = (r-1)(c-1). But I don't understand why I sum all fours cell (four squared standard normals) and compare this value with distribution of only one squared standard normal.
 A: Here is one kind of chi-squared test based on a $2 \times 2$ table.
We have 350 women and 320 men selected at random from the population of a city. We want to know whether the probability of having a college degree is the same in the two groups. 
Let $p_w$ and $p_m$ be the respective probabilities. Under the null hypothesis $p_w = p_m.$ Let's suppose both probabilities are $1/5.$ 
We can use binomial distributions to simulate data.
Here is how to simulate data for a single chi-squared test (using parameter cor=F to avoid the Yates continuity correction, which does not exactly use a chi-squared statistic).
set.seed(310)
x = rbinom(1, 350, 1/5)
y = rbinom(1, 320, 1/5)
DTA = rbind(c(x, 350-x), c(y, 320-y))
DTA
     [,1] [,2]        # 2 x 2 table
[1,]   54  296
[2,]   71  249

chisq.test(DTA, cor=F)

        Pearson's Chi-squared test

data:  DTA
X-squared = 1.5776, df = 1, p-value = 0.2091

Here is now to get chi-squared statistics from 100,000 such tests:
set.seed(2019)
m = 10^5;  q = numeric(m)
for(i in 1:m) {
  x = rbinom(1, 350, 1/5);  y = rbinom(1, 320, 1/5)
  DTA = rbind(c(x, 350-x), c(y, 320-y))
  q[i] = chisq.test(DTA, cor=F)$stat
  }
mean(q);  var(q)
[1] 0.9990056     # aprx E(Q) = 1
[1] 2.002622      # aprx Var(Q) = 2

lbl = "Simulated Chi-sq Statistics with CHISQ(1) Density"
hist(q, prob=T, br=40, col="skyblue2", main=lbl)
 curve(dchisq(x,1), add=T, lwd=2, col="red", n=1001)

Under the null hypothesis that the two probabilities are equal,
the chi-squared statistic $Q$ (X-squared in the output) has nearly the distribution
$\mathsf{Chisq}(1),$ for which the mean is $1$ and the variance is $2.$
The figure below shows a histogram of the 100,000 simulated
values of $Q$ along with the closely-matching density function of 
$\mathsf{Chisq}(1).$

