I have the following data (`prop` is like empirical CDF): 

```r
td <- data.frame(a = 3:14, prop=c(0, 0, 0.026, 0.143, 0.21, 0.361, 0.535, 
                                  0.719, 0.814, 0.874, 0.950, 0.964))

```

I want to fit a normal CDF using an appropriate mean and standard deviation. How can I find such a mean and a standard deviation value that fits the data above? 

```r
mean <- 8.8 # <-- How can I find a best fitting number?
sd <- 2.3 # <-- How can I find a best fitting number?
x <- seq(from = 2, to = 15, by = .1)
cdf <- data.frame(x = x, y = pnorm(q = x, mean = mean,  sd = sd))
library(ggplot2)
ggplot(data = td, aes(x = a, y = prop)) + geom_point() +
  geom_line(data = cdf, aes(x = x, y = y))

```

[![enter image description here](https://i.stack.imgur.com/jqUxx.png)](https://i.stack.imgur.com/jqUxx.png) 

**EDIT:**

The motivation behind this question is to replicate a graph I saw on a book. The book used the same proportions and fitted the normal ogive to the data. It looks like normal ogive fitted so well and I couldn't replicate it. There is no way the author used the raw data because the data is from a 100 years old book and the author's book published about 17 years ago. Here is the graph with its caption:

[![enter image description here](https://i.stack.imgur.com/6iBsQ.png)](https://i.stack.imgur.com/6iBsQ.png)

*Caption*: "Proportions correct on item 46, plotted against age, with a fitted normal ogive."