Normal curve for normal distribution

Question

I'm reading Linear Regression and correlation by Scott Hartshorn. There is Regression example for a television show Modern Family. We calculated equation for a regression line to predict number of viewers of that show. Then there is the example predicting errors between predicted number of viewers and actual value. So we have

____________________________________________________________________________
|Normalized   |      Predicted normalized               |   Regression error |
|viewers (Nv) |  viewers with our Regression line(PNv)  |      (PNv - Nv)    |
|____________________________________________________________________________|
|1            |              0.915                      |        -0.085      |
|0.79         |              0.907                      |         0.114      |
|...          |              ...                        |         ...        |
|_____________|_________________________________________|____________________|

We made a histogram of the regression errors.

Then the author drew normal curve(it is on the histogram). And I don't understand the formula is being used to draw that curve.

We have standard deviation of Regression errors = 0.074 There is table with data for provided histogram above.

_______________________________________________________________________________
Histogram bin index   |    Histogram bins |          Normal curve              |
______________________|___________________|____________________________________|
       1              |         -0.2      |      =NORM.S.DIST(-0.2/0.074, 0)*  |
                      |                   |   Number_of_episodes*(0.02/0.074)=0.458  |
______________________|___________________|____________________________________|
       2              |         -0.18     |      =NORM.S.DIST(-0.18/0.074, 0)* |
                      |                   |   Number_of_episodes*(0.02/0.074)=0.9194  |
       ...            |         ...       |            ...                     |
______________________|___________________|____________________________________|

Per my understanding normal curve consists of product of 3 parts:

1. NORM.S.DIST(-0.2/0.074, 0) That excel function is used to calculate probability density function value for standard distribution. -0.2/0.074 - Z score and 0 - is just a marker that we need to use density.

2.Number_of_episodes as far as I understand it is size of sample. It is sum of values from y axis on the histogram above.

3.(0.02/0.074) not sure what it is. Based on the table and graph above, 0.02 is distance between bins on histogram. 0.074 is standard deviation.

Can you please explain what is the formula being used by author to calculate normal curve? I though that NORM.S.DIST(-0.2/0.074, 0) is enough.

I'm kind of newbie in statistic and probability, sorry if I missed some data to be shown. I will edit the message and add it.

Answer 1

A normal distribution is completely characterized by its mean $\mu$ and its standard deviation $\sigma$ (equivalently, its variance $\sigma^2.$) Once you compute the mean (I assume you know how to compute that) and the (sample) standard deviation: $$\sigma=\sqrt{\frac{1}{N-1}\sum_{i=1}^N(y_i-\overline{y})^2},$$ then the formula for a normal distribution is $$f(y)=\frac{1}{\sigma\sqrt{2\pi}}\,\exp\left[-\frac{(y-\mu)^2}{2\sigma^2}\right].$$ However, as whuber pointed out in his comment, the histogram shown is not that of a distribution, since the area underneath it is clearly greater than $1.$ To get an approximate height for the curve, you can simply multiply the formula above by the sum of all the values (an approximation to the area under the curve) times the horizontal differential: $$\tilde{f}(y)=\frac{1}{\sigma\sqrt{2\pi}}\,\exp\left[-\frac{(y-\mu)^2}{2\sigma^2}\right]\times\Delta x\times\sum_{i=1}^ny_i.$$

This is my guess for how the author made the normal curve.