Draw histogram with normal distribution overlay from data

Question

I was asked to draw a histogram with normal distribution overlay over our data. Our data is an array of doubles with unlimited range. The idea is next:

1. Split all my values into buckets (I call them steps in my code)
2. Find all values that happen to be inside each bucket
3. Calculate the number of items in the bucket and divide them on the number of the items overall
4. Calculate mu as avg(values)
5. Calculate variance as avg([(each value - mu)^2])
6. Draw overlay with formula: 1. / Sqrt(2 * Pi * var)* e^((-(x - mean)^2 / 2 / var)

Here is what I wrote so far:

double[] values;
const int StepsNumber = 30;
// Choosing the size of each bucket
double step = (values.Max() - values.Min())/StepsNumber;

double mean = values.Average();
double deviationSq = values.Select(x => Math.Pow(x - mean, 2)).Average();

var bucketeer = new Dictionary<double, double>();
for (double curr = values.Min(); curr <= values.Max(); curr += step)
{
        // Counting the values that can be put in the bucket and dividing them on values.Count()
        var count = values.Where(x => x >= fromVal && x < fromVal + step).Count();
        bucketeer.Add(fromVal, count / values.Count());
}

// Then I build normal distribution overlay 
var overlayData = new LineSeries();
int x0 = values.Min();
int x1 = values.Max();
for (int i = 0; i < n; i++)
        {
            double x = x0 + (x1 - x0) * i / (n - 1);
            double f = 1.0 / Math.Sqrt(2 * Math.PI * varianceSq) * Math.Exp(-(x - mean) * (x - mean) / 2 / varianceSq);
            overlayData .Points.Add(new DataPoint(x, f));
        }

// And draw everything

plotModel.Series.Add(overlayData);
        foreach (var pair in bucketeer.OrderBy(x => x.Key))
        {
            columnSeries.Items.Add(new RectangleBarItem(pair.Key, 0, pair.Key + step, pair.Value));
        }
plotModel.Series.Add(columnSeries);

But the result looks a bit strange:

The histogram does not seem to match the overlay. It feels like I'm missing something - either calculating buckets wrong, or have a mistake in the math.

Answer 1

This question's pretty stale now, but I found it while trying to do something similar, so I'll offer this advice:

Firstly, the varianceSq variable should actually be the variance (or standard deviation squared).

Secondly, the standard formula to calculate f from the mean and standard deviation makes a curve which has an area of 1 below it. To match the histogram you need to scale the values up by the total area of the histogram rectangles

i.e. y = f * (bar width * total of bar heights).