I have calculated a type of mean which considers how the sum of the values is shared between individual values, but do not know if it has a name. I've previously been told it is the weighted median, but do not think it is. It is equivalent to if you were to sum values within classes and fit a normal distribution to the totals in each class, but can be calculated when there are only a few values.
Consider values of $3, 7, 15, 30, 40, 60$
Cumulative values: $3, 10, 25, 55, 95, 155$
The arithmetic mean is $\frac{155}{6} = 25.83$, but most of the total is present in the final $3$ values and I want a value that accounts for this and that would be relatively unchanged if we were to add a value of 1 at the start of the series. If we plot the cumulative weights against the individual weights, the value I calculate is where $50\text{%}$ of the total ($77.5$) intersects with the individual values. There is a graphical representation here.
I calculate it in the following way:
$$30+\frac{(77.5-55)}{40}*(40-30)$$
giving $35.625$. I'll try to explain it mathematically:
$$v_{low}+\frac{(m - c_{low})}{v_{high}} * (v_{high}-v_{low})$$
$v_{low} = 50\text{%}$ of the total intersects above this value
$m = 50\text{%}$ of the total
$c_{low}$ = the cumulative value for $v_{low}$
$v_{high} = 50\text{%}$ of the total intersects below this value