Iβm getting different answers for median(v,w::AnalyticWeights) and quantile(v,w::AnalyticWeights,0.5) and not sure why. Any ideas?

Please provide an example

```
using StatsBase; using Distributions
v=[1; 4; 3; 2; 2.5; 7];w=[0.1;0.3;0.05;0.05;0.2;0.3]
median(v,weights(w)::AbstractWeights)
quantile(v,weights(w)::AbstractWeights,0.5)
```

Median returns 4.0 and Quantile returns 3.5.

Sometimes a quantile isnβt uniquely defined (often solved by taking the average of the endpoint of the interval of quantile points). However, it only makes sense to use a definition that ensures that median and 0.5 quantile are identical.

In this case, it seems that things are worse. To me, it seems that the result of `quantile`

is just wrong. The 0.5 quantile and the median, say m, *is* the same thing and should satisfy P(X \geqslant m) \geqslant \frac{1}{2} and P(X \leqslant m) \geqslant \frac{1}{2}. For your inputs, I get

```
julia> x = [1; 4; 3; 2; 2.5; 7];
julia> w = [0.1;0.3;0.05;0.05;0.2;0.3];
julia> sum(w[x .<= 3.5])
0.4
```

so 3.5 is not a median. To see that 4 is the unique median, you can create the following table and see that the row with x=4 is the only one that has probabilities higher than \frac{1}{2}.

```
julia> p = sortperm(x);
julia> table(cumsum(w[p]), reverse(cumsum(reverse(w[p]))), x[p], names = [Symbol("P(X<=x)"), Symbol("P(X>=x)"), :x])
Table with 6 rows, 3 columns:
P(X<=x) P(X>=x) x
βββββββββββββββββββββ
0.1 1.0 1.0
0.15 0.9 2.0
0.35 0.85 2.5
0.4 0.65 3.0
0.7 0.6 4.0
1.0 0.3 7.0
```

See also https://github.com/JuliaStats/StatsBase.jl/pull/316 and discussion at https://github.com/JuliaStats/StatsBase.jl/issues/313. Matthieu Gomez is the person to contact about this, but he isnβt on Discourse AFAICT.

Note that `quantile(v, fweights(w))`

gives yet another answer (`7.0`

).

Itβs not clear why 3 should be considered a central element here though given the weight vector.

The inconsistency has been fixed by having `median`

call `quantile(x, w, 0.5)`

. See this issue and the associated PR.