MedianCI[y_,alpha_]:=(

(* This procedure implements David Olive's simple median confidence interval*)
(* First,compute the median *)

n=Length[y];
ysort=Sort[y];
nhalf=Floor[n/2];

If [2*nhalf<n,
med=ysort[[nhalf+1]];,
med=(ysort[[nhalf]]+ysort[[nhalf+1]])/2;
];

(* Next,compute Olive’s standard error for the median *)

Ln=nhalf-Ceiling[Sqrt[n/4]];
Un=n-Ln;
SE=0.5*(ysort[[Un]]-ysort[[Ln+1]]);

(* Compute the confidence interval based on Student’s t-distribution.*) 
(* The degrees of freedom parameter p is discussed in Olive’s paper *)

df=Un-Ln-1;
p=1-alpha/2;

(*t=qt(p=1-alpha/2,df=p)*)

t=Quantile[StudentTDistribution[df],p];
medLCI=med-t*SE;
medUCI=med+t*SE;
Return[{medLCI,medUCI}];
)

Needs["HypothesisTesting`"]

data=RandomVariate[SkewNormalDistribution[0,2,5],10^4];

SmoothHistogram[data]

BoxWhiskerChart[data]

Print["Mean: ", Mean[data]];
Print["Median: ", Median[data]];
Print["StDEV: ", StandardDeviation[data]];

Print["Mean 95% confidence interval: ", MeanCI[data]];
Print["Median 95% confidence interval: ", MedianCI[data, 0.05]];