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For instance, "Diamond" and "Outliers" in single plot.

BoxWhiskerChart[RandomVariate[LogNormalDistribution[1, 0.5], 100], "Outliers",  "Diamond"]

generates an error, as well as

BoxWhiskerChart[RandomVariate[LogNormalDistribution[1, 0.5], 100], {"Outliers",  "Diamond"}]

Thanks

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SeedRandom[1]
data= RandomVariate[LogNormalDistribution[1, 0.5], 100] ;
BoxWhiskerChart[data, {"Diamond",{ "Outliers"} }, ImageSize->300] 

enter image description here

"Diamond" and "MeanDiamond" both represent "mean confidence interval", but the latter can be used with parameters width and style.

Row[{BoxWhiskerChart[data, {{"MeanDiamond"}, {"Outliers"}}, ImageSize->300],
  BoxWhiskerChart[data, {"Diamond", {"Outliers"} }, ImageSize -> 300] }]

enter image description here

Same is true for "Notched" and "MedianNotch".

The documentation is not clear about how different elements can be combined. From checking various combinations it seems that at most one of the

"Basic" 
"Outliers"
"Notched"
"Median"
"Mean"
"Diamond"

can appear in a specification. Each of these elements can be combined with any number of the following:

 {"Fences", width, style}
 {"MeanDiamond",width,style}
 {"MeanMarker",width,style}
 {"MedianNotch",width,style}
 {"MedianMarker",width,style}
 {"Outliers",marker,style}
 {"FarOutliers",marker,style}
 {"Whiskers",style}
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