How to create time series animation with unequal time interval in the data in ArcGIS? for eg I have population data for the years 2013,2012,2010,2004,1990,1950,1900, so how can I create time series animation with this kind of unequal time interval?
There are two ways to animate data like you're talking about. The first is the Time Slider, which relies on time attributes in the data to create the span of the animation. You can make overall adjustments to playback, but it relies a constant, regular time interval - for instance show each year for one second. For this reason, it won't work well for what you want to do. 1900 would be shown for 50 seconds, then 40 seconds of 1950, 14 seconds of 1990, and so on. You can change what the interval is (days, years, months), but it has to be regular based on time.
The second way is actually several options/variations on the main them of using the Animation Toolbar (or an outside program). One method is simply to create a map for each year, save it as an image, and bring it into an outside program to turn into a video (or even a timed 'power point' type presentation). You then control how long each image is shown.
Within Arc the easiest way to do this is a group layer animation. I'll assume your data is all in one shapefile. You have some geometry (points, polygons) with an attribute for each year. You'll add that same shapefile multiple times to your document - once for each year. Symbolize each layer using the different year attributes; likely with the same symbology but based on different attributes so they maintain a consistent look and the change in the data is what stands out. You don't want one year red and one year blue unless you're showing them at the same time, adding more and more to the map as time progresses. This also works if each year is it's own file. If all in one file, a shortcut is to add it once, set up the symbology, then copy the layer and just alter which field (year attribute) the symbology is based on.
Once you have all the layers symbolized, in the animation track window you control when each layer is visible for the animation. As one turns off another turns on (or you turn them on in sequence if you want to go that route). The animation track controls how long they're visible for, whether they fade in/out, cross-fade, etc. The help file linked above should cover all the details.