Is Wi-Fi connection fast on your iPhone 5?

I've got an iPhone 5 and its Wi-Fi connection should be very fast, as Apple said. Nevertheless, my iPhone 5 in Wi-Fi is slower than my pc (Windows) that is 6-7 years old. Especially in YouTube, it's often impossible to me watching HD videos, because it loads them too slowly. Is it normal? If I took my iPhone to an Apple Store, they wouldn't find any problem, because it's an occasional problem; some days it's "fast" (normal), some days it's slow. That's very annoying! I've also installed today iOS 6.1 but nothing has changed.

Contrarily, in 3G it's very very fast.

Added (1). @Marquez: yeah, my modem is old, but my pc loads videos fast, even in HD.

Yeah i had the same problem but with my iPhone 4.It has nothing to do with your phone. Its your computer's modem. If you have had your modem for a long time as in your case 6-7 years you have to replace it. They tend to die slowly over time. Just contact you ISP and they will tell you the same thing. ISP is internet service provider. ATT, Comcast, clear, etc.

Yes, its standard.