When a photo indicates that it is, say, 300 DPI, is that based on a standard sized print?
i.e. is there a specific standard size that the DPI is based on, and if so what is that size?
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The manufacturer's use of 300 PPI is not based on any calculation, it is purely arbitrary and selected purely based on historical convention that hi-res printers were 300PPI.
Not all devices even record resolution (PPI) values, the raw files for my Nikon D70, D90 and D600 cameras don't report resolution in Adobe Bridge at all.
.jpg files based on those raw files do, suggesting there is a requirement in the jpeg metadata for resolution to be present (I may check and update later.)
Nope. The number is purely fictitious. The vast majority of cameras always put the same number. You can even check it out by changing the size of image output and you will see that the DPI stays exactly the same. So, if your 24 MP camera outputs a 24 MP image at 6000x4000 which specifies 300 DPI, it should correspond to a 20" x 13.3" but if you set it to 12 MP, it records at 3000x2000 which at 300 DPI is a 10" x 6.6" print.
A few digital cameras, mostly from Olympus as far as I can remember right now, let you set the DPI in camera. Still, the same number is stored in the metadata of all images, even if you change resolution, until you change the number again.
I think the aproach of a photographer and a designer is in this case the opposite. (I am both)
A Photographer needs a print of some size, for example an 8x10 print, and you can just send your photo on the Megapixels the photo has. The PPI are defined by the size of your print. It can be xxx, yyy or any number depending on the print size. The photo is streched to that size.
Is that based on a standard sized print?
No. The 300 ppi is based on a specific pre-press process for print in a specific paper and a specific quality on a specific plate resolution. (Technically the range is between 212 and 300 ppi.
The other resolution that is a myth is the 72 ppi for screen. Thoose 72 ppi was on one of the very first Macintoch computer, The number also had a coincidence on the value of the point vs an inch. 1 pt = 1/72 inch. So the value stays, but for screen you do not need that resolution value.
Is there a specific standard size that the DPI is based on, and if so what is that size?
No. The ppi alone do not tell you the physical dimension. It also needs the total pixel dimensions, so no. There are diferent Megapixel values.
If a camera has a resolution of 2400 x 3000 pixels then it simply has that many pixels, regardless of the PPI. It's not until you print that the PPI comes into play. At 300 PPI you'd get 8" x 10" (2400/300 and 3000/300). At 200 PPI you'd get 12" x 15".
So for a fixed resolution, if a camera reports 300 PPI, then yes that equates to a specific print size, but the PPI can be changed to whatever you feel appropriate, depending on the size of print you want, the viewing distance etc.