### by Bobby Situkangpoles with contributions from +Don Takahashi

During the launch of the iPhone 4, Steve Jobs said that the human eye, viewing a display from 12 inches away, can’t discern individual pixels if the density is over 300 pixels per inch.

This resulted in PPI being used in almost every device reviews that comes after the 2010 launch of the iPhone 4.

Nowadays, we have smartphones and tablets available with screen sizes ranging from 3 inches all the way up to 8 inches. With such diverse screen size options available in the market right now, is PPI still has a place in reviews and spec sheets?

To get an answer to this let's take a look at our solar system.

The sun is about 400 times larger than the Moon, yet when viewed from Earth, both celestial bodies have similar angular diameter. The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle.

Here's another example,

Even though Pluto is physically larger than Ceres, when viewed from Earth (e.g., through the Hubble Space Telescope) Ceres has a much larger apparent size.

If we apply the two analogies above to individual pixels on our devices' screens, we can start to see PPI measurement's deficiency when used in comparing screens with differing sizes.

Devices with larger screens will be used at further distances from the eyes than smaller ones. Hence, there's reason to argue that even though a larger screen has a lower PPI, thus larger pixels, at different normal viewing distances those pixels can actually be perceived to be smaller by our eyes compared to the physically smaller pixels of a higher PPI smaller screen.

Devices with larger screens will be used at further distances from the eyes than smaller ones. Hence, there's reason to argue that even though a larger screen has a lower PPI, thus larger pixels, at different normal viewing distances those pixels can actually be perceived to be smaller by our eyes compared to the physically smaller pixels of a higher PPI smaller screen.

Below is the formula astronomers use to measure visual diameter of objects

δ = 2 arctan (d/2D)

Where

d = actual diameter

D = Viewing distance

The formula showed that in order to measure the angular diameter or apparent size of individual pixels on a mobile device, we need to know the actual diameter of the actual pixels as well the normal viewing distance for the device.

When Apple launched the iPad 4 with its 264 PPI 9.7 inch display, Apple said it qualifies to be called a "retina" display because they assume that people will hold the iPad between 15-18 inches away from their eyes instead the iPhone's 10-12 inches.

Considering the above fact, I went with the following formula to determine normal viewing distances of devices:

Normal viewing distance = (7 x Screen diagonal)/4

This puts normal viewing distance for the 9.7 inch iPad at 17 inches, in line with Apple’s expected range as stated above, while the normal viewing distance for the 4 inch iPhone 5S is at 7 inches. Let’s take these numbers as the closest people would normally hold their devices according to their respective screen sizes.

Altering the formula to (11 x Screen diagonal)/4 returns normal viewing distance for the iPhone to 11 inches, which is right in the middle of Apple’s 10-12 inch normal viewing distance but it puts the iPad’s normal viewing distance at 26.7 inches. That’s a bit unrealistic but for a fair comparison we incorporated the results from both viewing angle calculations in the spreadsheet provided.

As for the actual physical diameter of each pixel, we can calculate it using each display’s PPI. 1/PPI gives us the approximate actual diameter for each pixels in inches.

If we do the aforementioned calculations then for the iPhone 5S we get:

0.0031 actual pixel diameter,

and,

7 inch of normal viewing distance

Putting these numbers into the formula for angular diameter we found that the apparent size of the individual pixels of the iPhone 5s, when viewed at 7 inches away, is 0.0251 degrees or 1.506 arc-minutes (since 1 degree = 60 arc-minutes)

To see the problem with comparing PPI numbers of today’s various display sizes, let’s compare the apparent size of individual pixels on the iPhone 5S at normal viewing distance to that of the Galaxy Mega 6.3’s.

233 PPI on a 6.3 inch screen might sound alarmingly deficient. A quick Google search of the phrase “Galaxy Mega 6.3 PPI” will give you links to numerous reviews that echo this prejudice with sentences such as the following:

“To have such a gigantic display, it would have been nice to have included a higher-resolution display. Offering 233 ppi, Galaxy Mega 6.3's display is a bit of an eyesore.”

“You'll malign it for its dismal DPI of 233”

Is the perceived screen density of the Galaxy Mega 6.3 really that bad when held at normal distance?

Our aggressive normal viewing distance formula returned 11 inches as the lower limit of the Galaxy Mega 6.3’s expected “normal viewing distance”. While the actual diameter of the Mega’s individual pixels is 0.0043 inch.

Running those numbers through the angular diameter formula gave us apparent pixel diameter of 0.0142 degrees or 0.852 arc-minutes.

This means the “low” pixel density screen of the Mega 6.3 actually has 11% smaller pixels than those of the iPhone 5S' when both devices are held at their respective normal viewing distances. This means that at normal viewing distance, the Mega's 6.3 inch screen is 11% sharper than the iPhone 5S'.

If 0.0142 degrees of apparent pixel size is considered “dismal”, I wonder what do the reviewers think about the iPhone 5S’ 0.0251 degrees apparent pixel size at its normal viewing distance, “horrific”?

We did the same process to other popular devices from the 4.7 inch 441 PPI HTC One to the 10.05 inch 300 PPI Nexus 10 and here are the results:

Here's the link to the spreadsheet

Here are the results in graphs:

As you can see, among the devices compared, the iPad Mini is the only one with larger apparent pixel size (the least sharp) when viewed at its normal viewing distance of 13.8 inches.

We can also see how devices with 1080p displays easily beats the iPhone 5S’s apparent pixel size by more than 40% while the 9.7 inch iPad 4 and the 10.05 inch Nexus 10 beat the yardstick by 49% and 57%, respectively.

Whereas 720p small tablets like the 2012 Nexus 7 and Samsung’s Galaxy Note 8 also beat the iPhone 5S in apparent size of individual pixels when held at their normal viewing distances by 14 and 15 percent respectively.

What does all this mean? Is the 326 PPI 4 inch display in the iPhone 5S a low resolution display? Certainly not, as when viewed at its normal viewing distance it would be very hard to tell the individual pixels apart. What this does mean is that bigger displays with significantly lower PPI are not the dismal pixelated mess that most reviews seemed to suggest. On the contrary, most of these larger devices, often labeled as having low PPI screens, actually display smaller apparent pixel size (thus better sharpness) when held at their normal viewing distance.

When trying to get an idea of how sharp a screen would likely be, PPI is as useful as peak horsepower in describing how fast a car would be in a drag race. Without knowing the weight of the car, there’s no way of getting any idea of how fast the car would go in a drag race.

Of course, this is an over simplification as any petrol head will tell you that there are many other variables involved in drag racing as they do in display quality, however, the analogy still serves us well as both horsepower and PPI are often misleadingly put front and center on reviews.

Without considering the size of the display and the normal viewing angles afforded by the size for each device, PPI does not give us any meaningful measure of a display's sharpness. So can we please stop using it straight up as one of the most important variables when comparing the quality of screens?