Did Anderson actually pass?

When Anderson releases the computer hacker guy and lets him walk free. Dredd tells her that releasing a felon is not only a fail but a criminal offense. Anderson explains that she has already failed because she lost her side arm (an automatic failure). She says "I'm not going to be a judge and I don't need to be a mind reader to know it" implying that she knew the outcome of her assessment without having to read Dredd's mind.

So technically she did fail, and fail multiple times.

I think Dredd passes Anderson on grounds that she represents what the judges need. A person who wants to make a difference. There are a couple of spots in the film where Anderson demonstrates why she wanted to become a judge.

• When they first enter Peach Tree. Dredd makes comments about how crime infested the building is (going as far as to cite statistics). Anderson shuts him up by telling him she was raised in one of these blocks. When asked why she wanted to become a judge, Anderson responds "Yes, I believe I can make a difference". Dredd responds with "admirable" implying a sense of hopelessness to her goals.
• In the scene where she releases the computer hacker. She says that "I'm still a judge, and I'll still dispense justice. At least while I can maybe I'll make a difference". This is also the only time in the film where you see Dredd smile (or as close to smiling as he can).

The corrupt four judges that Ma-ma calls in to have Dredd killed represent that current state of corruption in the system. One of the judges says to Dredd that "the city is a meat grinder". Implying that the law is hopeless and a lost cause.

Anderson doesn't read minds constantly. It's an action she has to willingly perform. When Dredd and Anderson are discussing what to do when they discover that Ma-ma is the king-pin for the slow-mo drug cartel in the city. Dredd says "You gave us two options. Defend or hide. What about we attack? Head straight for Ma-ma". Anderson says "Is that an option?" Dredd replies "Well she's guilty. We're judges." Anderson says worriedly, "sir, I think we should wait for backup". Dredd just frowns. Anderson continues "Wrong answer?" Dredd answers "You're the psychic." From this it is clear that Anderson only reads minds when she needs to, otherwise she would have said to go after Ma-ma in the first place.

So to summarize:

Dredd knew she failed multiple times. Anderson knew she failed, and that's why she gave him her badge. Dredd passed Anderson anyway cause he saw hope that she might make a difference. Chief Judge says "I knew she would" because she saw it too.

Actually, my take on it was that while she acknowledged she technically failed, she ultimately turned in her badge because she couldn't handle the questionable ethics and the emotional intensity of the job. Case in point: she executes one of the Ma-Ma clan members who was begging for his life, then later discovers he was the husband and father of a Peach Trees block wife and kid.

When she releases the hacker, she also says

"I'm not going to be a Judge and I don't have to be a mind reader to know it."

She goes on,

"He's a victim not a perp and until my assessment is formally over I'm still entitled to dispense justice and that's what I just did, by letting him go. Maybe that will be the one difference I do make. The code to Ma-Ma's chambers is 49436. Let's finish this."

I don't think it gets any clearer than this as to why she turned in her badge. A Judge must be strict, no mercy, the law is black and white, but because she's a psychic she sees the grey areas. "Let's finish this" is another reflection of her realization that she is not cut out for the harsh code of being a Judge. Her spirit is too kind.

At the end, when Dredd says,

"Your assessment is now over."

This is far from an answer on whether she passed or failed and she knows it. The fact that she didn't even wait for the official answer suggests that she had already made up her mind.

I think when he said she passed, he actually meant she passed up being a judge. But it ended too early to determine if he literally passed her or if it was a figure of speech.

Obviously she passed and becomes a judge. The last scene of the movie is a silhouette (or back lit) view of Anderson in her Judge uniform ... at least this is what it appeared to be to me.

Dredd did pass her, it was up to Anderson after that. At the very ending scene you see Anderson carrying a helmet and a new gun implying she learned from Dredd and decided to become a Judge, plus in the comics she was the lead Judge is the pyi division.

She's a Psychic, the sidearm isn't her primary weapon, her brain is. So the loss of her brain, or having it taken from her is a fail, the gun is just another tool to her, but she'd have failed if she lost the psychic duel with Kay.

With the judgment call on the hacker being a victim, whilst unusual for a judge to be lenient, given her psychic powers she's far more likely to assess a victim correctly than a regular judge, additionally she could just as easily pick up on a perp playing the victim.

And we see leniency from Dredd himself with the vagrant at the beginning.

So at no point did she actually fail, she merely judged herself along the same lines as Dredd, which was silly, cos hey, psychic.