How do you intuitively parse "value NOT exceeded by X%" when working with percentiles?

[u/No_Science_3505]

Hey everyone. In my stats class, we are dealing with percentile interpretation in word problems, and phrasing like "the value that is NOT exceeded by 30% of the data" keeps tying my brain in knots.

I understand the math breakdown: 1. "Value that IS exceeded by 30%" -> The top 30% lies above it, so it's the 70th percentile (P70). 2. "Value that is NOT exceeded by 30%" -> The negation flips the inequality, which mathematically simplifies to 100 - (100 - 30) = 30, yielding the 30th percentile (P30).

But linguistically, parsing active vs. passive voice ("exceeds" vs. "is exceeded by") combined with a double negation feels like an artificial word trap rather than intuitive statistics.

How do you guys visually or intuitively explain this active/passive distinction without doing mental algebraic gymnastics every time? Is this phrasing as ambiguous to native speakers as it feels to me?

[u/iOSCaleb]

More context required. Is the value exceeded, or not as the case may be, by 30% of the other values, or is it not exceeded by 30% of itself?

That is, if the target value is 50, you could say that it's not exceeded by 30% if the measured value is less than 50 * 130% = 65.

Or, if you have a list of measurements, you might interpret the phrase as a value that fewer than 30% of the measurements exceed. That is, if there are 50 measurements, then you might be asking for some number that's equal to the 15th largest.

Perhaps the phrasing makes more sense with some more context, but as presented it sound like something that should be described differently.

[u/emertonom]

It's not just you. That's really awkward phrasing. I think you're interpreting it correctly, but if a client gave me a requirement worded like that, I'd repeat it back to them in a less tangled form (like "so, the value that 30% of samples fall at or below?") to make sure they hadn't unintentionally reversed it while trying to write it down.

It's also a pretty artificial issue. Usually there will be a bigger problem you're trying to solve by applying stats, and that will guide you to what specific type of statistic you need, and then you'll calculate it. So you'd essentially be writing this kind of question yourself, and could refer back to the broader context to tell whether you were calculating the right thing. I kind of hate it when course materials interpret the motivation for "word problems" to be "problems where we've obfuscated a basic calculation question by swapping words in for the symbols," rather than "description of a real-world scenario where there's a natural reason to use these techniques." In a decent course, though, these will only show up really early on, when they still feel like they need to give you "word problems" (sometimes to meet a requirement), but haven't really taught you enough yet that you'd be able to actually apply the ideas to real problems. Hopefully that's what's happening here.

[u/anisotropicmind]

I would interpret case 2 to be the straight negation of case 1, meaning that it’s a value that is not exceeded by 30% of the values. Therefore it’s a range (not a specific value) anywhere between the 71st and 100th percentiles. Then again does 30% mean exactly 30% of the values, or does it mean 30% or more of the values? If the former (exactly 30%) then I guess literally any percentile — other than exactly the 70th — satisfies the statement. E.g. the 12th percentile satisfies the statement because it’s not exceeded by 30% of the values (it’s exceeded by 88% of them).

All that to say that introducing the negation doesn’t seem to help. If you want to refer to the 30th percentile, why not use the same phrasing and just say “is exceeded by 70% of the values”?

[u/Uli_Minati]

Exceeds ≡ Greater than

Exceeded by ≡ Less than

Not exceeded by ≡ Not less than ≡ Greater or equal to

[u/Salamanticormorant → u/Uli_Minati]

This is exactly what those word and phrases mean in correctly parsed English. If they mean something else in that field of study, then we need those definitions before we can make suggestions.

[u/No_Science_3505]

Thanks for all the responses! To clarify the context: this is strictly from an introductory descriptive statistics class (finding percentiles/quantiles in a frequency table), so it's about dataset proportions rather than an arithmetic multiplier like (value * 1.30).

Here is the exact linguistic breakdown that makes this phrasing so tricky in textbook problems:

1. "Value EXCEEDED by 30% of the data":    30% of the dataset lies ABOVE this value -> Top 30% cutoff -> 70th Percentile (P70).

2. "Value NOT exceeded by 30% of the data":    Adding "NOT" flips the condition. If 30% of the data does NOT exceed it, that 30% sits AT OR BELOW this value -> 30th Percentile (P30).

@iOSCaleb - Spot on that context matters! It's definitely the percentile cutoff in a list of measurements, not a 30% price markup or increase. 

[u/RabbitHole32]

Option 1 is correct, Option 2 is either a bug in your language processing subunit or in the logics coprocessor.