If you score 55th percentile on an exam, which of these are true? I'm not sure where I went wrong here... I've tried multiple combinations but still can't find the right combination.  Technically 3 and 5 are the ones that should be true right?  I've also tried 1,3,5, and 6 as well thinking maybe it was a trick question since 45% of a population of 100 could also be in the lower 55th percentile?  I'm lost on this. Any help/explanation is greatly appreciated. This is for statistics class.


45 % of the class scored worse than I did
I earned a 55% on the exam
55% of the class scored as good or worse than I did
I earned an F
45% of the class scored better than I did
45% of the class scored as good or worse than I did


Thanks
Example from R statistical software.  I generate a random sample of size 100
from $Norm(\mu=150, \sigma=20).$ Then round to integers, and sort.
The brackets in the output show the index of the first observation on each line.

x = sort(round(rnorm(100, 150, 20)))
&gt; x
 [1] 117 118 120 125 126 127 127 128 128 129 130 130 131 132 133 133 133 135 135 136
[21] 137 137 139 140 140 141 142 142 142 142 143 143 144 144 144 145 146 146 146 146
[41] 147 148 149 151 151 151 151 151 152 152 153 153 153 154 154 155 155 155 155 156
[61] 156 156 156 157 157 158 158 158 159 159 159 159 160 160 161 161 163 165 165 166
[81] 167 167 167 167 167 167 168 170 170 171 171 173 174 180 180 181 181 185 187 215


The 55th and 56th ordered observations are 154 and 155, respectively.

&gt; x[55]
[1] 154
&gt; x[56]
[1] 155


According to the default method of finding quantiles in R, the 55th percentile
is 154.45 (and there are two correct answers to your question).

&gt; quantile(x, .55)
   55% 
154.45 


If you want to see particulars of several more methods of finding quantiles,
type ? quantile at the prompt in the Session window of R, and look at Types.
The default type in R is 9. (There may be more information here than is worth knowing.)

However, if we specify type 1, then we get a
different result. This is the inverse of the ECDF (empirical CDF), a
definition used in some textbooks. (Here, you would have only one correct answer.)

&gt; quantile(x, .55, type=1)
55% 
155 


Each of the nine methods mentioned on the R documentation page has
optimal properties in some particular situation or for some particular
purpose. There seems no possibility of general agreement.

The good news is that percentiles are usually used for very large datasets
and for descriptive purposes. All of the definitions give answers that lie in a closed
interval between two consecutive sorted observations. So in practice
(except for poorly-constructed multiple choice questions),
it does not much matter which definition is used.
Percentiles can be a mite confusing, so this is how I remember it: there is no 100th percentile. That means if I score in the 99th percentile, then 99% of the scores are below mine. Extrapolating, if you score in the 55th percentile, then 54% are below you, and 45% did as well as you or better $(99 - 54 = 45)$. Using that definition, then, none of the answers are true -- except for #6, as $45 &lt; 54$, but it is misleading and shouldn't have gotten past the review stage. Of course, the use of "good" in place of "well" means it probably wasn't reviewed in the first place.