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# 85th Percentile Speed Question

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Took a practice exam today and had this question. Seemed pretty simple I didn't agree with the solution..

Q:

The findings of a speed survey is summarized below

Speed Interval (mph) Frequency

20 to 24 2

25 to 29 11

30 to 34 18

35 to 39 29

40 to 44 24

45 to 49 13

50 to 54 3

The 85 percentile speed, in mile per hour, is most nearly:

A) 37

B) 40

C) 43

D) 45

Solution says its C but I answered D.

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Not a traffic engineer but

40-44 is 84%

45-49 is 97%

43 mph is much closer to 85% (-1%) than 45 mph is (+12%), plus 43 mph would be more conservative from a regulatory perspective.

I guess 45 mph would be more conservative from a design perspective. Curious to see how a PTOE or Traffic PE would answer this.

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That's what I found too and since 85 would be into the next speed range the 45 MPH answer seemed logical.

In the solutions they graphed the frequency vs. the speed. However when they drew a horizontal line from 85% they didn't intersect the graphed information and called it 43.5 MPH and I was confused because if they intersected the graphed line with the 85% line it would've been higher that 43.5 MPH. Seems like a fine line between answers C and D.

The traffic engineer in my office told me there is an equation for questions like that but couldn't remember what it was. I would think if we had to graph anything in the actual test the answer would be a lot clearer and have a larger difference in the answers than 2 MPH.

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The answer is undoubtedly 43. Here's why:

On the cumulative speed distribution curve, the vehicule count percentile is conveniently the same as the accumulated vehicle count (100). At 40-44 MPH, the subtotal/percentile is 84%. When I graph this, I use the midpoint between 40 and 45 MPH (i.e. 42.5 MPH) as the representative speed at 84%. Thus, for 85%, it's incrementally higher than 42.5, but not as high 45 MPH.

Sometimes a little guesstimating is essential during exam time.

Good luck!

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That explains it! I assumed they were graphing the high point on each interval so it confused me.

Thanks for the help!

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The answer is undoubtedly 43. Here's why:

On the cumulative speed distribution curve, the vehicule count percentile is conveniently the same as the accumulated vehicle count (100). At 40-44 MPH, the subtotal/percentile is 84%. When I graph this, I use the midpoint between 40 and 45 MPH (i.e. 42.5 MPH) as the representative speed at 84%. Thus, for 85%, it's incrementally higher than 42.5, but not as high 45 MPH.

Sometimes a little guesstimating is essential during exam time.

Good luck!

I'm not so sure... here's my understanding of the 85th percentile:

If there were 100 cars sampled and a histogram created of their speeds (whole numbers of mph), you're looking for the #85 (from the lowest speed) and that car's speed is the 85th percentile. In other words, put in order from smallest to largest, what speed is #85.

For this problem,

the 84th percentile is 44 mph or less... it could be as low as 40 mph.

the 85th percentile is 45 mph or more... it could be as high as 49 mph.

That's all we know for sure. You can assume the distribution of actual mph in the ranges, but regardless, 43 mph is LESS THAN the 85th percentile. I like answer d, too!

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^ Disagree because you need to take a representative average for each speed range when graphing the cumulative distribution curve. You do not use the lowest or highest value in each range because the question doesn't imply to determine a design speed, just the 85th percentile based on the set of given data.

For each speed range, it is implied that the average speed per vehicle count is the midpoint of each range. It's the only way to construct a cumulative distribution curve.

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Every vehicle counted does have a speed (measured in whole MPHs), even if the histogram bins them in 5 MPH increments. 100 vehicles were sampled and their speeds are all somewhere between 20 and 54 MPH. The two lowest speed vehicles are somewhere in the 20-24 bin. The three highest speed vehicles are somewhere in the 50-54 bin. Do you agree?

If you agree, then answer this question: What bin is the third lowest speed vehicle in? I'm sure you'll answer the 25-29 bin.

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Give the level of debate among all of these PE's, I would say that it is a poorly written question. Understand the concept and try another problem.

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I don't think it's poorly written question because total number of cars is equal to 100 with a classic distribution curve which makes this question much easier than it could be. It's an exercise to see if the exam taker can properly sketch out a cumulative distribution curve and pull out 85th percentile speed which itself is not a range of speed values, but a specific one.

Remember, the PE exam isn't about overthinking, but using critical analysis skills to properly deduce an answer.

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Remember, the PE exam isn't about overthinking, but using critical analysis skills to properly deduce an answer.

Will you answer my questions above?

You're so sure you're right... but I'll bet a beer you're wrong. Collecting will be hard, though.

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Your question has no relevence to the original problem. Also, the sample question isn't asking for a range of values that would represent the 85th percentile, but a specific one. So I'm not sure what your trying to accomplish with your argument. It's time to move on.

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Your question has no relevence to the original problem. Also, the sample question isn't asking for a range of values that would represent the 85th percentile, but a specific one. So I'm not sure what your trying to accomplish with your argument. It's time to move on.

Sure... move on. But you're not right just because you say your answer is "undoubtedly" right. I believe you're likely wrong but just think too highly of yourself to imagine the possibility. You're trying to guess what the speed distribution is within a bin but you can't possible know.

And I'm not trying to accomplish anything with an argument. I'm just want you to see you're looking at the problem differently than I am. The fact that I think I'm right is of little importance.

If you'd have answered the question, I'd have followed up with a second question asking in which bin is the 3rd percentile.

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From my point if view the answer is D:

# of observations: 100

85th percentile: total # of observations x 0.85 = 85th observation. The 85th observation is in the 45 to 49 mph speed group, the answer should actually be the associated assumed speed for that group: 47mph. The closest answer is D:44mph.

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From reading the question and all the back and forth, I agree it is a poorly written problem. It is both baited and deceptive and probably written by a professor having little practical experience. Let's just all agree that we understand the concepts and move forward.

Jason

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From my point if view the answer is D:

# of observations: 100

85th percentile: total # of observations x 0.85 = 85th observation. The 85th observation is in the 45 to 49 mph speed group, the answer should actually be the associated assumed speed for that group: 47mph. The closest answer is D:44mph.

I mean D: 45 mph

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I am positive the answer to this question is 43 mph.

I looked a similar problem in Traffic Engineering text book by Roess, Prassas and McShane pg. 210

I have attached the solution, I hope it explains it.

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^ I agree!

The problem needs to be solved using the average of each speed group. There is a belief among many that the percentile is associated with the speed group rather than the actual speed. Even though I can see some validity to this belief, it doesn't mean that their justification is correct when it is required to use conventional methods to solve the problem during the exam.

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I am positive the answer to this question is 43 mph.

I looked a similar problem in Traffic Engineering text book by Roess, Prassas and McShane pg. 210

I have attached the solution, I hope it explains it.

Can you elaborate a bit more about why it is 43? (85 percentile is in the 45-49 range not in the 40-44). or maybe is it because the question actually asks " The 85 percentile speed, in mile per hour, is most nearly" meaning that 85th is closest to (looking at your .jpg solution) 84th than 97%? it is getting confusing. (sigh)

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Here's how I look at this, worded differently this time.

If the question asked what was the 84th percentile speed, I'd say we know it's somewhere in the 40-44 bin and BY DEFINITION it's the highest speed observed in that bin. Anyone disagree?

Assuming a normal distribution of traffic speeds, it is fair to assume a linear distribution of speeds within the bin... so 44 mph is most likely correct. But it could be 43, 42, 41, or 40. Given the 24 observations, though, it would not be likely to be low in the bin.

The 85th highest speed observed MUST be in the 45-49 bin. It's clearly not in the 40-44 bin or the number of observations would be 25 instead of 24.

sac_engineer: I hope we all agree the 85th percentile speed, by definition, is a speed and not a range of speeds. When there are 100 observations, it is an observed speed rather than a calculated speed. If there were only 50 observations, it would have to be calculated because you never observed the 85th percentile.

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I looked a similar problem in Traffic Engineering text book by Roess, Prassas and McShane pg. 210

Could you post the similar problem?

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I think the confusion lies in the way we are interpreting the numbers. The number of vehicles in the sampled set is equal to 100. We think we need to directly convert those numbers to percentiles since the question provides that apparent convenience; however, when solving the problem, the 85th percentile should not be confused with the 85th vehicle. I agree that in the data provided, the 85th vehicle is in the 45-49 bin, but the question isn't asking for that. The 85th percentile is determined from a linear relationship between the accumulated vehicle count at specific speeds (i.e the average of each bin) and the vehicle count percentile. If the bin data were explicit, then the graph would be a stepped, which cannot be used to determine speeds based on percentiles.

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Could it be that the open intervals have created all the confusion? After all, speeds don't have to be either 25 or 26, they can be in between. Consider the upper ends of the intervals modified as:

Interval Frequency Cumulative

20-25 2 2

25-30 11 13

30-35 18 31

35-40 29 60

40-45 24 84

45-50 13 97

50-55 3 100

Where each interval such as 20-25 basically means >25 - 30

In that case, wouldn't the 84th cumulative frequency correspond to the end of the interval i.e. 45 mph, and the 85th percentile be expected to be just above 45?

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when solving the problem, the 85th percentile should not be confused with the 85th vehicle.

Here is where we disagree clearly, and I invite you to find any definition of percentile which supports your belief.

I suggest a definition of: "A percentile is a measure that tells us what percent of the total frequency is at or below that measure. A percentile rank is the percentage of measures that fall at or below a given measure."

We could argue whether it is "at or below" or just "below" but that won't make a difference in this case.

There is certainly some interpretation on how to handle percentiles when the sample set is small, but for a sample of 100 it is VERY straightforward.

No one knows how the speeds are distributed in the bin, but can you at least agree that if every car speed was recorded individually (well, they were but for ease of writing the problem it was decided to put them in bins), the 85th percentile would have to be 45 or more?

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I looked a similar problem in Traffic Engineering text book by Roess, Prassas and McShane pg. 210

Could you post the similar problem?