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# On a particular test whose scores are distributed normally,

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On a particular test whose scores are distributed normally, [#permalink]  29 Jul 2018, 04:51
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On a particular test whose scores are distributed normally, the 2nd percentile is 1,720, while the 84th percentile is 1,990. What score, rounded to the nearest 10, most closely corresponds to the 16th percentile?

(A) 1,750
(B) 1,770
(C) 1,790
(D) 1,810
(E) 1,830
[Reveal] Spoiler: OA

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Re: On a particular test whose scores are distributed normally, [#permalink]  07 Aug 2018, 22:48
sandy wrote:
On a particular test whose scores are distributed normally, the 2nd percentile is 1,720, while the 84th percentile is 1,990. What score, rounded to the nearest 10, most closely corresponds to the 16th percentile?

(A) 1,750
(B) 1,770
(C) 1,790
(D) 1,810
(E) 1,830

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Re: On a particular test whose scores are distributed normally, [#permalink]  11 Jan 2019, 06:42
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ShubhiNigam29 wrote:
Quote:
On a particular test whose scores are distributed normally, the 2nd percentile is 1,720, while the 84th percentile is 1,990. What score, rounded to the nearest 10, most closely corresponds to the 16th percentile?

(A) 1,750
(B) 1,770
(C) 1,790
(D) 1,810
(E) 1,830

I found the following answer with explanation. I hope this will be helpful.
Attachment:

normdist.png [ 5.65 KiB | Viewed 1307 times ]

Quickly sketch a normal distribution: 0 ----2----16----50----84----96----100

A score at 84th percentile is three standard deviations away from a score at 2nd percentile

1,990 - 1,720 = 270

$$\frac{270}{3 SDs}$$ = 90 = one SD

16th percentile score is +1 SD greater than 2nd percentile score
16th percentile score would be:
1,720 + 90 = 1810

Explanation
Data is clustered around the mean, and percent distributed in any segment has a fixed number

"Distributed normally" means the data fall into the normal distribution curve, the top figure in the diagram.
68 percent of the data fall within one standard deviation from the mean -- one deviation above, one deviation below (so 34%/34%)

The percent of data underneath any part of the curve always follows the distribution seen in the top figure (figures in green ink):
2 (percent of the data), then 14 (percent of data) , then 34, 34, 14, 2
Memorize these numbers 2, 14, 34 <-[mean] -> 34, 14, 2

If you memorize those numbers and start with 50 as the mean, just add to get percentiles:
RIGHT OF THE MEAN: (50+34) = 84 | (84+14) = 98 | (98+2) = 100
LEFT OF THE MEAN: (50-34) = 16 | (16-14) = 2 | (2-2) = 0

Standard deviation corresponds with percentiles in a normal distribution
See middle figure, that pattern will always hold:
0th percentile is -3 SDs from mean
2nd percentile is -2 SDs from mean
. . .
84th percentile is + 1 SD from mean
100th percentile is +3 SDs from mean

What is score at 16th percentile here?

Score at 2nd percentile = 1,720
Score at 84th percentils = 1990

Score at the 16th percentile?

The score at the 84th percentile is THREE standard deviations from the score at the 2nd percentile (just count segments)

Find the difference between the scores
Divide by 3 (for three standard deviations)

(1,990 - 1,720) = 270

$$\frac{270}{3SDs}$$ = 90

So 90 = ONE standard deviation

The 16th percentile is +1 SD away from the 2nd percentile (see top and middle figures)

So, 16th percentile score = the given 2nd percentile score (1,720) + ONE standard deviation (90)

1,720 + 90 = 1810

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Re: On a particular test whose scores are distributed normally, [#permalink]  14 Jul 2019, 04:53
sandy wrote:
On a particular test whose scores are distributed normally, the 2nd percentile is 1,720, while the 84th percentile is 1,990. What score, rounded to the nearest 10, most closely corresponds to the 16th percentile?

(A) 1,750
(B) 1,770
(C) 1,790
(D) 1,810
(E) 1,830

Explanation:
The diagram below shows the standard distribution curve for any normally distributed variable. The percent figures correspond roughly to the standard percentiles both 1 and 2 standard deviations (SD) away from the mean: The 2nd percentile is 1,720, roughly corresponding to 2 standard deviations below the mean.

Therefore, the mean –2 standard deviations = 1,720.

Likewise, the 84th percentile is 1,990: 84% of a normally distributed set of data falls below the mean + 1 standard deviation, so the mean + 1 standard deviation = 1,990.

Call the mean M and the standard deviation S. Solve for these variables:
M – 2S = 1,720
M + S = 1,990

Subtract the first equation from the second equation:
3S = 270
S = 90

The question asks for the 16th percentile, which is the mean – 1 standard deviation or M – S. (It’s a fact to memorize that approximately 2% of normally distributed data falls below M – 2S, and approximately 14% of normally distributed data falls between M – 2S and M – S.)

Since M – 2S = 1,720, add another S to get M – S:
(M – 2S) + S = 1,720 + 90 = 1,810

Notice that the percentiles are not linearly spaced. The normal distribution is hump-shaped, so percentiles are bunched up around the hump and spread out farther away.

Attachment:

PERCENTILE.png [ 70.78 KiB | Viewed 1000 times ]

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Re: On a particular test whose scores are distributed normally,   [#permalink] 14 Jul 2019, 04:53
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