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Math Expert V
Joined: 02 Sep 2009
Posts: 59730
A particular characteristic or a feature in a large population has a  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 55% (01:17) correct 45% (01:25) wrong based on 39 sessions

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A particular characteristic or a feature in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than (m + d)?

(A) 18%
(B) 32%
(C) 48%
(D) 84%
(E) 94%

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Senior SC Moderator V
Joined: 22 May 2016
Posts: 3747
A particular characteristic or a feature in a large population has a  [#permalink]

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Bunuel wrote:
A particular characteristic or a feature in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than (m + d)?

(A) 18%
(B) 32%
(C) 48%
(D) 84%
(E) 94%

Attachment: normdist10.2.18.jpg [ 46.01 KiB | Viewed 548 times ]

A distribution that is symmetric about the mean with 68 percent of its data
within one standard deviation of the mean is a normal distribution.

$$\frac{68}{2} = 34$$% on each side of the mean

The mean plus one deviation (m + 1d) = (m + d) above the mean is (50% + 34%) = 84%
The percent of the distribution that is below (m + d) is 84%

Or: 50% of the data lies below the mean.
Another 34% lies within one standard deviation above the mean.
50% + 34% = 84% of the distribution lies below the mean.

The percent of the distribution that is less than (m+d) = 84%

Memorizing these three numbers will make life easy in normal distribution questions:
34%, 14%, 2%
Standard deviation = SD = d

In normal distribution questions, the mean is 50%
34%, 14%, and 2% correspond with 1SD, 2SD, and 3SD in the distribution

Above the mean, add the percents (to 50%). Below the mean, subtract the percents (from 50%).

Knowing the percents that correlate with the distribution helps, too: 0-2-16-50-84-98-100

In a normal distribution:

• 68 percent of data lies within 1 standard deviation of the mean
(i.e., within one deviation on EACH side of the mean)
34% below the mean, 34% above the mean
See the diagram.
ABOVE the mean: 50% + 34% = 84%
BELOW the mean: 50% - 34% = 16%

• 95 percent of data lies within 2 standard deviations of the mean (95%-68%) = 27%. It is standard to approximate.
Roughly half of 27% is 14%, so after the first deviation
ABOVE the mean: (84% + 14%) = 98%
BELOW the mean (16% - 14%) = 2%

• $$99.7 \approx 100$$% of data lies within 3 standard deviations of the mean (100 - 95) = 5%
Roughly half of 5% is 2.5%. I use 2%
ABOVE the mean (98% + 2%) = 100%
BELOW the mean (2% - 2%) = 0%

Note: some distributions use 2.5% (3 SDs from mean) and 13.5% (2 SDs from mean) instead of 2% and 14%
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Re: A particular characteristic or a feature in a large population has a  [#permalink]

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Bunuel wrote:
A particular characteristic or a feature in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than (m + d)?

(A) 18%
(B) 32%
(C) 48%
(D) 84%
(E) 94%

Characteristics about the symmetric distribution is that it's equally distributed about the mean of the distribution
i.e. 1) 68% should be distributed around the line of mean equally i.e. 34% is greater than mean and less than one standard deviation above the mean and rest 34% is great one standard deviation below the mean and less than the mean
2) rest 100-68=33% also needs to be equally distributed beyond the lines of one standard deviation from mean i.e. 16% should be greater than m+d and 16% should be less than m-d

i.e. Total percentage below m+d = 68%+16% = 84%

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Re: A particular characteristic or a feature in a large population has a  [#permalink]

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1
Bunuel wrote:
A particular characteristic or a feature in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than (m + d)?

(A) 18%
(B) 32%
(C) 48%
(D) 84%
(E) 94% Re: A particular characteristic or a feature in a large population has a   [#permalink] 03 Oct 2018, 11:33
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