A normal distribution is a common probability distribution . It has a shape often referred to together a \"bell curve.\"

many everyday data sets typically follow a normal distribution: because that example, the heights the adult humans, the scores top top a test given to a large class, errors in measurements.

The normal distribution is constantly symmetrical around the mean.

The typical deviation is the measure up of exactly how spread out a normally distributed collection of data is. That is a statistic that tells you exactly how closely every one of the examples are gathered around the mean in a data set. The shape of a normal distribution is figured out by the mean and the typical deviation. The steeper the bell curve, the smaller the conventional deviation. If the examples are spread much apart, the bell curve will be much flatter, meaning the standard deviation is large.


In general, about 68 % of the area under a normal circulation curve lies in ~ one conventional deviation that the mean.

that is, if x ¯ is the mean and σ is the conventional deviation that the distribution, then 68 % of the values loss in the range between ( x ¯ − σ ) and ( x ¯ + σ ) . In the figure below, this coincides to the an ar shaded pink.


about 95 % of the values lie in ~ two conventional deviations the the mean, that is, between ( x ¯ − 2 σ ) and ( x ¯ + 2 σ ) .

(In the figure, this is the amount of the pink and blue regions: 34 % + 34 % + 13.5 % + 13.5 % = 95 % .)

about 99.7 % of the values lie within 3 standard deviations of the mean, the is, between ( x ¯ − 3 σ ) and also ( x ¯ + 3 σ ) .

(The pink, blue, and also green areas in the figure.)

(Note the these values room approximate.)

You are watching: Which one of the following is defined by its mean and its standard deviation?

instance 1:

A collection of data is normally spread with a median of 5 . What percent the the data is less than 5 ?

A normal circulation is symmetric about the mean. So, fifty percent of the data will certainly be much less than the median and fifty percent of the data will be better than the mean.

Therefore, 50 % percent the the data is less than 5 .

example 2:

The life that a fully-charged mobile battery is normally dispersed with a typical of 14 hrs with a conventional deviation the 1 hour. What is the probability that a battery large at the very least 13 hours?

The mean is 14 and the conventional deviation is 1 .

50 % of the normal distribution lies come the ideal of the mean, so 50 % of the time, the battery will certainly last longer than 14 hours.

The interval native 13 come 14 hours represents one standard deviation come the left of the mean. So, around 34 % the time, the battery will certainly last in between 13 and also 14 hours.

Therefore, the probability that the battery big at the very least 13 hrs is around 34 % + 50 % or 0.84 .

example 3:

The average weight of a raspberry is 4.4 gm with a traditional deviation the 1.3 gm. What is the probability the a randomly selected raspberry would weigh at least 3.1 gm however not much more than 7.0 gm?

The average is 4.4 and the conventional deviation is 1.3 .

note that

4.4 − 1.3 = 3.1


4.4 + 2 ( 1.3 ) = 7.0

So, the expression 3.1 ≤ x ≤ 7.0 is actually between one conventional deviation listed below the mean and 2 traditional deviations over the mean.

In normally dispersed data, around 34 % the the worths lie between the mean and also one traditional deviation listed below the mean, and 34 % in between the mean and also one conventional deviation over the mean.

In addition, 13.5 % that the worths lie between the very first and 2nd standard deviations above the mean.

adding the areas, we obtain 34 % + 34 % + 13.5 % = 81.5 % .

Therefore, the probability the a randomly selected raspberry will weigh at least 3.1 gm but not more than 7.0 gm is 81.5 % or 0.815 .

instance 4:

A town has 330,000 adults. Their heights space normally distributed with a mean of 175 cm and also a variance the 100 cm 2 .How many people would you intend to it is in taller 보다 205 cm?

The variance the the data set is offered to be 100 centimeter 2 . So, the typical deviation is 100 or 10 cm.

Now, 175 + 3 ( 10 ) = 205 , so the number of people taller than 205 cm corresponds to the subset of data i beg your pardon lies more than 3 standard deviations above the mean.

The graph over shows the this represents about 0.15 % of the data. However, this percent is approximate, and in this case, we need an ext precision. The really percentage, exactly to 4 decimal places, is 0.1318 % .

330 , 000 × 0.001318 ≈ 435

So, there will be about 435 people in the town taller than 205 cm.

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