What percentage falls between the mean and −1 to +1 standard deviation from the mean?

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

Many everyday data sets typically follow a normal distribution: for example, the heights of adult humans, the scores on a test given to a large class, errors in measurements.

The normal distribution is always symmetrical about the mean.

The standard deviation is the measure of how spread out a normally distributed set of data is.  It is a statistic that tells you how closely all of the examples are gathered around the mean in a data set.  The shape of a normal distribution is determined by the mean and the standard deviation. The steeper the bell curve, the smaller the standard deviation.  If the examples are spread far apart, the bell curve will be much flatter, meaning the standard deviation is large. 

In general, about 68 % of the area under a normal distribution curve lies within one standard deviation of the mean.

That is, if x ¯ is the mean and σ is the standard deviation of the distribution, then 68 % of the values fall in the range between ( x ¯ − σ ) and ( x ¯ + σ ) . In the figure below, this corresponds to the region shaded pink.

About 95 % of the values lie within two standard deviations of the mean, that is, between ( x ¯ − 2 σ ) and ( x ¯ + 2 σ ) .

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

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

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

(Note that these values are approximate.)

Example 1:

A set of data is normally distributed with a mean of 5 . What percent of the data is less than 5 ?

A normal distribution is symmetric about the mean. So, half of the data will be less than the mean and half of the data will be greater than the mean.

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

Example 2:

The life of a fully-charged cell phone battery is normally distributed with a mean of 14 hours with a standard deviation of 1 hour. What is the probability that a battery lasts at least 13 hours?

The mean is 14 and the standard deviation is 1 .

50 % of the normal distribution lies to the right of the mean, so 50 % of the time, the battery will last longer than 14 hours.

The interval from 13 to 14 hours represents one standard deviation to the left of the mean. So, about 34 % of time, the battery will last between 13 and 14 hours.

Therefore, the probability that the battery lasts at least 13 hours is about 34 % + 50 % or 0.84 .

Example 3:

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

The mean is 4.4 and the standard deviation is 1.3 .

Note that

4.4 − 1.3 = 3.1

and

4.4 + 2 ( 1.3 ) = 7.0

So, the interval 3.1 ≤ x ≤ 7.0 is actually between one standard deviation below the mean and 2 standard deviations above the mean.

In normally distributed data, about 34 % of the values lie between the mean and one standard deviation below the mean, and 34 % between the mean and one standard deviation above the mean.

In addition, 13.5 % of the values lie between the first and second standard deviations above the mean.

Adding the areas, we get 34 % + 34 % + 13.5 % = 81.5 % .

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

Example 4:

A town has 330,000 adults. Their heights are normally distributed with a mean of 175 cm and a variance of 100 cm 2 .How many people would you expect to be taller than 205 cm?

The variance of the data set is given to be 100 cm 2 . So, the standard 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 which lies more than 3 standard deviations above the mean.

The graph above shows that this represents about 0.15 % of the data. However, this percentage is approximate, and in this case, we need more precision. The actual percentage, correct 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.

What percentage falls between the mean and − 1 to +1 standard deviation from the mean?

What percentage of the population falls between 1 and 1 standard deviation?

What percent of the normal distribution is between 1 and +1 standard deviations from the mean?

What percentage is 1 standard deviations from the mean?