How To Tell If Data Is Normally Distributed

Normal Distribution of Data

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

In full general, about $68\%$ of the area under a normal distribution curve lies within i standard divergence of the hateful.

That is, if $\overline{\mathrm{10}}$ is the mean and $\sigma$ is the standard difference of the distribution, then $68\%$ of the values fall in the range between $\left(\overline{x}-\sigma \right)$ and $\left(\overline{x}+\sigma \right)$ . In the figure below, this corresponds to the region shaded pink.

About $95\%$ of the values lie inside two standard deviations of the mean, that is, between $\left(\overline{\mathrm{ten}}-2\sigma \right)$ and $\left(\overline{\mathrm{10}}+\mathrm{two}\sigma \right)$ .

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

About

$99.7\%$

of the values lie inside 3 standard deviations of the mean, that is, betwixt

$\left(\overline{\mathrm{ten}}-3\sigma \right)$

and

$\left(\overline{x}+3\sigma \right)$

.

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

(Note that these values are approximate.)

Example ane:

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

A normal distribution is symmetric about the mean. And so, half of the data will exist 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 jail cell phone battery is ordinarily distributed with a mean of $14$ hours with a standard difference 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 correct of the mean, so $50\%$ of the time, the battery volition last longer than $14$ hours.

The interval from $13$ to $14$ hours represents one standard deviation to the left of the hateful. Then, about $34\%$ of time, the bombardment will last between $\mathrm{xiii}$ and $14$ hours.

Therefore, the probability that the bombardment lasts at to the lowest degree $\mathrm{thirteen}$ hours is about $34\%+\mathrm{fifty}\%$ or $0.84$ .

Example iii:

The average weight of a raspberry is $\mathrm{iv.4}$ gm with a standard deviation of $\mathrm{ane.3}$ gm. What is the probability that a randomly selected raspberry would weigh at least $\mathrm{three.1}$ gm simply non more than $7.0$ gm?

The mean is $\mathrm{4.iv}$ and the standard divergence is $\mathrm{one.iii}$ .

Note that

$\mathrm{iv.four}-1.3=3.1$

and

$\mathrm{4.four}+2\left(1.3\right)=\mathrm{seven.0}$

So, the interval $3.1\le \mathrm{10}\le 7.0$ is actually betwixt ane standard difference below the mean and $2$ standard deviations above the mean.

In normally distributed data, about $34\%$ of the values lie between the mean and 1 standard difference beneath the mean, and $34\%$ between the mean and one standard deviation to a higher place the hateful.

In addition, $\mathrm{13.v}\%$ of the values prevarication between the first and 2d standard deviations in a higher place the mean.

Adding the areas, nosotros get $34\%+34\%+\mathrm{13.five}\%=81.5\%$ .

Therefore, the probability that a randomly selected raspberry will counterbalance at least $3.1$ gm just non more $7.0$ gm is $81.5\%$ or $0.815$ .

Example four:

A town has $\mathrm{330,000}$ adults. Their heights are usually distributed with a mean of $175$ cm and a variance of $100$ cm ^$2$ .How many people would y'all expect to exist taller than $205$ cm?

The variance of the data prepare is given to be $100$ cm ^$2$ . So, the standard divergence is $\sqrt{100}$ or $10$ cm.

Now, $175+\mathrm{iii}\left(10\right)=205$ , so the number of people taller than $205$ cm corresponds to the subset of data which lies more $3$ standard deviations above the mean.

The graph above shows that this represents near $\mathrm{0.fifteen}\%$ of the data. Withal, this percentage is judge, and in this case, nosotros demand more than precision. The bodily percentage, right to $\mathrm{four}$ decimal places, is $0.1318\%$ .

$330,000\times 0.001318\approx 435$

Then, there will be about $435$ people in the boondocks taller than $205$ cm.

How To Tell If Data Is Normally Distributed,

Source: https://www.varsitytutors.com/hotmath/hotmath_help/topics/normal-distribution-of-data

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