Question

In a normally distributed data set a mean of 55 where 99.7% of the data fall between 47.5 and 62.5, what would be the standard deviation of that data set? Devry course

Answer 1

The standard deviation of the normally distributed data set is 2.5.

Step-by-step explanation:

To find the standard deviation of a normally distributed data set, we can use the empirical rule. According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

In this case, we know that 99.7% of the data falls between 47.5 and 62.5. Since this range corresponds to three standard deviations from the mean, we can set up the following equation:

(3 * standard deviation) = (62.5 - 55)

Solving for the standard deviation, we find that:

standard deviation = (62.5 - 55) / 3 = 2.5

Therefore, the standard deviation of the data set is 2.5.

Answer 2

Given that normally distributed data set has a mean of 55 and 99.7% of data fall between 47.5 and 62.5.

Let s be the standard deviation of data set.

Since 99.7% data fall within 3 standard deviations of mean, z-value for 47.5 and 62.5 has an absolute value of 3.

That is |z|=3

But z=
`(x-mean)/(standard deviation)`

Let us plugin x=47.5 and mean =55 and equate it to 3.

That is
`|(47.5-55)/(s)|  = 3`

`|(-7.5)/(s) | =3`

Since x is always positive ( being standard deviation),
`|(-7.5)/(s) | = (|-7.5|)/(s) = (7.5)/(s)`

Hence
`(7.5)/(s)= 3`

`s=(7.5)/(3)  = 2.5`

We will get same value with 62.5 as well.

Hence standard deviation of data set is 2.5.