Question

For a standard normal distribution, how do you find the percentage of data that are between 3 standard deviations below the mean and 1 standard above the mean?

Answer 1

Hint: In the above question, it follows the concept of standard normal distribution. Such distribution follows the shape of the bell curve where the percentage of the data points under the curve can be calculated by adding the percentage before and after the mean.

Complete step by step answer:
The normal distribution has a bell shape and under the curve has all the data points with mean at the center. Standard deviation is a measure of dispersion where the standard deviation gives the average distance of each point to the mean and the
The normal distribution has few properties in order to find the percentage of data
68% of the data falls from the mean for standard deviation 1
95% of the data falls from the mean for standard deviation 2
99.7% of the data falls from the mean for standard deviation 3
So, in the question, we have been asked to find the percentage between standard deviation 3 below the mean and standard deviation 1 above the mean.
So, for below the with Standard deviation as 3, we divide the by 2 because we want to find till the mean
\[\dfrac{{99.7}}{2} = 49.85\% \] and for 1 above the mean \[\dfrac{{68}}{2} = 34\% \]
Therefore, by adding both we get
\[49.85 + 34 = 83.8\% \]
Hence,\[83.8\% \] is the percentage of data.

Note:
 An important thing to note is that the data point is according to the standard deviation value as a difference applied before and after the mean. Suppose, if the value is 10 with standard deviation has
\[1\] then point below mean will be \[9,8,7..\] and after will be \[11,12,13..\]