Finding the standard deviation and mean

Question

M = 30 and 47.5% is between 24 and 30. What is the standard deviation?  Also, if Q1 (quartile 1) is 45 and standard deviation is 3.5 when what is the mean?

Answer 1

Hello, thank you for taking the time to post your question!

The key on this type of question is using the given information as well as the Z score formula in order to compute the missing variable.

So you’re given mean = 30 and that 47.5% lies between 24 and 30 … what that means is that a score of x = 24 corresponds to the 95% percentile and a Z score of -1.96

Plugging that all into the Z score formula yields

Z = (x – mean) / (standard deviation)

-1.96 = (24 - 30) / (standard deviation)

standard deviation = (24 – 30) / -1.96 = 3.06

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

B,

 

You have to convert the data to standardized normal variables to use the probability data under the Normal distribution in your book.

 

Let x = a value from the unknown non-standardized Normal distribution

     m = the mean for this distribution

     s  = the standard deviation for this distribution

     z  = the standardized version of x, where z has m=0 and s=1

 

z = (x - m) / s 

 

(a) Find s given x = 24 and m = 30

So z = (24 - 30) / s  and s = (24 -30) / z  = -6 / z

Now we're told that 47.5% of area under the Normal distribution is between 24 & 30.  We know that the area from the mean to the -1s point corresponds to 34.13% only so the 24 must be more than 1s away from the mean at 30.  Looking up the standardized Normal table, 1.96s corresponds to 0.475 area.  Thus z = -1.96 since we're to the left of the mean.

 

And s = -6 / -1.96 = 3.06

 

(b) I don't know what you mean by quartile? It could represent one-fourth of the area under the Normal distribution, or it could represent the -1.5 s point since that is one-fourth of 6 standard deviations which covers 99.79% of the area under the Normal distribution.  In either case, you work with the z equation above.

 

If it's one-fourth the area, that first 25% point corresponds to z = -0.675

So, z = - 0.675 = (45 - m) / 3.5  and m = 0.675(3.5) + 45 = 47.36

 

If it's the second case, then (45 - m) / 3.5 = -1.5   and m = 1.5(3.5) + 45 = 50.25

 

Take your pick!