The values 248 and 332 can be either on different sides of the mean or on the same side of the mean.
The classmate is wrong, see solution.
Practice makes perfect
Let's look at the given data set to see if the classmate's claim is correct. There are two possible cases for the placement of 248 and 332. The first case is when 248 and 332 are on the different sides of the mean.
The other case is when both 248 and 332 are on the same side of the mean.
To show that the mean and standard deviation are, in fact, different in each case, we will calculate them. Let's start with the first case.
Case 1
Look at the graph of the normal curve, assuming that 248 and 332 are on the different sides of the mean. The value 248 is one standard deviation away from the mean, and 332 is three standard deviations away.
In this case we can write the following system of equations.
mean-1 deviation=248 & (I) mean+3 deviations=332 & (II)
For convenience, the mean will be represented by m and the standard deviation will be represented by s.
m-s=248 & (I) m+3 s=332 & (II)
Let's solve this system using substitution.
We found that the mean m can equal to 269. Now to find the value of the standard deviation s, we can substitute m=269 into either of the equations in the given system. Let's use the first equation.
The mean is equal to 269 and the standard deviation is 21.
Case 2
This time, 248 and 332 are on the same side of the mean. The value 248 is one standard deviation away from the mean and 332 is three standard deviations away.
We can write the following system of equations.
mean+1 deviation=248 & (I) mean+3 deviations=332 & (II)
For convenience, the mean will be represented by m and the standard deviation will be represented by s.
m+s=248 & (I) m+3 s=332 & (II)
To solve this system, we can again use substitution.
This time, the mean m is equal to 206. Now to find the value of the standard deviation s, we can substitute m=206 into either of the equations in the given system. Let's use the first equation.
