Here is the question:

2. After one experiment where 4 dice were rolled 1,000 times, the observed distribution of averages was seen to be as follows:

Average 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

Frequency 0 5 16 16 21 35 53 95 91 111 118

Average 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6

Frequency 103 92 67 71 50 29 15 8 4 0

Compute the mean and standard deviation of this distribution and compare to the estimates given by the Central Limit Theorem.

Here is where I am so far with my work. The mean I have calculated seems correct.

Mean: 3.5063

1*0/1000+1.25*5/1000+1.5*16/1000+1.75*16/1000+2*21/1000+2.25*35/1000+2.5*53/1000+2.75*95/1000+3*91/1000+3.25*111/1000+3.5*118/1000+3.75*103/1000+4*92/1000+4.25*67/1000+4.5*71/1000+4.75*50/1000+5*29/1000+5.25*15/1000+5.5*8/1000+5.75*4/1000+6*0/1000= 3.5063

Standard Deviation: 4.76519

√((1^2*0/1000+〖1.25〗^2*5/1000+〖1.5〗^2*16/1000+〖1.75〗^2*16/1000+2^2*21/1000+〖2.25〗^2*35/1000+〖2.5〗^2*53/1000+〖2.75〗^2*95/1000+3^2*91/1000+〖3.25〗^2*111/1000+〖3.5〗^2*118/1000+〖3.75〗^2*103/1000+4^2*92/100+〖4.25〗^2*67/1000+〖4.5〗^2*71/1000+〖4.75〗^2*50/1000+5^2*29/100+〖5.25〗^2*15/1000+〖5.5〗^2*8/100+〖5.75〗^2*4/1000+6^2*0/1000)-(3.5063)^2 )=4.76519

I'm stuck as to where I have gone wrong.