Finding the sample mean and the sample standard deviation

Question

X|26 |27| 28 | 29 | 30|31 |32 f| 3 | 4 | 16 | 12 | 6 | 2 | 1 Compute the sample mean and the sample standard deviation. Show your work.

Answer 1

mean = sum(xf)/sumf = (26(3) + 27(4) + 28(16) + 29(12)+ 30(6) +31(2) +32(1))/(3+4+16+12+6+2+1)=

(78+ 108+ 448+ 348+ 180+62+32)/44

= (39+54+224+174+90+31+16)/22

= (93+398+137)/22

= (492+137)/22

= 629/22

= 314.5/11

= 28 10/11 which is between 26 and 32 as it has to be, highest frequency is for 28 and 29, so ean should be near 28 and 29

now take the sum of squared differences from the mean, then divide by n-1 = sum of frequencies -1 = 43, then take the square root = standard deviation

you use 43 instead of 44 because this is a sample and not the entire population. this involves very tedious calculations which is very error prone. best to use a calculator with statistics functions, where you just plug in the data and out pops the solution. if the mean is near 28 and 29 you feel confident it's correct

standard deviation from the mean has to be less than 3 (32-26)/2 = 3 if your calculations or calculator give you more than 3, you did something wrong or you plugged in the numbers wrong into the calculator or you have a defective calculator. rough guess would be sd=about 1 or less