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24 Feb 2012, 12:10
Kudos
Bookmarks
Hi Can someone please explain the below answer
Question : -The following table shows the grouped data, in classes, for the heights of 50 people.
Calculate the mean and standard deviation of the group of 50 people.
height (in cm) - classes frequency
120 <- 130 2
130 <- 140 5
140 <- 150 25
150 <- 160 10
160 <- 170 8
Answer :-
Let mi be the midpoint of the i th clss and fi be the corresponding frequency.
mean of grouped data = μ = (Σmi*fi) / Σfi
= (125*2 + 135*5 + 145*25 + 155*10 + 165*8) /(2+5+25+10+8)
= 148.4
b) standard deviation of grouped data = √[ (Σ(mi-μ)2*fi) / Σfi ]
= √[ (2*(125-148.4)2+5*(135-148.4)2+25*(145-148.4)2+10*(155-148.4)2+8*(165-148.4)2) /(50) ]
= 9.9
Question : -The following table shows the grouped data, in classes, for the heights of 50 people.
Calculate the mean and standard deviation of the group of 50 people.
height (in cm) - classes frequency
120 <- 130 2
130 <- 140 5
140 <- 150 25
150 <- 160 10
160 <- 170 8
Answer :-
Let mi be the midpoint of the i th clss and fi be the corresponding frequency.
mean of grouped data = μ = (Σmi*fi) / Σfi
= (125*2 + 135*5 + 145*25 + 155*10 + 165*8) /(2+5+25+10+8)
= 148.4
b) standard deviation of grouped data = √[ (Σ(mi-μ)2*fi) / Σfi ]
= √[ (2*(125-148.4)2+5*(135-148.4)2+25*(145-148.4)2+10*(155-148.4)2+8*(165-148.4)2) /(50) ]
= 9.9
24 Feb 2012, 17:09
Kudos
Bookmarks
Hi, there. I'm happy to help with this. 
Question : -The following table shows the grouped data, in classes, for the heights of 50 people. Calculate the mean and standard deviation of the group of 50 people. height (in cm) - classes frequency
from 120 to 130, 2 values
from 130 to 140, 5 values
from 140 to 150, 25 values
from 150 to 160, 10 values
from 160 to 170, 8 values
First of all, we have 2 + 5 + 25 + 10 + 8 = 50
Well, notice, we don't have the exact values --- for example, we know there are two values between 120 and 130 -- they both could be 121, they could be 123 and 127, or they both could be 129. We don't know. Any value we get for the mean and standard deviation will necessarily be an estimate.
Because two numbers are somewhere between 120 and 130, we are going to estimate that they are both smack-dab in the middle, at 125. This is our estimation. Similarly, we are going to estimate that the numbers in every other range are right at the center of their respective ranges -- thus, 5 values of 135, 25 values of 145, etc.
Now average. We have two values of 125, so I will express that as 2*125. We have 5values of 135, so I will express that as a 5*135. The full expression for the average is thus:
Mean = (125*2 + 135*5 + 145*25 + 155*10 + 165*8) /(50) = 148.4
Now, to find the standard deviation, there are
1) subtract the mean from each entry --- this is that value's "deviation"
2) square the list of deviations
3) average the list of square deviations
4) take the square-root: that's the standard deviation.
For the 2 values of 125, the deviation is (125-148.4) = -23.4
For the 5 values of 135, the deviation is (135-148.4) = -13.4
For the 25 values of 145, the deviation is (145-148.4) = -3.4
For the 10 values of 155, the deviation is (155-148.4) = 6.6
For the 8 values of 165, the deviation is (165-148.4) = 16.6
For the 2 values of 125, the squared deviation is (125-148.4)^2 = 547.56
For the 5 values of 135, the squared deviation is (135-148.4)^2 = 179.56
For the 25 values of 145, the squared deviation is (145-148.4)^2 = 11.56
For the 10 values of 155, the squared deviation is (155-148.4)^2 = 43.56
For the 8 values of 165, the squared deviation is (165-148.4)^2 = 275.56
Now, take the average of that list:
(2*547.56 + 5*179.56 + 25*11.56 + 10*43.56 + 8*275.56)/50 = 98.44
Take the square root now: standard deviation = sqrt(98.44) = 9.92169
No one on earth could expect you to do this problem without a calculator. Therefore, a problem of this complexity is beyond what you will see on the GMAT. Nevertheless, the principle of how to find the mean of a list like this and how to find the standard deviation are important.
Here's a question about standard deviation that is much much more representative of what you could see on the GMAT.
https://gmat.magoosh.com/questions/348
When you submit an answer to the question at that link, it will be followed by a full video explanation.
Let me know if you have any questions about this.
Mike
Question : -The following table shows the grouped data, in classes, for the heights of 50 people. Calculate the mean and standard deviation of the group of 50 people. height (in cm) - classes frequency
from 120 to 130, 2 values
from 130 to 140, 5 values
from 140 to 150, 25 values
from 150 to 160, 10 values
from 160 to 170, 8 values
First of all, we have 2 + 5 + 25 + 10 + 8 = 50
Well, notice, we don't have the exact values --- for example, we know there are two values between 120 and 130 -- they both could be 121, they could be 123 and 127, or they both could be 129. We don't know. Any value we get for the mean and standard deviation will necessarily be an estimate.
Because two numbers are somewhere between 120 and 130, we are going to estimate that they are both smack-dab in the middle, at 125. This is our estimation. Similarly, we are going to estimate that the numbers in every other range are right at the center of their respective ranges -- thus, 5 values of 135, 25 values of 145, etc.
Now average. We have two values of 125, so I will express that as 2*125. We have 5values of 135, so I will express that as a 5*135. The full expression for the average is thus:
Mean = (125*2 + 135*5 + 145*25 + 155*10 + 165*8) /(50) = 148.4
Now, to find the standard deviation, there are
1) subtract the mean from each entry --- this is that value's "deviation"
2) square the list of deviations
3) average the list of square deviations
4) take the square-root: that's the standard deviation.
For the 2 values of 125, the deviation is (125-148.4) = -23.4
For the 5 values of 135, the deviation is (135-148.4) = -13.4
For the 25 values of 145, the deviation is (145-148.4) = -3.4
For the 10 values of 155, the deviation is (155-148.4) = 6.6
For the 8 values of 165, the deviation is (165-148.4) = 16.6
For the 2 values of 125, the squared deviation is (125-148.4)^2 = 547.56
For the 5 values of 135, the squared deviation is (135-148.4)^2 = 179.56
For the 25 values of 145, the squared deviation is (145-148.4)^2 = 11.56
For the 10 values of 155, the squared deviation is (155-148.4)^2 = 43.56
For the 8 values of 165, the squared deviation is (165-148.4)^2 = 275.56
Now, take the average of that list:
(2*547.56 + 5*179.56 + 25*11.56 + 10*43.56 + 8*275.56)/50 = 98.44
Take the square root now: standard deviation = sqrt(98.44) = 9.92169
No one on earth could expect you to do this problem without a calculator. Therefore, a problem of this complexity is beyond what you will see on the GMAT. Nevertheless, the principle of how to find the mean of a list like this and how to find the standard deviation are important.
Here's a question about standard deviation that is much much more representative of what you could see on the GMAT.
https://gmat.magoosh.com/questions/348
When you submit an answer to the question at that link, it will be followed by a full video explanation.
Let me know if you have any questions about this.
Mike
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