Find the mean and standard deviation of the heights of 13 boys

In summary, the textbook problem is that there is a problem with the value of σ and the textbook solution is to use the correct formula for the variance.
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
statistics
Find the textbook problem here;

1641291258768.png


Find the textbook solution here:

1641291303879.png
Now, to my question, did the textbook guys make an error on the value of ##σ?##, see my working;

Mean (##13## boys)=##\dfrac{153.4+(148.8×12)}{13}=149.15##
We know that,
##29.16##=##\dfrac{\sum x^2}{12}##-##(148.8)^2##
##\sum x^2=266,047.2##
Now, it follows that, ##266,047.2+153.4^2=289,578.76##
Variance=##\dfrac{289,578.76}{13}##-##(149.15)^2##
Variance=##29.56673077##
Standard deviation=##\sqrt {29.56673077}=5.4375##
 
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  • #2
It seems you forgot to include your work for the standard deviation
 
  • #3
It is generally a bad idea to post before your post is ready and then continue editing it as people will not get notifications of your edits.

Regarding your work, you have used the wrong formula for the variance in a statistical sample where the mean is also estimated from the sample.
 
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  • #4
Ok let me check that...
 
  • #5
I am unable to see what i have done wrong...i got it! let me post my working...hmmmm

Variance=##\dfrac{\sum (X-x)^2}{N-1}##
##5.4^2##=##\dfrac{\sum (X-x)^2}{11}##
##\sum (X-x)^2##=##320.76##
Therefore,
Variance=##\dfrac{320.76+(153.4-148.8)^2}{12}##
=##\dfrac{341.92}{12}##
=##28.49333333##
Standard deviation=##\sqrt {28.4933333}=5.33##

cheers orodruin...i will make sure to type all my working before posting, my apologies...there may be different approach to this of which i will appreciate...i guess i need more practice here...
 
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  • #6
What is N?
 
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  • #7
Orodruin said:
What is N?
hope you can see it mate:smile:
 
  • #8
Just a question guys, to clear my doubt...the initial number of the boys i.e ##12##, is it considered as a sample population or population ?...is sample/population definition dependant on the question? or are we using the fact that data that is less than ##30## items is sample data...
Let me rephrase my question. Supposing the whole class had a total of ##12## boys only. Then is this population or sample data? I can see that the question indicates ' 12 boys in a class' implying population...
cheers
 
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  • #9
Orodruin said:
you have used the wrong formula for the variance in a statistical sample
I don’t understand. We are not estimating the mean and variance of a large population of boys from a sample of 12 or 13. In each case, the set of boys is the population. In particular, the mean is exactly known.
I'd say the given answer is wrong.
 
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  • #10
haruspex said:
I don’t understand. We are not estimating the mean and variance of a large population of boys from a sample of 12 or 13. In each case, the set of boys is the population. In particular, the mean is exactly known.
I'd say the given answer is wrong.
Upon rereading the wording of the problem, I agree. It would have been different if the problem had asked for an estimate of the mean and standard deviation in the height of boys in the age group of the class.
 
  • #11
So this means the textbook solution is wrong and my initial solution in post ##1## right?
 
  • #12
chwala said:
So this means the textbook solution is wrong and my initial solution in post ##1## right?
Yes.
 
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  • #13
haruspex said:
Yes.
Bingo! :cool:
 
  • #14
I was just looking at this problem again on my textbook...now i need to amend my textbook solution in Red so as not to fidget around trying to check where i went wrong in my working... i had to check my posts as the problem looked familiar. Hmmmm :oops:
 
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Related to Find the mean and standard deviation of the heights of 13 boys

1. What is the mean height of the 13 boys?

The mean height of the 13 boys can be calculated by adding up all of their heights and dividing by 13, which is the total number of boys. This will give you the average or mean height of the group.

2. How do you calculate the standard deviation of the heights?

To calculate the standard deviation, you first need to find the mean height. Then, subtract the mean from each individual height and square the result. Next, add up all of these squared differences and divide by the total number of boys. Finally, take the square root of this number to get the standard deviation.

3. Why is it important to find the mean and standard deviation of the heights?

Finding the mean and standard deviation of the heights allows us to understand the distribution of heights among the 13 boys. It gives us a measure of the average height and how much the individual heights vary from this average. This information can be useful in making comparisons or drawing conclusions about the group.

4. Can the mean and standard deviation be affected by outliers?

Yes, the mean and standard deviation can be affected by outliers. Outliers are extreme values that are significantly different from the rest of the data. These values can skew the mean and increase the standard deviation, making it less representative of the overall group.

5. How can the mean and standard deviation be used in further analysis?

The mean and standard deviation can be used in further analysis to compare the heights of the 13 boys to other groups or to a standard height. It can also be used to identify any patterns or trends in the data. Additionally, the standard deviation can help determine the range of heights that are considered typical for the group.

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