the sample variance of hourly wages was 10. what is the sample standard deviation? This is a topic that many people are looking for. amritsang.org is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, amritsang.org would like to introduce to you Standard Deviation Formula, Statistics, Variance, Sample and Population Mean. Following along are instructions in the video below:
This video. Were going to calculate the standard deviation of set of numbers. Now now theres two formulas you need to be aware of the first one is population standard deviation.
Now this formula is represented by the letter sigma thats the span deviation. Its equal to the sum of all the differences between every point in the data set and the population mean the population mean is mu. Which is this symbol here and then you need to square it divided by n.
Which is all of the numbers in the set. And then you got to take the square root of the whole result so thats the population standard deviation. The next formula is the sample standard deviation.
So lets say if you have just a sample of a population not the entire population. If you just have a sample data out of the entire data. Then you want to use this formula f.
Which is the standard deviation is equal to sigma the sum of all of the differences between every point and the mean thats the sample mean and the other equation. We have the population mean represent by mu. But this is the sample mode.
Which is basically the average of all the data points in the set. And then you have to square. It.
But its going to be divided by n minus 1. As opposed to n. And so thats how you calculate the standard deviation of the sample now lets work on an example.
Lets say. If we have two set of numbers. 4.
5. And 6. And also 3 5.
And. 7. Which one has a greater standard deviation lets use the population standard deviation formula.
But we have to guess. Which set of numbers has the greater standard deviation is it the one on the left or the one on the right. What would you say we need to understand the basic idea of standard deviation you need to know what it measures standard deviation tells you how far apart.
The numbers are related to each other so the more spread out.
They are the greater the standard deviation four five and six are closer to each other than three five and seven and you could tell if we plot them on a number line. Lets put five in the middle. So four five and six here they are on a number line now in contrast.
Lets put the same numbers on this number line. Were going to have three five and seven. So if you look at the red point the points in red are further apart the points in blue.
Theyre very close together. So theyre four four five and six has a lower standard deviation in three five and seven. So sigma is low and here.
The sigma value is high now go ahead and calculate the population standard deviation for this for this set of numbers three five and seven. So whats the first thing that we should do the first thing that we should do is calculate the move to find the mean. Its going to be the sum of all the numbers divided by three now because the three numbers are evenly spaced apart.
The mean is going to be the middle numbers. Five three plus five is eight eight plus seven is 15 15. Divided by 3 is 5.
So thats the mean now what should we do next now that we have a move now think of the formula. Its going to be sigma of every point. Minus.
The mean squared divided by n. And then all of this is within the square root. So heres how to use the equation first were going to use the first point three and subtracted by the mean and then squared next.
Were going to take the second point five subtract it from the mean squared and then its going to be seven minus five squared so each of these three points youre going to plug in to x sub. I and then youre going to square the differences between each of those dollars and the sigma represents some youre going to add every difference that you get or you can add the square of every difference that you get and now lets divided by n. So n is the number of numbers that we have in this set there are three numbers inside so n is three and then were going to take the square root of the entire thing three minus five is negative two negative 2 squared is 4 5 minus.
5. Is 0 7 minus 5. Is 2 2.
Squared is 4 4. Plus. 4 is 8.
So we have the square root of 8 divided by 3 and at this point. Were going to use the calculator. 8.
Divided by 3 is about two point six seven and if you take the square root of that youre going to get one point.
Six three. So thats the standard deviation for 3 5. 7.
Now lets calculate the standard deviation for the other set of numbers. 4 5. 6.
So why dont you go ahead. And pause. The video and try this example.
Calculate the standard deviation using the same formula. So lets go ahead and begin lets calculate the population mean. Its going to be 4 plus.
5. Plus. 6.
Divided by the number of numbers that we have which is 3 4. Plus. 6.
Is 10 10 plus. 5. Is 15.
When we know that 15 divided by 3 is 5. So once again anytime. The numbers are evenly spread apart the mean is going to be the middle number.
So now we can calculate the standard deviation. So sigma is going to equal the square root. But before you do that lets calculate the differences so the first difference that we have the first number is going to be 4.
And were going to subtract it from the mean and then square it the next number is 5 subtracted from the mean and then square it and then after that the last number is 6. This is going to be 6 minus. 5.
Squared now its divided by n. And lets not forget to take the square root of the entire thing 4 minus 5 is negative 1 negative 1 squared is simply 1 5 minus 5 is 0 6 minus 5 is 1. And its all divided by 3 1.
1 is 2. So we have the square root of 2 3. Now 2 3.
As a decimal is about point. 67. And the square root of point.
67. Is point 8 1. 6.
So as you can see the standard deviation is less because these numbers are closer to each other theyre not far apart from the mean and the other example 3 5. 7. Theyre further apart from the mean.
Which is 5 3. Is 2 units away from 5 4. Is only one unit away from 5.
And thats why the standard deviation is so much less now lets go back to the first example. We said that the population standard deviation is approximately 1 point 6 3. So given this information how can you calculate the variance var.
I am. Seeing how can we find the variance. The variance is simply the square of the standard deviation so 1 point 6 3.
Squared is equal to now keep in mind. This is a rounded answer. I dont remember what the exact answer was but once you.
Square its about 266. So thats how you can calculate the variance. The formula for variance is basically the sum of all the squared differences between every point and the population divided by n.
Its basically the same formula without the square root symbol well thats it for this video. So now you know how to calculate the population standard deviation and also the sample standard deviation even though we did just one of them the process is the same of finding. The other one.
The only difference is you have n minus 1. Instead of you often know how to calculate the variance as well so that concludes this video by the way. If you want to find more of my videos you can check out my channel or visit.
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