Comparing Standard Deviations The standard deviation of battin...

Question

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A research team records the systolic blood pressure in units of millimeters of mercury and resting heart rates in units of beats per minute. Of 8 marathon runners, the blood pressures are 118, 122, 120, 116, 124, 119, 121, and 117. The heart rates are 58, 62, 60, 56, 64, 59, 61, and 57. Calculate the coefficient of variation for both blood pressure and heart rate, which measurement is more variable relative to its mean. Awesome. So it appears for this particular problem we're asked to solve for three separate answers. Our first two answers we're asked to calculate the coefficient of variation for blood pressure and heart rate. And our third and final answer is we're asked to determine which measurement, is it the blood pressure measurement or the heart rate measurement that is more variable relative to its mean. Awesome. So now that we know what we're ultimately trying to solve for, we need to recall a note once again that We need to determine which measurement, blood pressure or heart rate is more variable relative to its mean, and we also need to calculate the coefficient of variation for each data point that we're trying to solve for, which once again our each of our data is for the blood pressure and the heart rate. Awesome. So our first step is to examine our data. So, our first The point of interest is we need to focus our attention on the blood pressure, which I'll call BP so our blood pressure, which once again is in units of milligrams of mercury. Is 118, 122, 120, 116, 124, 119, 121, and 117. And then we have our heart rate, which I'll call HR for short, for heart rate. And our heart rate, once again, that's in units of beats per minute. Is 58 62 60, 56 645961 and 57. Awesome. So now our next step is we need to recall and use our formula, the solve for the coefficient of variation, which we'll call CV for short, and the coefficient of variation is equal to the standard deviation divided by the mean. All multiplied by 100 to get our value into a percentage. Fantastic. So once again, our coefficient of variation is equal to the standard deviation, divided by the mean, multiplied by 100 to convert our decimal value into a percentage. Fantastic. Our second step now is we need to perform our blood pressure calculations, starting with the mean. So we need to note, solving for the mean, we need to take 118 plus 122 plus 120 plus 116 plus 124 plus 119 plus 121 plus 117. And then we need to divide everything by the number of values, which there's 8. And when we plug that into our calculator, we will find that it's equal to 119.625. Fantastic. And once again, that's when we plug in that big old long expression into our calculator and we will find once again that it's equal to 119.625. Fantastic. So now, And that's for our mean. So let's make a note that M or mean is equal to 119.625. So now we need to calculate our standard deviation, which we'll just call SD for short. So solving for our standard deviation, which we can organize our data by putting it into a table. So I've gone ahead and created a table. And as you can see, For our table, we have 3 separate columns. Our far left column is for X, our middle column is X minus X bar, and our far right column is X minus X bar, all to the power of 2. Fantastic. And let us quickly note that SD, our standard deviation. It's going to be equal to the square root of 49.875 divided by 8 minus 1. And when we plug that into our calculator, you'll find that our standard deviation is approximately equal to. 2.669269563. Fantastic. And once again, this is using our data table because we need to take our sum, which we're told because I've gone ahead and used like an Excel spreadsheet to do all the math for me. So if you were to use Excel spreadsheet, you can just plug in all your values and then have the spreadsheet do the math for you. But when you sum all the values together for X minus X bar squared, you will find that it's equal to 49.875. And once again, you need to note that our values for X are once again, Like as you can see, 118, 122, that's our blood pressure values, and if you minus that by your X bar values, you will receive the following results, which at the same time, I won't read off each individual data point. So moving right along now, now that we've determined our value for the standard deviation, we can now calculate our coefficient of variation. So CV. It's going to be equal to parenthesis 2.6692, so 2.669269563. Divided by 119.625. And don't forget to multiply by 100 to convert your decimal to a percentage, so you'll find that your coefficient of variation when you round to two decimal places is approximately equal to 2.23%, and that is one of our final answers. Hooray, we did it. So now we can move along to our third step. So, moving right along here, so our 3rd step now, we need to perform our heart rate calculations, starting with the mean first. So the means, so we need to do our heart rate calculations, so we need to take 58 plus 62 plus 60 + 56. So once again, to quickly recap, we're calculating the mean for our heart rates, so we need to take 58 plus 62 plus 60 plus 56. Plus 64 + 59 plus 61 + 57. Then we need to divide by 8 means there's 8 values total. So when we plug that into our calculator, we will find that our mean, when we round to 3 decimal places, it's gonna be equal to 59.625. Fantastic. So now we need to solve for the standard deviation. And using our second table now means we need to make another table to help us solve for the standard deviation. For this particular Set of data. Awesome. So, once again, our values for X, our X column change because we're using once again our heart rate values. And when we look at our final column, which is X minus X squad, we will find that the sum is also 49.875. So that will mean that when we solve for our standard deviation SD will be equal to the square root of 49.875 divided by 8 minus 1, so that will mean that our standard deviation is approximately equal to. 2.669269563. And solving for our coefficient of variation for the heart rate, we will find that it's equal to parenthesis 2.669269563. Divided by 59.625, and don't forget to multiply by 100 to convert your decimal to a percentage. So that will mean when we plug this into our calculator, our coefficient of variation when we around the two decimal places will be approximately equal to 4.48%. And that is our 2nd final answer. Hooray, we did it. So, that means in conclusion, That our blood pressure coefficient of variation will be equal to 2.23%, and our coefficient of variation for the heart rate will be 4.48%. And to determine our 3rd and final answer, which value is more variable, It is important to recall and note that since heart rate is more variable. Because it has a higher coefficient of variation. So, since the heart rate has a higher coefficient of variation, that will make the heart rate value more variable. So going back to look at our multiple choice, or actually there's no multiple choice answers for this wrong, but to recap what we've learned, Our final set of answers without a doubt will have to be. The following. So once again, blood pressure is 2.23%, heart rate is 4.48%, and heart rate is more variable. And that's it. Those are our final three answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

Key Concepts

Standard Deviation

Aggregation and Averaging Effects

Within-Group vs. Between-Group Variability

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