The data set shows the number of tomatoes growing on each tomato plant in a garden. Given this data, we have three tasks: calculate the range of the data, calculate the interquartile range of the data, and calculate to the nearest hundredth the population standard deviation.
Our first task, calculating the range of the data, and our second task, calculating the interquartile range, will require sorted data. We need to list it from least to greatest or greatest to least. Starting with the least, our 𝑥 values are zero, three, four, four, five, six, seven, eight, and 12.
We know that the range is the distance from the smallest piece of data to the largest. 12 minus zero is 12. The range of this data is 12. To find the interquartile range, we’ll need to find the median. Five is the median of the data. Half of the date falls before, and half of the data comes after. And then we need to find the middle of both the upper half and the lower half of the data.
The median between these four points is between three and four. Three and a half is then what we would call quartile one. We follow the same procedure on the upper half of our data. Between these four points, we find the middle; that is, halfway between seven and eight. Seven and a half is our third quartile. 𝑄 one, three and a half; 𝑄 three, seven and a half. The interquartile range, the IQR, is the distance between 𝑄 three and 𝑄 one.
Seven and a half minus three and a half equals four. The interquartile range of this data is four. Next, we have to calculate the standard deviation. Standard deviation is found by taking the square root of the summation of every 𝑥 value minus the mean squared divided by 𝑛, the number of data pieces you have. Calculating the standard deviation is not really hard, but it is a little bit meticulous. And if you’re not careful, one small mistake can throw off your standard deviation.
To help prevent small mistakes, I like to solve it this way. We’re going to set up a table and make sure we leave ourselves plenty of room. The first thing we need is found in the middle of this formula. And it’s written as 𝑥 bar. 𝑥 bar is the mean of our numbers. We need to know what the average is. So we take all of our values from zero to 12 and add them up. The sum of all our 𝑥 values divided by 𝑛, the number of values we have.
When we add up all of our values, we get 49. We can divide that by nine the total number of data points. This gives us five and four repeating. We’re working to the nearest hundredth, so we’ll round that to 5.44 so we can use for 𝑥 bar. Now that we know what our average is, we’re going to work on the next piece of the function: 𝑥 minus 𝑥 bar squared. For every single 𝑥 value, we have to subtract the average and then square it.
We have to take zero, subtract 5.44, and then square that value. The answer that my calculator tells me when I subtract 5.44 from zero and then square that value is 29.5936. But again, I’m working with the nearest hundredth. So I’m going to go ahead and round this value to the nearest hundredth. It would look like this: 29.59. The next value, three, minus the average, 5.44, Squared equals 5.95 when we rounded to the nearest hundredth.
Four minus 5.44 squared equals 2.07. That would be the same for our second value of four, same process for the number five: 0.19. For the number six, we will get 0.31, for the number seven 2.43, 6.55 for the number eight, 43.03 for the number 12. This is what I mean when I say it’s not a difficult process, but it is a meticulous process.
The next value we’ll find will be the numerator for the formula for standard deviation. It is the sum of all the values we just found. The sum of all of these differences squared. The sum of these values is 92.19. We take that value plug it into the standard deviation formula: 92.19 over nine, because that’s the number of values we had. 92.19 divided by nine and then the square root of that value equals 3.2005 and continues to repeat. Rounding that to the nearest hundredth, and we can say that the standard deviation for this data set is 3.20.