Back To CourseCollege Algebra: Help and Review
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Yolanda has taught college Psychology and Ethics, and has a doctorate of philosophy in counselor education and supervision.
Imagine that you are teaching a psychology course and you want to examine your students' performance on the midterm and final exams. The grades of your students are as follows:
Notice that there are thirteen students in the class and both the midterm and final grades are listed for each student. Notice that there is only one student who received the same grade for both the midterm and the final. You now want to know if the students' scores on each exam are similar to each other, or if the scores are spread out. This is called variability.
Variability refers to how spread out a group of data is. In other words, variability measures how much your scores differ from each other. Variability is also referred to as dispersion or spread. Data sets with similar values are said to have little variability, while data sets that have values that are spread out have high variability.
The range is the simplest measure of variability. You take the smallest number and subtract it from the largest number to calculate the range. This shows the spread of our data. The range is sensitive to outliers, or values that are significantly higher or lower than the rest of the data set, and should not be used when outliers are present.
Let's calculate the range for midterm exam grades. The midterm grades listed in numerical order are shown here. Since the range is equal to the highest midterm grade minus the lowest midterm grade, we can easily find the range for this data set. Plugging in 100 for our highest midterm grade and 52 for our lowest midterm grade, we find that the range is equal to 100 minus 52, or 48.
So what measure of variability can we use when working with sets of data that contain outliers? One solution is to use the interquartile range (IQR). The IQR, or the middle fifty, is the range for the middle fifty percent of the data. The IQR only considers middle values, so it is not affected by the outliers.
To calculate the IQR, use the following steps:
1) List the data in numerical order. Listing the data in numerical order is necessary for finding the range and median.
2) Find the median. In a set of odd data, the median is the middle value that cuts the data into half. For example, in a set of 13 data, the median is the number in the seventh place. In a set of even data, the median is the mean of the two middle values. For this data set, the median is 85.
3) Place brackets around the numbers above and below the median but not around the median. The brackets will separate the median from Q1 and Q3. So for the midterm grades, our data will now look like this.
4) Locate the quartiles. For the midterm grades, Q1 is 52, 55, 71, 75, 81, 83. And Q3 is 89, 90, 90, 99, 100, 100.
5) Find the median of the data in Q1. We've got an even number of data points in Q1, so our median will be the average of the middle two numbers. In other words, we add 71 to 75 and then divide the sum by 2. The median is 73. And this is our Q1 value.
6) Find the median of the data in Q3. To find the median, repeat what was done for Q1 but with the values in Q3. We can find that Q3 is 94.5.
7) Find the interquartile range using the formula IQR = Q3 - Q1. Plugging 73 for Q1 and 94.5 for Q3, we'll find that IQR = 94.5 - 73 = 21.5. So the interquartile range for this data set is 21.5.
The variance is a measure of how close the scores in the data set are to the mean. The variance is mainly used to calculate the standard deviation and other statistics. There are four steps to calculate the variance. Let's use the midterm exam grades again, but this time we'll calculate the variance.
Take another look at our midterm data.
1) Find the mean of the data set. To find the mean, add to get the sum of all the numbers in the data set. Divide the sum by 13. The mean of the midterm exam grades is 82.31.
2) Subtract the mean from each value in the data set. To do this, subtract 82.31 from each number in the data set. For example, the student's score is 71 and 71 minus 82.31 is negative 11.31. Continue subtracting 82.31 from each number in the data set.
3) Now square each of the values so that you now have all positive values. For example, the difference we got before was negative 11.31. That number squared gives us 127.86. Continue to square each value, then add the squared values together. The sum of the squared values is 2922.77.
4) Finally, divide the sum of the squares by the total number of values in the set to find the variance. This data set has 13 numbers, so divide the sum of the squared differences by 13. The variance is 2922.77 divided by 13, or 224.83.
The square root of the variance is known as the standard deviation. Like the variance, the standard deviation measures how close the scores in the data set are to the mean. However, the standard deviation is measured in the exact same unit as the data set. Let's find the standard deviation of the midterm exam grades.
With the midterm exam grades, the variance was 224.83. The square root of 224.83 = 14.99.
Let's practice calculating the range, IQR, variance, and standard deviation by using the final exam grades in the chart.
When finding the range, remember to first list the data in numerical order. Then we subtract the smallest value from the largest value. The range is the largest number in the data set subtracted from the smallest, which is 100 - 69, or 31.
To find the IQR, we first have to find the median and locate Q1 and Q3. For the data set shown, 88 is the median. Then separate the quartiles with brackets. After determining Q1 and Q3, find the medians of those quartiles which will be our values for Q1 and Q3. In each quartile look for the two middle numbers since each quartile has an even data set. We should find that Q1 = 79 and Q3 = 98. Now we can plug in Q1 and Q3 into the formula. If we subtract the median for Q3 from Q1 we'll get 98 - 79, or 19 for the IQR.
In order to the find the variance, start by finding the mean for the final exam grades and then subtract the mean from each value in the data set. Then square each value and find the sum of the squares. To find the variance, divide the sum of the squares by 13. The variance for this data set is 103.51.
To find the standard deviation, all we need to do is take the square root of the variance. Since we know the variance is 103.51 we can quickly calculate the standard deviation to be 10.17.
Midterm grades compared to final grades look like this: midterm variance is 224.83, the range is 48 and 14.99 is the standard deviation. For the final grades, the variance is 103.51, 31 is the range and 10.17 is the standard deviation.
So after analyzing the students' grades, it can be determined that the midterm grades have a higher variance, range, and standard deviation than the final grades. We can conclude that the midterm grades have more variability than the final grades. We can also conclude that the final exam grades are more similar to each other than the midterm grades.
Because measures of variability are a form of descriptive statistics, they can only be used to describe the data in our study. They cannot be used to draw conclusions or make inferences that go beyond our data set.
Variability refers to how spread out a group of data is. The common measures of variability are the range, IQR, variance, and standard deviation. Data sets with similar values are said to have little variability while data sets that have values that are spread out have high variability. When working to find variability, you'll also need to find the mean and median. Measures of variability are descriptive statistics that can only be used to describe the data in a given data set or study.
As the lesson on statistical variability concludes, you might find it easy to:
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Back To CourseCollege Algebra: Help and Review
27 chapters | 230 lessons | 1 flashcard set
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