Back To CourseMath 102: College Mathematics
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Chad has taught Math for the last 9 years in Middle School. He has a M.S. in Instructional Technology and Elementary Education.
In this lesson, we're looking at the measures of central tendencies. The measures of central tendency provide us with statistical information about a set of data. The four primary measurements that we use are the mean, median, mode and range. Each one of these measurements can provide us with information about our data set. This information can then be used to define how the set of data points are connected. To really examine these data points, let's take a look at a football game between the Green River Ducks and the Southland Bears.
The first measure is the mean, which means average. To calculate the mean, add together all of the numbers in your data set. Then divide that sum by the number of addends.
For example, let's look at the Bears' first offensive series. They had plays of 16, 14, 12 and 18 yards on their way to scoring a touchdown. To find the average number of yards, or the mean, we would first add all four of these values together: 16 + 14 + 12 + 18 = 60. Since there were four numbers in our data set, you would divide that sum by 4: 60 ÷ 4 = 15. So, the average number of yards that the Bears gained was 15 yards. The mean is used to show us the true average of a set of data.
Another measure of central tendency is the median, which is the middle number when listed in order from least to greatest. You may have heard the word median before, and it was likely on a highway. On a highway, you have opposite lanes of traffic. In the middle of the lanes, there is typically a grassy area or a turning area. This area in the middle of the highway is referred to as a median.
Let's return to our game in progress and see how the Green River Ducks are doing. On the Ducks' first offensive series, they had plays of 10, 6, 19, 21 and 4 yards before scoring a touchdown. Let's find the median number of yards gained by the Ducks. The first thing you need to do with this list of yardages is put the numbers in order from least to greatest. So, in order from least to greatest, we would have 4, 6, 10, 19 and 21. Now that your yardages are in order from least to greatest, find the middle number. Since there are five numbers, the middle number would be the third value. The median value of this set of data is 10. On the first offensive series for the Ducks, their median yards gained were 10 yards.
Occasionally there may be an even number of values, which would provide you with two numbers in the middle. If this occurs, you will need to average the two values. At halftime of our game, the Bears quarterback has passes of 3, 8, 9, 12, 12 and 15 yards. Let's find the median pass thrown by the Bears quarterback. The first step is to make sure your numbers are in order from least to greatest, which they are in this problem. The next step is to find the middle number. Since there are six numbers in this set, the middle numbers would be the third and fourth values. Since there are two numbers in the middle, you will average them together. Add the two numbers together, 9 + 12 = 21. Then, divide by 2: 21 ÷ 2 = 10.5. The median of this set of data is 10.5. In the first half, the Bears quarterback had a median passing yardage of 10.5 yards.
When looking at a data set, the median is used when there is an outlier, which is a number that is significantly greater or smaller than the rest of the data. In the second quarter, the Ducks had plays that were 21, 24, 26, 20, 56 and 20 yards. You can see that the value 56 is significantly larger than the other values. 56 would be an example of an outlier. Compared to the other yards that the Ducks gained, 56 yards was much greater than their other gains.
The mode is another measure of central tendency that tells us the number that occurred the most often in your data set. When looking for the mode, there can be more than one mode or no mode. The mode can tell us the most popular choice.
The Bears threw the ball to the following jersey numbers in the third quarter: 5, 6, 6, 3 and 4. You can see that there was only one receiver that had the ball thrown to him more than once. The mode of this data set would be 6. The Bears receiver #6 was the most popular choice to throw the ball to in the third quarter.
The Ducks threw the ball to the following receivers: 12, 13, 15, 17, 19 and 20. You can see that none of these receivers caught more than one pass. This data set has no mode.
Entering the fourth quarter, the Bears had scored the following points: 6, 7, 3, 0, 7, 3, 7 and 3. You can see there are two values that repeat three times each. The mode of this data set is both 3 and 7, which can sometimes be referred to as bimodal. This means that the most popular scoring values for the Bears were 3 and 7.
The last measure of central tendency is the range. The range is the difference between the highest and lowest values. Simply put, find the largest and smallest numbers and then subtract them. The range tells us the distance between the values in our data set.
At the end of the game, the Ducks' kickers had kicked field goals of 10, 14, 17, 19, 21 and 30 yards. Find the range. The smallest value is 10 and the largest value is 30. To calculate the range, subtract the two values: 30 - 10 = 20. The range of this data set is 20.
Let's put our new skills into practice with an example. Let's find the mean, median, mode and range of how many medals the U.S. has won over the last six summer Olympics.
To find the mean of this data set, we would add 104 + 110 + 101 + 94 + 101 + 108, and then divide by 6 because there are six values. So, 104 + 110 + 101 + 94 + 101 + 108 = 618. And, 618 ÷ 6 = 103. So, over the past six Summer Olympics, the United States has been awarded an average of 103 medals.
To find the median, we must first put the data in order from least to greatest. So our numbers in order from least to greatest would be 94, 101, 101, 104, 108, 110. The middle of this data set is actually two numbers (101 and 104). To find the median, we will need to add these two numbers together and divide by 2. 101 + 104 = 205, then dividing by 2 makes the median 102.5.
Looking at this data set, we can see that there is only one number that repeats itself, which is 101. This means that the mode of the data set is 101.
The range of this data set is found by taking the largest value (110) and the smallest value (94) and subtracting. So, 110 - 94 = 16. The range of this data set is 16.
In this lesson, we've discussed four measures of central tendency. These measurements can provide you with important information about a set of data. These four measures are the mean, median, mode and range.
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Back To CourseMath 102: College Mathematics
14 chapters | 108 lessons